Strongly correlated quantum fluids: ultracold quantum gases, quantum chromodynamic plasmas and holographic duality

Historically, atomic Fermi gases were first brought to degeneracy at JILA in 1999 [23], using a mixture of two hyperfine states in ⁴⁰K to enable s-wave scattering in a magnetic trap. Dual species radio-frequency (RF)-induced evaporation produced a degenerate, weakly interacting sample, with T/T_F ≃ 0.5. Later, degeneracy was achieved in fermionic ⁶Li by direct evaporation from a magneto-optical trap (MOT)-loaded optical trap [26] and by sympathetic cooling with another species [24, 25, 27], producing a lower T/T_F. However, the minimum reduced temperature was initially limited to T/T_F ≃ 0.2, which may have been a consequence of trap-noise-induced heating [40] or, at the lowest temperatures, Fermi hole heating [38] in combination with evaporative cooling [39].

Optical traps enabled a dramatic improvement in the efficiency of evaporation and the creation of strongly interacting Fermi gases, through the use of magnetically tunable collisional resonances or Feshbach resonances [41]. Feshbach resonances in fermionic atoms were initially characterized in 2002 by several groups [42–44]. For a recent review of Feshbach resonances see [32]. In a Feshbach resonance, a bias magnetic field tunes the total energy of a pair of colliding atoms in the incoming open (triplet) channel into resonance with a molecular bound state in an energetically closed (singlet) channel. At resonance, the zero-energy s-wave scattering length a diverges and the collision cross section attains the unitary limit, proportional to the square of the de Broglie wavelength, i.e. σ = 4π/k², where ℏk is the relative momentum. The collision cross section therefore increases with decreasing temperature, enabling highly efficient evaporative cooling in optical traps and much lower reduced temperatures T/T_F ≃ 0.05.

An optical trap is formed by a focused laser beam. Atoms are polarized by the field and attracted to the high-intensity region when the laser is detuned below resonance with the resonant optical transition, so that the induced dipole moment is in phase with the field. For large detunings, obtained using infrared lasers, the trapping potential is independent of the atomic hyperfine state, enabling several species to be trapped, which is ideal for Fermi gases [45]. Forced evaporation is accomplished by slowly lowering the intensity of the optical trap laser beam. Near a Feshbach resonance, a highly degenerate sample can be produced in a few seconds [46].

A milestone in the Fermi gas field was the observation in 2002 of a strongly interacting, degenerate Fermi gas [46], in the so-called BEC–BCS crossover regime, using this method. In contrast to Bose gases, which are unstable and undergo three-body loss on millisecond time scales near Feshbach resonances, the two-component Fermi gas was found to be stable, as the Pauli principle suppresses three-body s-wave scattering [47]. Released from the cigar-shaped trapping potential of the focused beam, the cloud was observed to expand much more rapidly in the initially narrow direction, compared to the long direction, as a consequence of the much larger pressure gradient along the narrow axis. Consequently, the aspect ratio inverts from a cigar to an ellipse, as shown in figure 4. Remarkably, the same type of elliptic flow is also observed in the momentum distribution of an expanding QGP produced in an off-center collision of two heavy ions; see section 3.5. There, the temperature is 19 orders of magnitude hotter and the particle density is 25 orders of magnitude greater than that of the Fermi gas. In both systems, however, this nearly perfect 'elliptic' flow, figure 4, is a consequence of extremely low-viscosity hydrodynamics, which persists in the normal, non-superfluid unitary gas and deeply connects these two apparently disparate fields.

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The creation of a degenerate Fermi gas near a Feshbach resonance was followed in 2003 by the first measurements of the interaction energy [48, 49] and the creation of Bose-condensed dimer molecules [50–53]. In 2004, condensed fermionic atom pairs were observed using a fast magnetic field sweep to project the pairs onto stable molecular dimers [54, 55]. Using this method, the first phase diagram in the crossover region was obtained (see figure 3) as a function of magnetic field and temperature, albeit using the temperature of the ideal gas obtained by an adiabatic sweep to a non-interacting regime [54]. Evidence for superfluidity in a Fermi gas was provided by the measurement of collective mode frequencies and damping rates versus temperature [56] and magnetic field [57], and by the measurement of the pairing gap by RF spectroscopy [58]. The observation of a vortex lattice in 2005 provided a direct verification of Fermi superfluid behavior [59].

Also in 2005, initial thermodynamic measurements were made by adding a controlled amount of energy to the cloud and measuring an empirical temperature from the corresponding cloud spatial profile [60]. However, the results were model dependent, as the calibration of the empirical temperature relied on comparing the measured cloud profiles with theoretical predictions. Model-independent measurements were soon to follow, based on universal behavior in the unitary regime, where the local thermodynamic quantities, such as pressure, are functions only of the density n and the temperature T [61].

Model-independent measurements of the total energy E of a resonantly interacting Fermi gas are based on the Virial theorem, which holds for a unitary gas as a consequence of universality and yields the energy directly from the cloud profile [62]. Using entropic cooling [63, 64], a model-independent measurement of the total entropy S was accomplished by an adiabatic sweep of the bias magnetic field from resonance to the weakly interacting regime, where the entropy can be calculated from the cloud profile [65]. As T = ∂E/∂S, these measurements enabled the first model-independent temperature calibration and estimates of the critical parameters in the strongly interacting regime [66]. A refined temperature calibration is used in the measurement of universal quantum viscosity, as described in this focus issue [2].

Measurements of global thermodynamic quantities from the cloud profiles in the strongly interacting regime are now superseded by model-independent measurements of local quantities [67, 68]. Using the Gibbs–Duhem relation

$$\begin{equation} {\mathrm{d}}P=n\,{\mathrm{d}}\mu \end{equation} \tag{ 4 }$$

at constant temperature, the local pressure is obtainable from the integrated column density, where the local chemical potential μ is determined by the known trap potential [69, 70]. Combined with a temperature measurement, the local equation of state P(μ,T) or P(n,T) is determined. The most precise local measurements avoid temperature measurement, which introduces the most uncertainty, by determining the pressure, density and compressibility from the cloud profiles. The resulting equation of state reveals clearly a lambda transition, and provides the best measurement of the critical parameters for a unitary Fermi gas [36], as described in detail in section 2.3. Measurements of equilibrium thermodynamic quantities are now used as stringent tests of predictions, as described in this focus issue by Hu and co-workers [71]. These thermodynamic measurements are connected to universal hydrodynamics and transport measurements, as described in [2].

We proceed to describe the all-optical methods developed at Duke in 2002 [26, 46], as one specific example of experimental techniques, which are closely tied to the theme of this focus issue, namely viscosity and transport measurements on Fermi gases in the universal regime [2]. A degenerate, strongly interacting Fermi gas is readily made by all-optical methods [46]. As sketched in the left panel of figure 5, an MOT is used to pre-cool a 50:50 mixture of spin-up and spin-down ⁶Li atoms, which is loaded into a CO₂ laser optical trap and magnetically tuned to an s-wave Feshbach resonance. Atoms from the source (lower right, green cylinder) form an atomic beam (blue arrow) that is slowed by radiation pressure forces from a resonant laser beam (top left, opposing red arrow). For ⁶Li, the deceleration is 2 × 10⁶ m s⁻², slowing the atoms from oven thermal speeds of 1 km s⁻¹ to tens of m s⁻¹ over a distance of a fraction of a meter. Six laser beams (three thick red lines) then propagate toward the center of the MOT (point of intersection of three thick red lines), creating inward damping forces that cool the atoms. Opposing magnetic fields, created by two coils (stacked black circles, top and bottom), spatially tune the atomic resonance frequency, causing the six beams to produce a harmonic restoring force at the MOT center. Typical MOT clouds are a few millimeters across and contain several hundreds of million atoms. A trapping laser beam (shown in yellow) is focused (lenses indicated by two light blue ovals) at the MOT center and loaded. After turning off the MOT beams and the MOT magnetic field, an additional bias magnetic field tunes the atoms to a collisional (Feshbach) resonance. Forced evaporation near resonance, by lowering the trap depth, rapidly cools the cloud to quantum degeneracy, i.e. T/T_F ≪ 1, producing a cigar-shaped cloud that is typically a few microns in diameter and several hundreds of microns long, containing several hundred thousands of atoms. In the right panel of figure 5 is shown the experimental apparatus from the Duke laboratory, currently located at North Carolina State University. From right to left in the photo: the oven assembly where hot fermions are produced (aluminum housing); the camera to produce density images (blue device in the foreground); a Zeeman slower to bring atoms into MOT (middle, behind camera); bias field magnets containing MOT in ultrahigh vacuum (white plastic housings); ZnSe lens for the CO₂ laser trapping beam and optical table (left).

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In the simplest case, the optical trap consists of a single laser beam, focused into the center of the MOT and detuned well below the atomic resonance frequency to suppress spontaneous scattering, which would otherwise heat the atoms. For an optical trapping laser detuned below the atomic resonance, the induced atomic dipole moment is in-phase with the trapping laser field, so that the atoms are attracted to the high-field region at the trap focus, i.e. the effective trapping potential is U = −α〈E²〉/2, where the polarizability α > 0 and 〈E²〉 is proportional to the trap laser intensity, time averaged over a few optical cycles. The effective potential then has the same spatial profile as the intensity of the focused trap laser. For ultracold atoms, the energy per particle is typically quite small compared to the depth of the optical trap, so that the atoms vibrate in a nearly harmonic confining potential. The vibration frequencies of the atoms in each direction ω_i, i = x,y,z, of the trap are readily determined by parametric resonance: the trap laser intensity is modulated and the size of the cloud is measured as a function of modulation frequency. When the modulation frequency is twice the harmonic oscillation frequency, the energy of the atoms increases, causing the density profile to increase in size. This method permits a precise characterization of the trap parameters.

All information about the cloud is obtained by absorption imaging: a spatially uniform short (several μs) low-intensity laser pulse is transmitted through the atom cloud, which partially absorbs the light. The shadow cast by the absorption is imaged onto a CCD (charge coupled device) array to record the image, which is analyzed to extract the column density, integrated along the line of sight. This 'laser flash photography' method provides real space images with a resolution of a few microns, on a time scale short compared to the time scale over which the atoms move significantly compared to the spatial resolution. Both non-destructive and destructive imaging techniques are possible. In the latter case the entire cloud is destroyed by the laser pulse in order to make the most complete possible image. One then runs the same experiment multiple times, with nearly identical initial conditions, to obtain an average picture of temporal evolution. While the CCD measures just the density or g⁽¹⁾ correlations, in fact it is possible to extract density–density or g⁽²⁾ correlations from the noise on an image [72, 73].

What do experimental measurements actually look like? In figure 4 are shown absorption images from the 2002 Duke experiment on elliptic flow [46]. In the experiments, N = 7.5 × 10⁴ atoms in each of the two lowest hyperfine states were cooled to degeneracy in a CO₂ laser trap, with a reduced temperature T/T_FI between 0.08 and 0.2. Here, $T_{\mathrm {FI}}=\hbar \bar {\omega }(6N)^{1/3}/k_{\mathrm {B}}$ is the Fermi temperature for an ideal gas at the center of a harmonic trap¹⁴, where $\bar {\omega }=2\pi \times 2160 (65)\,{\mathrm {Hz}}$ is the geometric mean of the trap oscillation frequencies, measured by parametric resonance as described above. For these parameters, the Fermi temperature is T_FI = 7.9 μK. For an ideal gas in the trap, the Fermi radii are σ_x = 3.6 μm in the narrow directions and σ_z = 103 μm in the long direction.

Studies of trapped ultracold Fermi gases have provided important information about the phase diagram, the equation of state, transport properties and quasiparticle properties of strongly correlated Fermi gases. This is possible because under the conditions typically encountered in the experiments, local properties of the trapped gas directly correspond to equilibrium properties of the homogeneous Fermi gas. Consider the ground state of N harmonically trapped fermions. Hohenberg and Kohn [74] showed that the solution of the N-body Schrödinger equation is equivalent to the minimum of the energy functional

$$\begin{equation} E[n(\mathbf{x})] = \int \mathrm{d}\mathbf{x} \left( {\cal E}(n(\mathbf{x})) + n(\mathbf{x})U(\mathbf{x})\right), \end{equation} \tag{ 5 }$$

where n(x) is the density, subject to the condition $\int \mathrm {d}\mathbf {x}\, n(\mathbf {x})=N$, ${\cal E}(n)$ is the energy density functional and U(x) is the trap potential. If the density is sufficiently smooth we can write ${\cal E}(n)$ as a function of the local density and its gradients. On dimensional grounds we have

$$\begin{equation} {\cal E}(n(\mathbf{x})) = \frac{c_{0}\hbar^{2}}{m}\, n(\mathbf{x})^{5/3} + \frac{c_{1}\hbar^{2}}{m}\frac{(\vec{\nabla} n(\mathbf{x})){^2}}{n({\mathbf{x}})} + O\left(\nabla^{4}n(\mathbf{x})\right), \end{equation} \tag{ 6 }$$

where c₀,c₁,... are dimensionless constants. At unitarity the coefficients c_i are pure numbers, but for a finite scattering length they become functions of na³. To first approximation we can neglect the gradient terms. Then the density is given by n(x) = n_eq(μ − U(x)), where n_eq(μ) is the equilibrium density at the chemical potential μ and zero temperature. This approximation is known as the local density approximation. Gradient terms are suppressed by (ω_⊥/μ)² ∼ 1/(λ_zN)^2/3, where λ_z = ω_z/ω_⊥ is the trap deformation. Typical experiments involve λ_z ≃ 0.025–0.1 and N ⩾ 10⁵, so corrections beyond the local density approximation are quite small. These arguments generalize to systems at non-zero temperature. In this case the density of the trapped system is n(x) = n_eq(μ − U(x),T).

The equilibrium density can be determined from the equation of state, P = P(μ,T), through the thermodynamic relation¹⁵ n = ∂P/∂μ. In the following we also frequently refer to the relation P = P(n,T) as the equation of state. At unitarity the interaction is scale invariant and the only scales in the many-body system are the interparticle distance n^−1/3 and the de Broglie wavelength, given in equation (1). Dimensional analysis implies that the equation of state must be of the form

$$\begin{equation} P(n,T) = \frac{\hbar^2 n^{5/3}}{m}f(n\lambda^3_{\mathrm{dB}}), \end{equation} \tag{ 7 }$$

where f(x) is a universal function. At zero temperature the pressure is proportional to n^5/3/m. This implies, in particular, that the pressure is given by a numerical constant times the pressure of a free Fermi gas. The same is true of the energy per particle and the chemical potential. It has become standard to denote the ratio of the energy per particle of the unitary gas and the free Fermi gas as the Bertsch parameter ξ,

$$\begin{equation} \frac{E}{N} = \xi\left(\frac{E}{N}\right)_0. \end{equation} \tag{ 8 }$$

Bertsch posed the calculation of the parameter ξ as a challenge problem to the many-body physics community in 1999 [75]. At the time, the problem was stated in the context of a toy model for dilute neutron matter; see section 3.1.

Using thermodynamic identities we can show that equation (7) implies that $P=\frac {2}{3}\epsilon$, where is the energy density. This relation is analogous to the equation of state of a scale-invariant relativistic gas, $P=\frac {1}{3}\epsilon$, as discussed in section 3.2. For a trapped gas the relation between pressure and energy density implies a Virial theorem: in a harmonic trap, the internal energy of the system is equal to the potential energy of the trapping potential [76, 77],

$$\begin{equation} \int \mathrm{d}\mathbf{x} \, \epsilon(\mathbf{x}) = \int \mathrm{d}\mathbf{x}\, n(\mathbf{x})U(\mathbf{x}). \end{equation} \tag{ 9 }$$

These universal relations have been extended in many ways; see [78] for a review. An important class of relations, discovered by Tan, connects the derivative of thermodynamic quantities with respect to 1/a to short-range correlation in the gas. Tan defined the contact density ${\cal C}$ via [79–81]

$$\begin{equation} \frac{\mathrm{d}\epsilon}{\mathrm{d}(a^{-1})} = - \frac{\hbar^2}{4\pi m}\, {\cal C}, \end{equation} \noindent \tag{ 10 }$$

where the derivative is taken at constant entropy density. The contact density appears in a number of thermodynamic relations. The universal equation of state, for example, is given by

$$\begin{equation} P =\frac{2}{3}\, \epsilon + \frac{\hbar^2}{12\pi ma} \, {\cal C}. \end{equation} \tag{ 11 }$$

More remarkable is the fact that ${\cal C}$ controls short-distance correlations in the dilute Fermi gas. The tail of the momentum distribution is given by

$$\begin{equation} n_\sigma(k)\to \frac{C}{k^4} \quad \left( |a|^{-1} \ll k \ll r_0^{-1} \right), \end{equation} \tag{ 12 }$$

where $C=\int \mathrm {d}^3x\, {\cal C}(x)$ is the integrated contact, n_σ(k) is the momentum distribution¹⁶ in the spin state σ and r₀ is the range of the interaction. There are similar expressions for the asymptotic behavior of other correlation observables such as the static and dynamic structure factors, and the dynamic shear viscosity; see [82] for a review. In this focus issue, Kuhnle et al [83] present a comprehensive set of measurements of the contact as a function of interaction strength and reduced temperature. These results can be compared to new theoretical predictions discussed by Hu and co-workers [71].

Below the critical temperature for superfluidity, the superfluid flow velocity v_s can be viewed as an additional thermodynamic variable. The response of the pressure to the superfluid velocity defines the superfluid mass density

$$\begin{equation} \rho_{\mathrm{s}} =mn_{\mathrm{s}} = -\left. \frac{\partial^2 P}{\partial v_{\mathrm{s}}^2} \right|_{v_n=0}, \end{equation} \tag{ 13 }$$

where the derivative is taken in the rest frame of the normal fluid, meaning that v_n = 0. The superfluid mass density can be measured using rotating clouds [84]. The second moment of the trap integrated value of the superfluid mass density determines the quenching of the moment of inertia. New measurements of the moment of inertia can be found in [85].

For small values of n|a|³ the equation of state P(n,T) can be computed in perturbation theory. This program was initiated by Lee and Yang [86] and Huang and Yang [87]. At unitarity, weak coupling methods can be used in the limit of high temperature. This is based on the observation that the binary cross section at unitarity is σ = 4π/k². At high temperature the mean momentum is large and the average thermal cross section is small. The equation of state can be written as an expansion in nλ³_dB, which is the well-known Virial expansion. We have

$$\begin{equation} P = n k_{\mathrm{B}} T \left\{ 1+b_2 (n\lambda_{\mathrm{dB}}^3) +O((n\lambda_{\mathrm{dB}}^3)^2) \right\}, \end{equation} \tag{ 14 }$$

with $b_2=-1/(2\sqrt {2})$ at unitarity [88, 89]. Analytic approaches in the non-perturbative regime nλ³_dB ∼ 1 are based on extrapolating to the unitary limit from different regimes in the phase diagram. For this purpose the phase diagram has been studied as a function of the strength of the interaction, the number of species and the number of spatial dimensions. The oldest theory of this type is the Noziéres–Schmitt–Rink (NSR) theory [90–92], which is based on a set of many-body diagrams that correctly describe both the BCS and BEC limits. NSR theory works surprisingly well, despite the formal lack of a small parameter at unitarity. For example, the basic form of the critical temperature sketched in figure 3 is correctly reproduced. Modern theories of this type are typically based on self-consistent T-matrix approximations; see [93, 94]. Another idea is to generalize the unitary Fermi gas to 2N spin states [95]. Mean field theory is reliable in the limit N → ∞, and the interesting case N = 1 can be studied by expanding in 1/N. This method is of interest in connection with holographic dualities, because the gravitational dual is expected to be classical in the limit that the number of degrees of freedom is large. Finally, it was proposed to use the number of dimensions as a control parameter. The unitary limit is perturbative in both D = 2 and 4 spatial dimensions [96]. The interesting case D = 3 can be studied as an expansion around D = 2 +  or 4 −  dimensions [97].

These methods are promising, but currently the only techniques that provide reliable and systematically improvable results in the regime nλ³_dB ≃ 1 are quantum Monte Carlo calculations. At zero temperature the standard technique is Green function Monte Carlo [98, 99]. This method relies on a variational initial wavefunction, which is used as the initial condition for an imaginary time diffusion process. The Monte Carlo method suffers from a fermion sign problem which is addressed using the fixed node approximation. At finite temperature a number of groups have performed imaginary time path integral Monte Carlo calculations [100–103]. These calculations do not rely on variational input, and they do not suffer from a sign problem, but they are formulated on a space–time lattice and require an extrapolation to zero lattice spacing. One new technique is bold diagrammatic Monte Carlo, which is based on sampling the sum of all Feynman diagrams. The method suffers from a sign problem, but convergence in the regime above the critical temperature for superfluidity was found to be very good [104].

The equation of state describes a functional relation between key thermodynamic variables, such as the pressure P(μ,T) as a function of chemical potential μ and temperature T. In Fermi gases, as stated in section 2.1, what we can actually measure is the density profile of a trapped cloud. There are various techniques for translating density measurements into thermodynamic quantities, all relying on the local density approximation.

The first thermodynamic measurements took place at Duke in 2005, where global thermodynamic variables were measured. In the initial experiments [60], a controlled amount of energy was added to the trapped cloud by abruptly releasing it from the optical trap, allowing it to expand hydrodynamically by a known factor, and then recapturing the cloud after a selected expansion time, thereby increasing the potential energy. After allowing the gas to equilibrate, the cloud profile was measured to determine an empirical temperature, which was later calibrated by comparing to theoretical cloud profiles, predicted as a function of reduced temperature. The resulting energy versus temperature curve was compared to that measured for an ideal Fermi gas in the same trap, and showed a departure from ideal gas behavior at a certain temperature, which yielded an estimate of the superfluid-to-normal fluid transition temperature. However, the results suffered from being model dependent, as calibration of the empirical temperature relied on a comparison of the measured cloud profiles with theoretical predictions. To avoid this model dependence, in 2006 the JILA group measured the potential energy of the strongly interacting cloud of ⁴⁰K as a function of the ideal Fermi gas temperature that was obtained after an adiabatic sweep of the bias field to the non-interacting regime above resonance [105].

Model-independent determination of thermodynamic quantities was done by the Duke group in 2007 [65], where the total energy E and the total entropy S of a trapped cloud were measured from cloud profiles, by exploiting the universal behavior of a unitary Fermi gas at a Feshbach resonance. From equation (9) we know that for a harmonic trapping potential the total energy is twice the average potential energy, E = 2〈U〉 = 3 mω²_z〈z²〉. Hence, by measuring the harmonic oscillator frequency and mean square cloud size, the total energy is readily determined from cloud images. This method was demonstrated experimentally in [62]. With the energy in the strongly interacting regime measured from cloud images, the entropy is determined by adiabatically sweeping the magnetic field to a weakly interacting regime. There, the entropy and mean square cloud size are readily calculated as a function of the temperature, yielding the entropy as a function of the mean square cloud size of the weakly interacting gas. Adiabatic behavior is verified by a round-trip sweep. Figure 6 shows the energy per particle as a function of the entropy per particle, in the universal regime. The temperature is determined by fitting a smooth curve [2, 66]. Hu et al [106] combined the demonstrated universal behavior of the global thermodynamic quantities by reanalyzing the measurements in ⁴⁰K and showing that the ⁴⁰K and ⁶Li data fit on a single thermodynamic curve.

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As already mentioned briefly in section 2.1, model-independent measurement of global thermodynamic variables was superseded in 2010 by model-independent measurement of local thermodynamic quantities, which can be directly compared to predictions, within the local density approximation. Equation (4) determines the local pressure P from the local density n and the local chemical potential, μ = μ_g − U, where μ_g is the global chemical potential and U is the known trapping potential. Absorption imaging directly yields the column density $\tilde {n}(x,z)=\int \, \mathrm {d}y\,n(x,y,z)$ for an imaging beam propagating along y. In a cylindrically symmetric harmonic trap U = mω²_⊥(x² + y²)/2 + mω²_z z²/2, where ω_⊥ and ω_z are the radial and axial trapping frequencies, respectively, we can write −dμ = dU = mω²_⊥ρ dρ = mω²_⊥ dx dy/(2π) from which we see that the pressure is determined from the doubly integrated density, i.e. the integrated column density $\bar {n}(z)=\int \,\mathrm {d}x\, \mathrm {d}y\,n(x,y,z)$. For a 50:50 mixture of spin-up and spin-down fermions with n the total density, the pressure is determined as a function of z, $P(z,T)=m\omega _\perp ^2\,\bar {n}(z)/(2\pi )$. For x = y = 0, the corresponding chemical potential is μ(z) = μ_g − mω²_z z²/2, so that P(μ,T) is determined if the temperature and global chemical potential can be determined.

To determine the temperature, the Tokyo group [67] used the temperature calibration by the Duke group to obtain T from the total energy and hence from the mean square cloud size [66]. An improved version of this calibration is described in this focus issue [2]. The ENS group [68] directly determined the temperature by using a weakly interacting ⁷Li impurity to measure the temperature of a strongly interacting ⁶Li gas. The global chemical potential was determined from the wings of the density profile using a fit based on a Virial expansion, yielding P(μ,T), from which all other local thermodynamic quantities, such as the energy density and entropy density, were determined. These results enable a direct comparison with predictions. Further, the total entropy S and the total energy E of the trapped cloud are readily determined by integration, and can be compared with the corresponding global quantities measured by the Duke group. The total energy versus temperature is shown in figure 7, where the improved temperature calibration of [2] from this focus issue is displayed for the Duke data. As quite different methods were employed to make the measurements, their close agreement indicates the correctness of the data.

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The latest studies of local thermodynamics by the MIT group [36] use refined methods, where measurements of the isothermal compressibility κ directly from the density profiles replace temperature measurements, yielding an equation of state n(P,κ). This eliminates the determination of the temperature and local chemical potential from fits at the edges of the cloud, which produced the most uncertainty in previous work. For this method, the three-dimensional (3D) density n(x,y,z) is determined by tomographic imaging, using an inverse Abel transform [107] to determine n from the measured column density. The trap potential is carefully characterized by determining the surfaces of constant density and the hence constant chemical potential, in a very shallow trap where the axial z trap potential is almost perfectly harmonic and therefore known. Equation (4) yields the pressure

$$\begin{equation} P(U,T)=\int_{-\infty}^\mu \mathrm{d}\mu\,n(\mu',T)=\int_U^\infty \mathrm{d}U\,n(U,T), \end{equation}\noindent \tag{ 15 }$$

where the unknown global potential μ_g is not needed, since the integral is over the known trap potential [36]. The compressibility is the change in density with respect to a change in the local trapping potential U, and is determined from the density profile by

$$\begin{equation} \kappa=\left.\frac{1}{n}\frac{\partial n}{\partial P}\right|_T =-\left.\frac{1}{n^2}\frac{\partial n}{\partial U}\right|_T, \end{equation} \noindent \tag{ 16 }$$

since dP = n dμ = −n dU at constant T. Thus, the actual observed equation of state is the functional relation n(κ,P), measured directly from the density distribution, the clear and direct experimental observable for ultracold quantum gases as discussed in section 2.1. From n(P,κ), all other local thermodynamic quantities are determined [36], as shown in figure 8.

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These experiments provide the best current value for the Bertsch parameter of ξ = 0.376(5), which is consistent with the value obtained in measurements of global quantities, ξ = 1 + β = 0.39(2) [66]. These measurements can also be compared with theory predictions. The two most recent quantum Monte Carlo calculations give ξ ⩽ 0.383(1) [108] and ξ = 0.3968^+0.0076_−0.0077 [109]. See [109] for an extensive compilation of analytic results and earlier Monte Carlo calculations.

Figure 8 shows several representations of the equation of state, and provides a glimpse of the level of precision that can be achieved in present experiments. In the left panel are shown the chemical potential μ, energy E and free energy F. The right panel shows the entropy per particle S/(Nk_B) versus T/T_F. The chemical potential (red solid circles) is normalized by the Fermi energy; energy (black solid circles) and free energy (green solid circles) are normalized by $E_0 = \frac {3}{5} N E_{\mathrm {F}}$, which is the energy per particle in a uniform Fermi gas. At high temperatures all quantities approximately track those for a non-interacting Fermi gas, shifted by ξ_n − 1 with ξ_n ≃ 0.45 (dashed curves). The peak in the chemical potential roughly coincides with the onset of superfluidity. In the very low-temperature regime, μ/E_F, E/E₀ and F/F₀ all approach ξ (blue dashed line). At high temperatures, the entropy closely tracks that of a non-interacting Fermi gas (black solid curve). The open squares are from the self-consistent T-matrix calculation [94]. A few representative error bars are shown, representing means ±standard deviation.

The first measurements of the phase transition at unitarity were made at JILA, shown in figure 9, by a pair projection technique. After the pairs were created in the BCS and unitary regimes, a rapid magnetic field sweep was used to pair-wise project the fermions onto molecules to protect them during subsequent expansion measurements, where the molecular momentum distribution was measured to determine the condensed pair fraction. The fraction of near-zero momentum molecules is interpreted as the percentage of condensed Fermi unitary or BCS pairs. In this early experiment, ⁴⁰K was used, and the measured temperature was that of an ideal Fermi gas, rather than the temperature of the interacting gas. This ideal Fermi gas temperature was obtained by ballistic expansion after an adiabatic sweep to the weakly interacting regime above resonance, yielding the condensed pair fraction versus ideal gas temperature and magnetic field. Despite the uncertainties regarding the temperature and the pair conversion efficiency we observe that the shape of the transition line is qualitatively consistent with theoretical expectations, as summarized in figure 3.

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Initial estimates of the critical parameters of the trapped gas were done by the Duke group, first based on model-dependent measurements of the energy versus temperature [60] and later based on model-independent measurements of the total energy E and entropy S, obtained as described above. Assuming that a phase transition would be manifested in a change in the scaling of E with S, two power laws were used to fit the E(S) data, one to fit the high-temperature data and the other to fit the low-temperature data, which joined at a point S_c. The continuity of the energy and temperature was used as a constraint, and the critical entropy S_c was estimated from the joining point that minimized the χ² for the fit [66].

This fit method is useful for calibrating the temperature, as described in [2] of this focus issue. However, the fitted critical parameters are dependent on the form of the fit function, making the uncertainty difficult to quantify. Further, the global observables suffer from the trap averaging that masks an abrupt local phase transition, initially near the trap center. Hence, the curve of total energy versus entropy is too smooth to obtain reliable estimates of the critical parameters.

Using the refined local measurements described in section 2.3, the MIT group has traced out the phase transition in the unitary regime, without any need for rapid magnetic sweeps, fitting parameters, thermometry, or indeed any kind of theory besides elementary thermodynamic considerations. The main goal of these experiments was to obtain a cusp-like signal of the phase transition to superfluidity in a unitary Fermi gas, by focusing on second-order derivatives of the pressure, where a cusp appears. Although superfluidity was established by the creation of vortex lattices [59], the actual phase transition itself had been only indirectly observed.

In figure 10 is shown the specific heat per particle, clearly displaying the superfluid phase transition. Experimentally, the specific heat is derived from the compressibility κ and the pressure P [36],

$$\begin{equation} \frac{C_V}{k_{\mathrm{B}} N} = \frac{5}{2}\frac{T_{\mathrm{F}}}{T} \left(\tilde{p}-\frac{1}{\tilde{\kappa}}\right), \end{equation} \noindent \tag{ 17 }$$

where $\tilde {\kappa } = \kappa /\kappa _0$ and $\tilde {p}\equiv P/P_0$ are normalized to the non-interacting Fermi gas compressibility $\kappa _0=\frac {3}{2}\frac {1}{nE_{\mathrm {F}}}$ and pressure $P_0=\frac {2}{5} n E_{\mathrm {F}}$, respectively. Using n = (∂P/∂μ)_T , the compressibility can be written as κ = (1/n²)(∂n/∂μ)_T  = (1/n²)(∂²P/∂²μ)_T . As κ is a second derivative of the pressure, the specific heat shows a clear cusp-like signature (figure 10). Qualitatively, the behavior of C_V can be understood as follows: as one approaches the phase transition from above, T/T_c > 1, the compressibility increases due to the attraction between fermions; below the phase transition, T/T_c < 1, the compressibility decreases because fermions are bound into pairs, and it becomes more difficult to squeeze the gas, i.e. to change the single-particle density.

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Transport properties of the unitary Fermi gas are of interest for several reasons. The first reason is related to the main theme of this review: holographic dualities suggest a new kind of universality in the transport properties of strongly interacting quantum fluids. We expect, in particular, that the shear viscosity to entropy density ratio is close to the value η/s ∼ ℏ/(4πk_B) originally discovered in the QGP, and first obtained theoretically using the AdS /CFT correspondence [11], where AdS is a special maximally symmetric space–time described in detail in section 4.2, and CFT stands for conformal field theory. The second reason is that transport properties are very sensitive to the strength of the interaction, and the types of quasiparticles present in the system. The Bertsch parameter, which characterizes the effect of interactions on the energy per particle, varies by about a factor of two between the weak coupling (BCS) and strong coupling (unitarity) limits. The shear viscosity, on the other hand, changes by many orders of magnitude. Finally, quantum-limited transport has also been observed in systems that are of great practical significance, in particular in the strange metal phase of high-T_c compounds; see the contribution by Guo et al to this focus issue [110].

Transport properties have been studied experimentally by exciting hydrodynamic modes, such as collective oscillations [56, 111–114], collective flow [46, 115], sound [116] and rotational modes [117]. In a system that can be described in terms of quasiparticles the hydrodynamic description is valid if the Knudsen number Kn = l_mfp/L, the ratio of the mean free path l_mfp to the system size L, is small¹⁷. In the unitary gas the mean free path is l_mfp = 1/(nσ), where n is the density and σ = 4π/k² is the universal cross section. In the high-temperature limit the thermal average cross section is σ = 4λ²_dB. The Knudsen number of a unitary gas confined in a cigar-shaped harmonic trap is

$$\begin{equation} Kn = \frac{3\pi^{1/2}}{4(3\lambda_z N)^{1/3}}\left(\frac{T}{T_{\mathrm{FI}}}\right)^2,\end{equation} \noindent \tag{ 18 }$$

where we have taken L as the radius in the narrow or z direction. Here, N is the number of particles, λ_z was defined previously as the aspect ratio of the trap and T_FI is the global Fermi temperature for a harmonically trapped ideal gas; see section 2.1. Using N = 2 × 10⁵ and λ_z = 0.045 as in [113], we conclude that hydrodynamics is expected to be valid for . For the Fermi gas viscosity measurements described in this focus issue [2, 115], the maximum temperature is T ≃ 1.5 T_FI and Kn ⩽ 0.09.

Nearly ideal hydrodynamic behavior was first observed in the expansion of a unitary Fermi gas after release from a deformed trap [46]; see figure 4. For a ballistic gas the expansion reflects the isotropic local momentum distribution in the trap. As a result the gas expands in all directions and the cloud slowly becomes spherical. For a hydrodynamic system the expansion is driven by gradients in the pressure. In the case of a deformed cloud the gradients are largest in the short direction of the trap, and the expansion takes place mostly in the transverse direction. As a result, the cloud eventually becomes elongated along what was originally the short direction. This phenomenon is analogous to the elliptic flow observed in heavy-ion collisions, as described in section 3.5. What is also remarkable is the fact that even though the gas becomes more dilute as it expands, this effect is compensated for by the growth in the mean cross section. As a consequence, the gas remains hydrodynamic throughout the expansion. Ballistic behavior sets in eventually only because of imperfections, such as the fact that the scattering length is not truly infinite.

The role of dissipative effects, in particular shear viscosity, was first studied in collective modes. The radial breathing mode can be excited by removing the confining potential, letting the gas expand for a short period of time, and then restoring the potential. Hydrodynamic behavior can be established by measuring the frequency of the breathing mode. For an ideal fluid $\omega =\sqrt {10/3}\,\omega _0$, whereas in a ballistic system ω = 2ω₀ [118, 119]. The transition from ballistic behavior in the BCS limit to hydrodynamics in the unitary limit was observed experimentally in [111, 112]. In the hydrodynamic regime damping is expected to be dominated by dissipative terms in the equations of fluid dynamics. The energy dissipation is given by

$$\begin{eqnarray} &&\fl \dot{E} = - \int \mathrm{d}\mathbf{x}\, \Bigg\{\frac{\eta(\mathbf{x})}{2}\, \left(\nabla_iv_j+\nabla_jv_i-\frac{2}{3}\delta_{ij} (\nabla\cdot \mathbf{v}) \right)^2 + \zeta(\mathbf{x})\, \big( \nabla\cdot \mathbf{v}\big)^2 + \frac{\kappa(\mathbf{x})}{T} (\nabla T)^2\Bigg\}, \end{eqnarray} \noindent \tag{ 19 }$$

where v_i is the fluid velocity, η is the shear viscosity, ζ is the bulk viscosity, κ is the thermal conductivity, and all derivatives, divergences and gradients are spatial. At unitarity the system is scale invariant and the bulk viscosity is expected to vanish [78, 120]. This prediction was experimentally checked by Cao et al [121] and Dusling and Schäfer [121]. Thermal conductivity is not important because the system remains isothermal in ideal hydrodynamics. Temperature gradients only appear due to shear viscosity, and their contribution to dissipation is higher order in the gradient expansion. This means that damping is dominated by shear viscosity.

Collective modes in ideal fluid dynamics are described by scaling solutions of Euler's equation. This means that the shape of the density profile does not change during the evolution, and that the velocity is linear in the coordinates. The solution is analogous to Hubble flows in cosmology and the Bjorken expansion of a QGP, as discussed in section 3.5. In the case of a scaling solution, the shear stresses ∂_iv_j are spatially constant, and the energy dissipated only depends on the spatial integral of η. On dimensional grounds we can write η = ℏnα_n. For a scale invariant system α_n is only a function of the dimensionless variable n^2/3ℏ²/(mk_BT), i.e. a function of the reduced temperature T/T_F. The reduced temperature varies across the trap, but for a given fluid element it remains approximately constant during the hydrodynamic evolution of the system. This implies that the damping constant of a collective mode is related to the spatial average of the shear viscosity in the initial equilibrium state

$$\begin{equation} \langle \alpha_n \rangle =\frac{1}{N} \int \mathrm{d}\mathbf{x} \, \eta(x). \end{equation} \tag{ 20 }$$

The extracted values of 〈α_n〉 can be converted into the trap averaged shear viscosity to entropy density ratio by using the measured entropy per particle. This type of analysis was originally carried out in [122, 123], where it was observed that in the vicinity of T_c. More recently, Cao et al [115] showed that in the high-temperature regime α_n exhibits the scaling behavior expected from the solution of the linearized Boltzmann equation. The shear viscosity due to elastic two-body scattering has the form η ≃ npl_mfp, where p ∼ λ⁻¹_dB is the mean quasiparticle momentum. As we saw above, l_mfp ∼ 1/(nσ) ∼ 1/(nλ²_dB). This implies that at high temperature η ∼ λ⁻³_dB. The coefficient of proportionality was determined by Bruun and co-workers [8] and Brun and Smith [124]. They find that

$$\begin{equation} \eta = \frac{15}{32\sqrt{\pi}}\frac{(mk_{\mathrm{B}}T)^{3/2}}{\hbar^2}. \end{equation} \tag{ 21 }$$

There is an important problem related to equation (21) that affects the extraction of the shear viscosity from experiments with scaling flows. Equation (21), which is reliable in the high-temperature or low-density part of the cloud, is independent of the density¹⁸. As a consequence the integral in equation (20) diverges in the low-density region. A solution to this problem was proposed in [126, 127]: in the low-density regime, the viscous relaxation time τ_R ≃ η/(nk_BT), which is the time it takes for the dissipative stresses to relax to the Navier–Stokes form $\eta (\nabla _i v_j +\nabla _j v_i-\frac {2}{3}\delta _{ij}\nabla _k v_k)$, becomes very large. Since the dissipative stresses are initially zero, taking relaxation into account suppresses the contribution from the dilute corona. A simplified version of this approach was used in Cao et al [2, 115]. The relaxation time and its relation to the spectral function of the shear tensor are discussed in the contribution by Braby et al [128].

The shear viscosity drops with temperature and is expected to reach a minimum near T_c. In this regime quantum effects are important. T-matrix calculations can be found in [129] and in the contributions by Guo et al and LeClair in this focus issue [110, 130]. It is also possible that in this regime quasiparticle descriptions break down completely, and the most efficient description of the unitary gas near T_c is in terms of a suitable weakly coupled holographic dual. Progress toward constructing holographic duals of non-relativistic quantum fluids is summarized in section 4.3.4.

In the low-temperature superfluid regime the appropriate description is superfluid (two-fluid) hydrodynamics [131]. Superfluid hydrodynamics predicts the existence of additional hydrodynamic modes, in particular second sound, and contains additional transport coefficients that come into play if there is relative motion between the superfluid and normal components of the fluid. There are proposals for exciting second sound modes in the literature [132], but these ideas have not yet been confirmed. At very low temperature the shear viscosity is expected to be dominated by phonons, similar to liquid helium or dilute Bose gases. Elastic phonon scattering gives η ∼ 1/T⁵ [133], whereas inelastic processes can give a slower increase at low temperature, η ∼ 1/T [9]. These predictions are difficult to verify experimentally because the phonon free path is quite large.

The short mean free path in the strongly interacting normal fluid suggests that not only the viscosity, but also other transport coefficients may exhibit universal behavior. The spin diffusion constant was recently studied by Sommer et al [134, 135]. In the first paper, Sommer et al observed collisions between polarized Fermi gas clouds. The colliding clouds are initially very far from equilibrium, but at late times the system relaxes diffusively. The corresponding relaxation time can be used to measure the spin drag and the spin diffusion constant D_s. The spin diffusion constant in the homogeneous system is defined by Fick's law,

$$\begin{equation} \mathbf{\jmath}_{\mathrm{s}} = -D_{\mathrm{s}}\nabla M,\end{equation}\noindent \tag{ 22 }$$

where $\mathbf{\jmath}_\mathrm{s}$ is the spin current, and M = n_↑ − n_↓ is the polarization. In this focus issue, Sommer et al follow up on these studies by measuring the damping of the spin dipole mode in strongly polarized gases [135]. A theoretical study of the spin drag relaxation rate for a repulsive gas is presented by Duine et al [136] in this focus issue. They show that spin fluctuations enhance the spin drag in the vicinity of the Stoner ferromagnetic transition. Similarities in the spin transport in the unitary gas and graphene are studied by Müller and Nguyen in this focus issue [137].

A calculation of the diffusion constant based on the two-body Boltzmann equation can be found in [134, 138]. They find that

$$\begin{equation} D_{\mathrm{s}} = \frac{9\pi^{3/2}}{32\sqrt{2}}\frac{\hbar}{m} \left(\frac{T}{T_{\mathrm{F}}}\right)^{3/2}. \end{equation} \tag{ 23 }$$

Similar to the shear viscosity, the spin diffusion constant drops with decreasing temperature. Near the critical temperature D_s is expected to approach the universal value D_s ∼ ℏ/m. This behavior is indeed observed in the experiment [134]; see also the recent analysis of Bruun and Pethick [139]. It is interesting to compare this result with the observed minimum of the shear viscosity. Shear viscosity governs the rate of momentum diffusion. The associated diffusion constant is D_η = η/(mn). Near T_c we have η/s ≃ 0.5ℏ/k_B and s/n ≃ k_B. This implies that D_η ≃ 0.5ℏ/m, comparable to what is seen in spin diffusion. A similar correlation between shear viscosity and diffusion was observed in the QGP, as discussed in section 3.7.

A question quite distinct from that of the BCS–BEC crossover in ultracold Fermi gases in the continuum is how fermions pair, and reach the unitary regime, in a discrete context. Optical lattices, or crystals of light, are created from interfering laser beams. They make a sinusoidal standing wave that traps fermions via the ac Stark effect. The strongly discretized regime, i.e. in the lowest band(s) and tight binding approximations, is quite distinct from that explored by both the QGP and continuum ultracold quantum gas experiments. The essential features of weakly interacting fermions in this context have already been explored and understood [140]; however, the strongly interacting regime remains in question. Moreover, the lattice regime presents an additional challenge for holographic duality, as it too has a unitary point where 1/(k_Fa) → 0. The question of how to correctly model this problem in the context of a lattice is subtle, as simply attaching a band index to fermionic fields leads to hundreds of bands deep in the BEC regime, and is therefore numerically and practically intractable.

In the continuum, a single-channel model treating only fermions qualitatively reproduces the phase diagram of figures 3 and 9. This qualitative approach can be taken, for instance, by using the generalized BCS ansatz, technically valid only for weak interaction and high density, at the mean-field level for arbitrary interaction and finding the chemical potential self-consistently by fixing the average number of particles [141]. This method was first used by Eagles in the context of superconductivity in low carrier concentration systems [142] and later by Leggett [143] and Noziéres and Schmitt-Rink [90] explicitly for the BCS–BEC crossover at zero temperature and finite temperature, respectively, together called NSR theory. In NSR one begins with a lattice, but then quickly takes a continuum or low-momentum limit. Interestingly, if this limit is not taken one obtains a completely incorrect prediction: instead of the BEC critical temperature tending to a constant non-zero value as 1/(k_Fa) → ∞, as in figure 3, the critical temperature tends to zero algebraically. This unphysical tendency is because in order to hop or tunnel between lattice sites, a pair of fermions must be broken and tunnel one by one. Then for very strong pair binding energy the process is prohibitively expensive. In fact, even on the BCS side such models utilizing a single band have been shown to fail quantitatively for quite small values of k_Fa < 0 [144, 145]. This model is called the Hubbard Hamiltonian [146] and is also a proposed model for high-temperature superconductivity [147].

A solution to this quandry is to introduce a two-channel model incorporating both fermions and bosons [148–151]. The fermions then represent the unbound atoms scattering at threshold (open channel), while the bosons represent the weakly bound molecule brought into resonance (closed channel). Such two-channel models are mentioned briefly in the condensed matter context as long ago as 1985 [152] and explored seriously in partial form starting in 1995 [153]; they continue to be explored as the Cooperon model, in current research on high-temperature superconductivity [154]. However, only in ultracold quantum gases have all hopping, interaction and interconversion terms been included to make a Fermi–Bose Hubbard Hamiltonian in the lattice [155]. A series of papers attempted this approach with steady improvement over time, starting from simply attaching a band index to each channel [155, 156] up to performing a renormalization procedure to produce effective molecules in the lattice and minimize the number of bands that must be included [157–161]. The latter method has led to a Hamiltonian that is so much far from the original Hubbard Hamiltonian that it has been given a new name, the Fermi resonance Hamiltonian.

The Fermi resonance Hamiltonian is

$$\begin{eqnarray} &&\fl \hat{H}_{\mathrm{eff}}=-t_f\sum_{\sigma\in\left\{\uparrow,\downarrow\right\}} \sum_{\langle i,j\rangle}\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma} +E_0\sum_{\sigma\in\left\{\uparrow,\downarrow\right\}}\sum_i \hat{n}_{i\sigma}^{\left(f\right)} -\sum_{\alpha\in\mathcal{M}}\sum_{i,j}t^{\alpha}_{i,j}\hat{d}_{i,\alpha}^{\dagger}\hat{d}_{j,\alpha} \nonumber\\ &&+\sum_{\alpha\in\mathcal{M}}\bar{\nu}_{\alpha}\sum_i\hat{n}_{i\alpha}^{\left(b\right)} +\sum_{\alpha\in\mathcal{M}}\sum_{ijk}g_{i-j,i-k}^{\alpha} \left[\hat{d}_{i,\alpha}^{\dagger}\hat{a}_{j,\uparrow}\hat{a}_{k,\downarrow}+\mathrm{h.c.}\right], \end{eqnarray} \tag{ 24 }$$

where $\hat {a}_{i\sigma }^{\dagger }$ creates a particle with spin σ in the lowest open channel band Wannier state centered at lattice site i; $\skew3\hat{d}_{i,\alpha }^{\dagger }$ creates a particle in the αth dressed molecule Wannier state centered at site i; $\hat {n}_{i\sigma }^{\left (f\right )}$ is the number operator for fermions in the lowest Bloch band; and $\hat {n}_{i\alpha }^{\left (b\right )}$ is the number operator for the αth dressed molecule state. The set of dressed molecules $\mathcal {M}$ which are included dynamically can be determined on energetic and symmetry grounds from the two-particle solution. In order, the terms in equation (24) represent tunneling of atoms in the lowest Bloch band between neighboring lattice sites i and j; the energy $E_0=\sum _{\mathbf {q}}E_{\mathbf {1},\mathbf {q}}/N^3$ of a fermion in the lowest band with respect to the zero of energy; tunneling of the dressed molecular center of mass between two lattice sites i and j, not necessarily nearest neighbors; detunings of the dressed molecules from the lowest band two-particle scattering continuum; and resonant coupling between the lowest band fermions at sites j and k in different internal states and a dressed molecule at site i. The Fermi resonance Hamiltonian is a two-channel resonance model, between unpaired fermions in the lowest band, and dressed molecules nearby in energy [161]. Among other unusual predictions it makes, in contrast to usual Hubbard physics, is a significant diagonal hopping, pairing between atoms which do not lie along a principal axis of the lattice to form a dressed molecule and multiple molecular bound states induced by the lattice. Thus, the lattice problem is indeed quite different from both the continuum crossover problem and the well-known Hubbard physics.

In practice, the solution of the crossover problem in the lattice must occur in three steps. Firstly, for a given lattice strength and interacting strength the two-body problem must be solved exactly, including a renormalization procedure to get effective Wannier states for the molecules. Secondly, the coefficients in equation (24) are calculated. Thirdly, the Hamiltonian itself must be solved. Potential solution methods range from matrix product state methods in one dimensional (1D) to mean field, quantum Monte Carlo and dynamical mean field theory methods in higher dimensions. Holographic duality may offer new approaches to this Hamiltonian in the future. The crossover problem on the lattice remains very much an open problem and therefore we return to it in section 5.

In this subsection, we cover some of the new directions in unitary Fermi gases and related systems not discussed in previous sections, as explored in this focus issue.

2.7.1. New experimental probes

2.7.2. Solitons to polarons

2.7.3. Disorder and quantum phase transitions

2.7.4. Trimers: from Efimov states to SU(3)

QCD describes the behavior of strongly interacting matter, such as protons, neutrons and nuclei as well as the hot and dense matter created in heavy-ion collisions, in terms of quarks and gluons and their interactions. The complicated phenomenology of the strong interaction is encoded in a deceptively simple Lagrangian. The Lagrangian is formulated in terms of quark fields q^c_α f and gluon fields A^a_μ. Here, α = 1,...,4 is a Dirac spinor index (corresponding, in the Dirac representation, to quarks and anti-quarks with spin up and down), c = 1,...,N_c is a color index and f = up, down, strange, charm, bottom, top is a flavor index. In our world the number of colors is N_c = 3, but as a theoretical laboratory it is useful to consider theories with N_c ≠ 3. The dynamics of the theory is governed by the color degrees of freedom. The gluon field A^a_μ is a vector field (such as the photon) labeled by an adjoint color index a = 1,...,8. The octet of gluon fields can be used to construct a matrix valued field $A_\mu =A_\mu ^a \frac {\lambda ^a}{2}$, where λ^a is a set of traceless, Hermitian, 3 × 3 matrices. Repeated indices are assumed to be summed throughout our treatment. The QCD Lagrangian is

$$\begin{equation} {\cal L } = - \frac{1}{4} G_{\mu\nu}^a G_{\mu\nu}^a + \sum_f^{N_f} \bar{q}_f (\mathrm{ i}\gamma^\mu D_\mu - m_f) q_f, \end{equation} \tag{ 25 }$$

where G^a_μν is the QCD field strength tensor:

$$\begin{equation} G_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc} A_\mu^b A_\nu^c, \end{equation} \noindent \tag{ 26 }$$

and f^abc = 4i Tr([λ^a,λ^b]λ^c) is a set of numbers called the SU(3) structure constants. The covariant derivative acting on the quark fields is

$$\begin{equation} \mathrm{i} D_\mu q = \left( \mathrm{i}\partial_\mu + g A_\mu^a \frac{\lambda^a}{2}\right) q, \end{equation} \tag{ 27 }$$

and m_f is the mass of the quarks. The different terms in equation (25) describe the interaction between quarks and gluons, as well as nonlinear three- and four-gluon interactions. It is important to note that, except for the number of flavors and their masses, the structure of the QCD Lagrangian is completely fixed by a local SU(3) color symmetry.

For many purposes, we can consider the light flavors (up, down and strange) to be approximately massless, and the heavy flavors (charm, bottom and top) to be infinitely massive. In this limit the QCD Lagrangian contains a single dimensionless parameter, the coupling constant g. If quantum effects are taken into account the coupling becomes scale dependent. At leading order the running coupling constant is

$$\begin{equation} g^2(q^2) = \frac{16\pi^2} {b_0\log(q^2/\Lambda_{\mathrm{QCD}}^2)}, \quad b_0=\frac{11}{3}N_c-\frac{2}{3}N_f, \end{equation} \tag{ 28 }$$

where q is a characteristic momentum and N_f is the number of active flavors (N_f = 3 if we consider charm, bottom and top to be infinitely heavy). The result in equation (28) implies that, as a quantum theory, QCD is not characterized by a dimensionless coupling but by a dimensionful scale, the QCD scale parameter Λ_QCD. This effect is called dimensional transmutation [191]. We also observe that the coupling decreases with increasing momentum. This is the phenomenon of asymptotic freedom [192, 193]. The flip side of asymptotic freedom is anti-screening or confinement: the effective interaction between quarks increases with distance.

In massless QCD the scale parameter is an arbitrary parameter (a QCD 'standard kilogram'), but once QCD is embedded in the standard model and quarks acquire masses by electroweak symmetry breaking the QCD scale is fixed by the choice of units in the standard model. The numerical value of Λ_QCD depends on the renormalization scheme used to derive equation (28). Physical masses, as well as the value of b₀, are independent of this choice. In the modified minimal subtraction ($\overline {MS}$), one finds that Λ_QCD ≃ 200 MeV; see the QCD section in [194].

Asymptotic freedom and the symmetries of QCD determine the main phases of strongly interacting matter that appear in the QCD phase diagram shown in figure 11. In this figure we show the phases of QCD as a function of the temperature T and the baryon chemical potential μ. The chemical potential μ controls the baryon density ρ, defined as 1/3 times the number density of quarks minus the number density of anti-quarks.

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At zero temperature and chemical potential the interaction between quarks is dominated by large distances and the effective coupling is large. As a consequence, quarks and gluons are permanently confined in color singlet hadrons, with masses of order Λ_QCD. For example, the proton has a mass of m_p = 935 MeV.²⁰

If we think of the proton as composed of three constituent quarks this implies that quarks have effective masses m_Q ≃ m_p/3 ≃ Λ_QCD. This should be compared to the bare up and down quark masses which are of the order of 10 MeV.

Strong interactions between quarks and anti-quarks lead to a vacuum condensate of $\bar {q}q$ pairs, $\langle \bar {q}q\rangle \simeq -\Lambda ^3_{\mathrm {QCD}}$ [195–197]. This vacuum condensate spontaneously breaks the approximate chiral SU(3)_L × SU(3)_R flavor symmetry of the QCD Lagrangian. Chiral symmetry breaking implies the existence of Goldstone bosons, particles with masses smaller than Λ_QCD. These particles are known as pions, kaons and etas²¹.

The quark–anti-quark condensate $\langle \bar {q}q\rangle$ is analogous to the di-fermion condensate 〈qq〉 that characterizes BCS superconductors [34]. BCS pairing involves particles with opposite momenta near the Fermi surface. The chiral condensate is an order parameter for the pairing of fermions and anti-fermions near the surface of the Dirac sea. The most important difference between BCS pairing and chiral condensation is that because of the finite density of states near the Fermi surface, BCS pairing can take place at weak coupling. Chiral condensation, on the other hand, only happens at strong coupling²². We will see below that in QCD at large baryon density there is a transition between $\bar {q}q$ and qq pairing. This transition is driven by the competition between the stronger coupling in the $\bar {q}q$ channel and the growing density of states for qq pairing.

At very high temperature quarks and gluons have thermal momenta p ∼ T ≫ Λ_QCD (note that we are writing momenta in units of energy, again setting c = 1.). Asymptotic freedom implies that these particles are weakly interacting and that they form a plasma of mobile color charges, the QGP [199, 200]. We note that the argument that the QGP at asymptotically high temperature is weakly coupled is somewhat more subtle than it might appear at first sight. If two particles in the plasma interact via large-angle scattering, then the momentum transfer is large, and the effective coupling is weak because of asymptotic freedom. However, the color Coulomb interaction is dominated by small-angle scattering, and it is not immediately clear why the effective interaction that governs small-angle scattering is weak. The important point is that in a high-temperature plasma there is a large thermal population (n ∼ T³) of mobile charges that screen the interaction at distances beyond the Debye length r_D ∼ 1/(gT). We also note that even in the limit T ≫ Λ_QCD the QGP contains a non-perturbative sector of static magnetic color fields [201]. This sector is strongly coupled, but it does not contribute to thermodynamic or transport properties of the plasma in the limit T → ∞.

The plasma phase exhibits neither color confinement nor chiral symmetry breaking. This means that the high-temperature QGP phase must be separated from the low-temperature hadronic phase by a phase transition. The nature of this transition is very sensitive to the values of the quark masses. In QCD with massless u,d and infinitely massive s,c,b,t quarks the transition is second order [202]. In the case of massless (or sufficiently light) u,d,s quarks the transition is first order. Lattice simulations show that for realistic quark masses, m_u ≃ m_d ≃ 10 MeV and m_s ≃ 120 MeV, the phase transition is a rapid crossover [203]. The transition temperature, defined in terms of the chiral susceptibility, is T_c ≃ 151 ± 3 ± 3 MeV [204, 205].

The transition is believed to strengthen as a function of chemical potential, so that there is a critical μ at which the crossover turns into a first-order phase transition [206]. This point is the critical endpoint of the chiral phase transition. Due to the fermion sign problem it is very difficult to locate the critical endpoint using simulations on the lattice. A number of exploratory calculations have been performed [207–210], but at this point it is not even clear whether the idea that the transition strengthens as the baryon chemical potential increases is correct [211]. The critical endpoint is interesting because it is the only point on the phase transition line at which the correlation length diverges (there is a similar endpoint on the nuclear liquid–gas transition line). This means that the critical point may manifest itself in heavy-ion collisions in terms of enhanced fluctuations [212]. The idea is that one can tune the baryon chemical potential by changing the collision energy, since lower beam energy allows for more stopping of the baryons in the initial state and therefore leads to higher chemical potential. As the collision energy is varied one looks for non-monotonic behavior of fluctuation and correlation observables. A typical observable is the variance (divided by the mean) of the net number of protons (protons minus anti-protons) in a finite sub-volume, 〈(ΔN_p)²〉/〈N_p〉 with ΔN_p = N_p − 〈N_p〉. A beam energy scan is now under way at RHIC, and similar scans have been performed at lower energy as part of the CERN fixed target program [213, 214].

At low temperature the first phase one encounters when the chemical potential is increased from zero is nuclear matter, a strongly correlated superfluid composed of approximately non-relativistic neutrons and protons. It is interesting to note that nuclear matter in weak $\mathrm {n}\leftrightarrow \mathrm {p}+\mathrm {e}+\bar \nu _{\mathrm {e}}$ equilibrium is neutron rich²³ and that the neutron–neutron scattering length²⁴ a_nn ≃ 20 fm is much larger than the average inter-particle spacing, r_nn ≃ 1.9 fm at nuclear matter saturation density. As a consequence, dilute nuclear matter is closely related to the unitary atomic Fermi gases discussed in section 2.

At very large chemical potential, we can use arguments similar to those in the high-temperature limit to establish that quarks and gluons are weakly coupled. The main difference between cold quark matter and the hot QGP is that because of the large density of states near the quark Fermi surface even weak interactions can cause qualitative changes in the ground state of dense matter. In particular, attractive interactions between quark pairs lead to color superconductivity and the formation of a 〈qq〉 condensate. Since quarks carry color, flavor and spin labels, many superconducting phases are possible. The most symmetric of these, known as the color–flavor locked (CFL) phase, is predicted to exist at very high density [215, 216]. In the CFL phase the diquark order parameter is 〈q^A_αfq^B_βg〉 ∼ _αβ^ABC_fgC. This order parameter has a number of interesting properties. It breaks the U(1) symmetry associated with baryon number, leading to superfluidity, and it breaks the chiral SU(3)_L × SU(3)_R symmetry. Except for Goldstone modes the spectrum is fully gapped; fermions acquire a BCS-pairing gap, and gauge fields are screened by the Meissner effect. This implies that the CFL phase, even though it arises from a superdense liquid of quarks, shares many properties of superfluid nuclear matter.

The CFL phase involves equal pair condensates 〈ud〉 = 〈us〉 = 〈ds〉 of all three light quark flavors. As the density is lowered, effects of the non-zero strange quark mass become more important, and less symmetric phases are likely to appear [180]. Calculations based on weak coupling suggest that the first non-CFL phase is a CFL-like phase with a Bose condensate of kaons, and the second phase involves a standing meson wave superimposed on the kaon condensate [217, 218]. Other possibilities which may appear in strong coupling include a phase with up–down pairing only (2SC), or separate spin-one condensates of up, down and strange quarks.

Some guidance an analyzing the competition between these phases may come from studying analogue models of the quark–hadron phase transition based on ultracold atomic gases. A number of authors have pointed out the possibility of a BCS–BEC crossover in dense quark matter [219]. The basic idea is that as the density is lowered, quark Cooper pairs become more strongly bound and form a diquark Bose condensate. At lower densities one may also encounter a mixture of condensed diquarks and unpaired quarks. The physics of this system can be studied using ultracold boson–fermion mixtures [24, 25, 220, 221]. Finally, it may be possible to study the transition from nucleons, bound states of three quarks, to color superconducting diquarks using three-species systems of ultracold fermions [177–179, 222].

Color superconductivity affects both the equation of state and the transport properties of dense quark matter. These effects may manifest themselves in the structure and evolution of neutron stars. Possible signatures of pairing appear in the mass–radius relation, the cooling curve and the relation between the spin-down rate and the magnetic field. There are two contributions in this focus issue that touch on these issues. Shovkovy and Wang [223] study the bulk viscosity of normal quark matter in the large-amplitude regime. Anglani et al [224] investigate the interaction of collective modes in the CFL phase. This calculation is a first step toward more accurate calculations of transport properties of the CFL phase [225, 226].

At asymptotically high temperature the coupling is weak and properties of the QGP can be systematically computed. One of the most basic properties is the equation of state. In the following we will consider a related quantity, the entropy density. The perturbative expansion of the entropy density is [227, 228]

$$\begin{equation} s = T^3 \left\{ c_0 + c_2\,g^2 + c_3\,g^3 + \cdots \right\}, \end{equation} \noindent \tag{ 29 }$$

where g is the QCD coupling constant from equation (28) and we have chosen units such that ℏ = k_B = 1. The expansion is performed with the coupling $g=g(\bar \mu )$ evaluated at a fixed scale $\bar \mu$. Higher order terms contain logarithms of the form $\log (2\pi T/ \bar \mu )$, which combine with lower order terms to eliminate the dependence on the arbitrary scale $\bar \mu$, and lead to an expansion in terms of the running coupling g(q) evaluated at a scale q ≃ 2πT. The first term in equation (29) corresponds to the Stefan–Boltzmann law, and c₀ is proportional to the number of degrees of freedom,

$$\begin{equation} c_0 = \frac{2\pi^2}{45}\left( 2\left( N_c^2-1 \right) + 4N_cN_f\frac{7}{8} \right), \end{equation} \tag{ 30 }$$

where 2(N²_c − 1) is the number of degrees of freedom from the gluons and 4N_cN_f from the quarks. The naive perturbative expansion is an expansion in powers of g². Odd powers of g appear because of infrared divergences. As we approach the phase transition the coupling becomes large and higher order terms are no longer small. Indeed, the convergence properties of the expansion in equation (29) are extremely poor: the series shows no signs of converging unless the coupling is taken to be much smaller than one, g ≪ 1, corresponding to completely unrealistic temperatures of the order of 1 TeV. The convergence can be improved significantly by using a self-consistent quasiparticle expansion [229]. This means that the perturbative expansion is formulated not in terms of free quarks and gluons, but in terms of quasiquarks and quasigluons which have effective masses and effective interactions.

Quasiparticle expansions fit lattice data quantitatively down to temperatures T ∼ 2T_c. In this regime s/s₀ ≃ 0.85, where s₀ is the entropy density of a non-interacting plasma. In the past this was frequently taken as evidence that quasiparticles are not strongly coupled at temperatures relevant to the early stages of heavy-ion collisions at RHIC or the LHC. A new perspective on this question is provided by holographic duality; see section 4. Holographic duality maps the strongly interacting quantum field theories onto one higher dimensional classical gravity, in particular anti-de Sitter space (AdS). In the original version of the mapping, the quantum field theory (QFT) is a QCD-like theory known as the N = 4 supersymmetric (SUSY) Yang–Mills theory [230]. This theory is conformal. This means that its coupling constant does not run, g²(q²) = const, and there is no analogue of Λ_QCD. It also does not have a phase transition, and so the theory is in the plasma phase for all values of the coupling. The particle content of the N = 4 SUSY Yang–Mills theory differs from that of QCD. The theory has no quarks, and in addition to gluons it contains supersymmetric fermionic partners of gluons called gluinos as well as additional colored scalar fields. Nevertheless, the plasma phase shares many features of the QGP, and by focusing on a ratio such as s/s₀ we can remove the difference in the number of degrees of freedom. Later versions of holographic duality relax conformal and supersymmetric requirements, but still do not map precisely onto QCD.

Using holographic duality it was found that in the limit of strong coupling and a large number of colors the entropy density of the SUSY Yang–Mills theory is s/s₀ = 0.75 [231]. This value is remarkably close to the result observed in lattice QCD calculations near the phase transition, casting doubt on an interpretation of the data in terms of weakly coupled quasiparticles. More generally, holographic duality demonstrates that thermodynamic properties of the plasma need not be very sensitive to the strength of the interaction.

Holographic duality also shows that, in contrast to equilibrium properties, transport properties of the plasma are very sensitive probes of the strength of the interaction. In perturbative QCD the shear viscosity of three-flavor QCD is [6, 232, 233]

$$\begin{equation} \eta = \frac{kT^3}{g^4\log(\mu^*/m_{\mathrm{D}})}, \end{equation} \noindent \tag{ 31 }$$

where k = 106.67, μ* = 2.96T and m_D ∼ gT is the screening mass. Shear viscosity is related to the rate of momentum diffusion. A simple estimate of the shear viscosity is [234]

$$\begin{equation} \eta\simeq \frac{1}{3}np\,l_{\mathrm{mfp}} \simeq \frac{1}{3}\frac{p}{\sigma_{\mathrm{T}}}, \end{equation}\noindent \tag{ 32 }$$

where n is the density, p is the mean momentum, l_mfp is the mean free path and σ_T is the transport cross section. In a perturbative QCD plasma the typical momentum is p ∼ T, and the transport cross section is σ_T ∼ g⁴log(g)T⁻². The time scale for momentum diffusion is η/(sT) ∼ 1/(g⁴ log(g)T). We note that in the weak coupling limit this number is parametrically large. In a QGP at T = 200 MeV, just above the phase transition, we have T⁻¹ ≃ 1 fm/c and η/s ≃ 9.2/g⁴. A typical value of the coupling is g ≃ 2 (corresponding to α_s ≃ 0.3), which implies that η/(sT) ≃ 0.6 fm/c.

The strong coupling limit of η/s in the SUSY Yang–Mills plasma was studied by Policastro et al [235]. They find that

$$\begin{equation} \frac{\eta}{s} = \frac{1}{4\pi}\left( 1 + O\left( \lambda^{-3/2} \right) \right) ,\end{equation}\noindent \tag{ 33 }$$

where λ = g²N_c is the 't Hooft coupling. This result implies that at strong coupling momentum equilibration is almost an order of magnitude more rapid than it is in weak coupling. The ratio η/s is also close to a bound that was argued to arise from the uncertainty relation [234]. The idea is that, because of quantum mechanics, the product p l_mfp in equation (32) cannot become smaller than Planck's constant ℏ (note that we are using units in which ℏ = 1). This implies that , where we have used the entropy per particle of a free gas [234]. The uncertainty argument has never been made precise, since the mean free path estimate is not valid at strong coupling. However, the holographic duality calculation was discovered to be quite general. It was shown that the strong coupling limit of η/s is universal in a large class of theories that have gravitational duals, and that the O(λ^−3/2) corrections are positive [236, 237]. These observations led KSS to make the conjecture that

$$\begin{equation} \frac{\eta}{s} \geqslant \frac{1}{4\pi}\end{equation}\noindent \tag{ 34 }$$

is a universal bound [11] that applies to all fluids. The status of this conjecture is discussed in more detail in section 4 and sketched in figure 2. There are some known counterexamples involving theories with gravitational duals described by higher derivative gravity. However, it seems clear that equation (34) applies to a large class of theories that include generalizations of QCD.

Experiments on collective flow in heavy-ion collisions at RHIC and the LHC (discussed in section 3.4) indicate that the viscosity of the QGP near the critical temperature is indeed close to the proposed bound and that equilibration must be very rapid. Together with the observation of a large energy loss of highly energetic probes of the plasma, these results led to the conclusion that the QGP produced at RHIC must be strongly coupled [185, 238].

In the case of the SUSY Yang–Mills plasma, one can show that strong coupling implies the absence of well-defined quasiparticles. This can be seen most clearly by studying the spectral function ρ(ω) of the stress tensor correlation function. Kubo's formula relates the value of the spectral function at zero energy to the shear viscosity. The spectral function at finite energy carries information about the physical excitations that contribute to momentum relaxation. In weak coupling the spectral function has a peak at zero energy. The width of the peak, Γ ∼ g⁴log(g)T, is related to the quasiparticle lifetime. The spectral function in the strong coupling limit was computed in [239, 240]. It was found that the spectral density is completely featureless: the intercept at zero energy smoothly connects to the continuum contribution ρ(ω) ∼ ω⁴.

In the case of the QGP, the evidence regarding the existence of quasiparticles is ambiguous. The stress tensor correlation function in lattice QCD was studied by Meyer [4]. The reconstructed spectral function is smooth, but the resolution was not sufficient to exclude the presence of quasiparticles. There are a number of results that have been interpreted as favoring the existence of quasiparticles. One is the fact that lattice calculations of fluctuations of conserved charges, such as baryon number and strangeness, are compatible with the behavior of a free gas, even near T_c [241]. An experimental observation that has been cited as evidence for the presence of quasiparticles is the approximate quark-number scaling of the elliptic flow parameter v₂ [242]. Another experimental observable that may shed some light on the quasiparticle structure is the heavy quark diffusion constant D, which we will discuss in section 3.7. The main observation is that kinetic theory and holographic duality make very different predictions about the relation between the momentum relaxation time η/(sT) and the heavy quark relaxation time m_QD/T [181].

The experimental study of heavy-ion collisions at relativistic energies, which were long seen as the best environments for the production of the QGP, began in the late 1980s with fixed target programs both at CERN near Geneva, and at BNL, outside of New York City; see [243–245] for historical overviews. These two programs were essential in building up a vital experimental community, formed of larger and larger collaborations as the experiments became more and more sophisticated. The programs at CERN and BNL demonstrated that heavy-ion collisions produce strongly interacting, approximately equilibrated matter. The next step was taken with the turn-on of the RHIC at Brookhaven in 2000, which brought heavy-ion physics into the collider era, with accessibility to processes calculatable with perturbative QCD. Colliders are accelerators with counter-circulating beams in which the entire collision energy is available for particle production. RHIC collides gold ions with a beam energy such that every nucleon in each of the beams has an energy of 100 GeV. Since the rest mass energy of a nucleon is about 1 GeV, this implies a relativistic γ factor of about 100. When two bunches of counter-rotating beams overlap in each of the RHIC experimental halls, at a rate of thousands of times per second, the 1 billion ions in each bunch typically induce less than one collision. In a fraction of these collisions, particularly the ones where the nuclei are head-on, thousands of particles are produced by the conversion of kinetic energy into mass energy.

The particles are recorded in detectors which were originally located in four experimental areas located around the ring. The early RHIC program had two large multipurpose detectors, STAR and PHENIX, with roughly 500 collaborators each, and two smaller detectors, BRAHMS and PHOBOS, with about 50 collaborators each [185]. STAR and PHENIX are both large spectrometers. STAR focuses primarily on charged hadrons using a time projection chamber of 4 m diameter that creates a 3D image of collision events at a relatively low rate. PHENIX is a combination of drift chambers, particle identification counters and calorimeters that make precise measurements of both charged particles and photons. Both of these detectors were inspired by the collider detectors of the previous generation and make measurements mainly at large angles with respect to the beam directions. To make sure physics was not missed in other regions of phase space, BRAHMS and PHOBOS were both designed to make more limited measurements near the beam axis, BRAHMS with a narrow-band spectrometer and PHOBOS with a large-coverage single-layer silicon detector, sensitive to roughly 75% of the total number of charged particles produced per event. A range of the data from these experiments will be shown in later sections, and in several of the contributions to this focus issue [186, 187, 189]. As of 2011, RHIC has completed its 11th experimental run, after colliding ions with a wide range of energies (from 7.7 to 200 GeV) and nuclear species (protons, deuterons, copper and gold).

As RHIC was taking data, the LHC at CERN was being built, along with its three large detector systems ALICE, ATLAS and CMS. The LHC provides heavy-ion collisions of primarily lead, slightly larger than gold, at center of mass energies up to 28 times that reached by RHIC. This translates into higher particle densities and temperatures and, even more significantly, into much higher rates for large momentum-transfer processes, such as jets, photons and heavy flavor, discussed below. The ALICE detector was designed around another large time projection chamber, similar to but larger than the one in STAR at RHIC, with a set of additional detectors to identify different hadron species as well as photons and electrons in limited angular regions. The layout of the ALICE detector is shown in figure 12. The two larger experiments, ATLAS and CMS, were general-purpose detectors deploying charged particle tracking and hermetic calorimetry over a large solid angle. However, the stringent design requirements to search for new high-mass particles led to detectors that were quite capable for the higher multiplicities expected in heavy-ion collisions at the LHC and which have excellent capabilities for high-energy processes, similar to that needed for Higgs and searches beyond the Standard Model. As of late 2011, the LHC has completed its second lead ion run, as well as a first feasibility study for future proton lead collisions.

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In the following, we will concentrate on the results from Au + Au at the top RHIC energy of 100 GeV per nucleon. The transverse radius of an Au nucleus is approximately 6 fm, and the duration of a heavy-ion event is τ ∼ 6–10 fm/c. The estimate for the lifetime comes from hydrodynamic simulations which we will describe in section 3.5. The simplest observable in a heavy-ion experiment, typically published very soon after a new machine becomes operational, is the total multiplicity of the produced particles. In Au + Au collisions at 100 GeV per nucleon the total multiplicity is about 7000; see section 3.4. Somewhat more detailed information is provided by the spectra dN/d³p of produced particles (e.g. [246, 247]). The momenta can be decomposed into a transverse momentum p²_T = p²_x + p²_y and a longitudinal momentum p_z; see figure 13. In the relativistic regime the natural variable to describe the motion in the z direction is the rapidity,

$$\begin{equation} y = \frac{1}{2}\log\left(\frac{E+p_z}{E-p_z}\right). \end{equation}\noindent \tag{ 35 }$$

At RHIC the energy of the colliding nuclei is 100 + 100 GeV per nucleon, and the separation in rapidity is Δy = 10.6.

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A simple picture of the initial state of the fireball created in the collision was suggested by Bjorken [249]. He proposed that the two highly Lorentz contracted nuclei pass through each other and create a longitudinally expanding fireball in which particles are produced. In the original model the number of produced particles is independent of rapidity, and the subsequent evolution is invariant under boosts along the z-axis. A simple model of the initial energy density in the transverse plane is the Glauber model [248, 250]. The Glauber model is based on the observation that high-energy scattering can be described in the eikonal (geometric optics) approximation. The initial entropy density in the transverse plane is

$$\begin{eqnarray} &&\fl s({\bf x}_\perp,b) \propto \, T_A({\bf x}_\perp+{\bf b}/2) \Big[\!1 - \exp\!\left(-\sigma_{\mathrm{NN}} \, T_A({\bf x}_\perp-{\bf b}/2) \right)\!\Big] \nonumber \\&&\ms\ms + \, T_A({\bf x}_\perp-{\bf b}/2) \Big[\!1 - \exp\!\left(-\sigma_{\mathrm{NN}} \, T_A({\bf x}_\perp+{\bf b}/2) \right)\! \Big] ,\end{eqnarray}\noindent \tag{ 36 }$$

where b is the impact parameter,

$$\begin{equation} T_A({\bf x}_\perp) = \int \mathrm{d}z \, \rho_A({\bf x})\end{equation}\noindent \tag{ 37 }$$

is the thickness function, x ≡ (x,y,z) and x_⊥ ≡ (x,y), and $\sigma _{\mathrm {NN}}(\sqrt {s})$ is the nucleon–nucleon cross section. We also define the nuclear density as ρ_A(x). The idea behind the Glauber model is that the initial entropy density is proportional to the number of nucleons per unit area which experience an inelastic collision, which we call the number of participants, N_part. Other variants exist. For instance, one can distribute the energy density according to the number of binary nucleon–nucleon collisions, N_coll; see [251] for a comparison. A more sophisticated theory of the initial energy density is provided by the color glass condensate [252, 253]. This model leads to somewhat steeper initial transverse energy density distributions.

It should also be noted that the simple Glauber model is typically not used by modern calculations, since it does not account for the large fluctuations which are currently attributed to the event-wise variations in the nucleon configuration coming from each nucleus and the collision process itself. Figure 14 illustrates this by means of a single event simulated by a Glauber Monte Carlo code, which counts participants and collisions by means of a simple prescription that collisions occur when two nucleons are within $d<\sqrt {\sigma _{\mathrm {NN}}/\pi }$, where σ_NN is the total nucleon–nucleon inelastic cross section [248].

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Particle multiplicities are important both as first measurements at new machines, but also as a means of testing the current theoretical approaches in their ability to predict the degrees of freedom available at higher energies.

From the theoretical perspective, multiplicities are argued to be a good proxy for the initial gluon density. This relies on one essential non-trivial argument that subsequent scatterings in the QGP, after the initial phase where hard collisions occur and energy is deposited in the interaction region, do not increase the overall entropy, and thus do not increase the overall multiplicity. This is certainly not exactly true, as both thermalization and viscous effects during the hydrodynamic evolution will produce some amount of entropy, but there is evidence that the total entropy is dominated by initial particle production [255]. Assuming that entropy is conserved one can derive the Bjorken estimate for the initial entropy density,

$$\begin{equation} s_0 = \frac{3.6}{\pi R^2\tau_0}\left(\frac{\mathrm{d}N}{\mathrm{d}y}\right), \end{equation} \noindent \tag{ 38 }$$

where τ₀ is the thermalization time, R is the nuclear radius and dN/dy is the total number of particles per unit rapidity. This estimate is based on a model of the hydrodynamic evolution which we will discuss in more detail in section 3.5. The left panel in figure 15 shows experimental results for dN/dy of charged particles at RHIC. Assuming that N(all) = 1.5N(charged) (most particles are pions) we obtain dN/dy|_y=0 ≃ 975 for Au + Au collisions as 100 GeV per nucleon. A conservative estimate for the equilibration time is τ₀ = 1 fm/c; see section 5.2. This corresponds to an initial entropy density s₀ ≃ 30 fm⁻³. For a weakly interacting QGP this implies an initial temperature T₀ ≃ 230 MeV, and an initial energy density ₀ = 5 GeV fm⁻³. This number is significantly larger than the critical energy density for forming a QGP, _crit ≃ 1 GeV fm⁻³ [256].

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Experiments typically focus on studying how multiplicities scale with beam energy and collision geometry. Empirically, the multiplicity per unit rapidity scales as a small power of the beam energy; see the right panel of figure 15. In nucleus–nucleus collisions, data ranging from the fixed target SPS program all the way to the LHC are described by dN/dy|_y=0 ∼ s^α, where s is the square of the center of mass energy in the nucleon–nucleon system and α ≃ 0.15. In nucleon–nucleon collisions the exponent is somewhat smaller, α ≃ 0.11.

As a first approximation, it is natural to expect that heavy-ion collisions can be built by a linear superposition of proton–proton or, more accurately, nucleon–nucleon collisions. This is the basis of the wounded nucleon model of the 1970s which was invoked to explain how multiplicities in proton–nucleus collisions tended to scale as N^p+A_ch = N^p+p_ch × N_part/2 [257, 258]. In this expression, N_part is the number of participating nucleons estimated either using a Glauber model or by counting the number of slow proton tracks, e.g. in emulsion. In this picture, soft particle production was expected to result from the excitation and subsequent independent decay of individual nucleons. Hard processes, in which single partons (quarks and gluons) from one nucleon scattered off partons of another nucleon, were also discovered in the 1970s. As these were not simply decay products of a soft excitation but were short-range elastic scatterings, they were expected to scale linearly with the number of binary collisions, also predictable using a Glauber model.

In order to compare to experimental results, calculations typically scale the produced multiplicity by the number of participants, since the multiplicity is dominated by soft particles, with transverse momenta under 2 GeV. The experiments use the observed distribution of multiplicities at a fixed beam energy to estimate the centrality of each collision, usually expressed as a percentage of the total inelastic cross section. The 10% highest multiplicity events are the 0–10% centrality bin, the next highest are the 10–20% and so on. These bins are then compared to the events with the 10% smallest impact parameter, or 10% highest multiplicity events, in either a simple Glauber model or one which includes some model of the experimental fluctuations [248].

The first comparisons of theory and experiment at RHIC using the multiplicity in the most central events were surprising. While many models predicted a combination of soft and semi-hard physics processes, it was the models that were predominantly composed of soft processes which came closest to the experimental data. This included particularly models based on parton saturation, which were preferred by the early data. The situation is a bit different at the LHC, as described briefly in the contribution by Steinberg [186]. At the higher energies, models which include a combination of hard and soft processes seem to do better at predicting the multiplicities in the most central collisions. However, the centrality dependence, i.e. the relationship between the number of measured charged particles and the number of participating nucleons (or effective nucleon–nucleon collisions), is essentially the same at RHIC and the LHC [259]. At present it is not clear how to reconcile this observation with the apparent increase in the fraction of bulk particle production stemming from hard processes.

The particle density measured by the early experiments is not only valuable to compare with different production models. It is also an essential empirical input that can be used to determine the initial conditions for hydrodynamical models of the subsequent evolution. The hydrodynamic description is valid if the system is in local thermal equilibrium and if the thermodynamic variables are varying smoothly. If the system has a kinetic description in terms of quasiparticles this implies that the ratio of the mean free path over the characteristic length scale of the system, known as the Knudsen number, must be small. There is evidence that this condition is satisfied during the early stages of the evolution, but it must eventually break down during the late, dilute, stage of the evolution, since the mean free path increases with diluteness. The time when the mean free path becomes larger than the system size, more precisely, the expansion time multiplied by the mean velocity, is called freezeout.

There are several experimental observations that provide evidence for the assumption that heavy-ion collisions create a thermalized state. The first observation is that the overall abundances of produced particles, including very rare ones, is described by a simple thermal model that contains only two parameters, the freezeout temperature T and the baryon chemical potential μ [183]. The second observation is that for momenta less than about 2 GeV the spectra dN/d³p of produced particles follow a Boltzmann distribution characterized by the freezeout temperature and a collective radial expansion velocity [182]. This radial flow is clearly seen from the fact that the spectra of heavy particles, which receive a larger momentum boost from the collective velocity field, have a larger apparent temperature than the spectra of light particles.

The third piece of evidence in support of not only thermalization, but early thermalization, which means equilibration significantly before freezeout, is the observation of strong azimuthal anisotropies, typically called elliptic flow, in non-central heavy-ion collisions. Elliptic flow represents the collective response of the system to pressure gradients in the initial state. The analogous phenomenon in ultracold atomic gases is discussed in section 2.1. At finite impact parameter the initial state has the shape of an ellipse, with the short axis along the x-direction, and the long axis along the y-direction; see figure 13. This implies that pressure gradients along the x-axis are larger than along the y-axis. Hydrodynamic evolution converts the initial pressure gradients into velocity gradients in the final state. Elliptic flow is sensitive to early thermalization because the initial anisotropy that drives elliptic flow disappears with time. Elliptic flow is quantified in terms of the second Fourier coefficient v₂ of the particle distribution in the transverse plane,

$$\begin{equation} \left. p_0\frac{\mathrm{d}N}{\mathrm{d}^3p}\right|_{p_z=0} = v_0(p_{\mathrm{T}})( 1 + 2v_1(p_{\mathrm{T}})\cos(\phi)+ 2v_2(p_{\mathrm{T}})\cos(2\phi) +\cdots), \end{equation}\noindent \tag{ 39 }$$

where ϕ is the angle between the momentum vector and the x-axis. Odd Fourier moments such as v₁ and v₃ arise mainly from fluctuations in the initial state [260, 261]. Techniques for measuring v₂ and systematic trends in the results are described in some detail in the contribution by Snellings [187]. Snellings emphasizes that measurements of v₂ constrain the equation of state and the transport properties of the QGP. A similar conclusion is reached in the contribution of Lisa et al [188] to this focus issue. These authors focus on direct measurements of the shape of the final state using HBT (Hanbury–Brown–Twiss) interferometry.

In this section we will explain how elliptic flow can be used to constrain the ratio η/s introduced in section 3.2. In a relativistic fluid the equations of energy and momentum conservation can be written as a single equation

$$\begin{equation} \partial_\mu T^{\mu\nu}= 0 , \end{equation}\noindent \tag{ 40 }$$

where T^μν is the energy momentum tensor. In ideal fluid dynamics the form of T_μν is completely fixed by the Lorentz invariance,

$$\begin{equation} T^{\mu\nu} = (\epsilon+P)u^\mu u^\nu + P \eta^{\mu\nu}, \end{equation}\noindent \tag{ 41 }$$

where u^μ is the fluid velocity (u² = −1) and η^μν = diag(−1,1,1,1) is the metric tensor. The hydrodynamic equations have to be supplemented by an equation of state P = P(). The four equations given in equation (40) can be split into longitudinal and transverse components relative to the fluid velocity. The longitudinal equation is equivalent to entropy conservation

$$\begin{equation} \partial_\mu (su^\mu)=0, \end{equation}\noindent \tag{ 42 }$$

and the transverse equation is the relativistic Euler equation

$$\begin{equation} Du_\mu = -\frac{1}{\epsilon+P}\nabla_\mu^\perp P, \end{equation}\noindent \tag{ 43 }$$

where D = u^μ∂_μ and ∇^⊥_μ = ∂_μ − u_μD. We observe that the inertia of a relativistic fluid is governed by  + P. In the non-relativistic limit we can have $u^\mu \simeq (1,\vec {v})$ and  + P ≃ ρ (energy density is dominated by rest mass energy). We also find that D ≃ ∂_t + v·∂ and $\vec {\nabla }^\perp \simeq \vec {\partial }$. These approximations led to the usual Euler equation. We also note that ∇^⊥_μP = c²_s∇^⊥_μ, where c_s is the speed of sound. This implies that for a given initial energy density gradient the resulting acceleration is determined by the speed of sound. Ideal fluid dynamics corresponds to the limit that variations in the hydrodynamic variables occur on scales much larger than the mean free path. Dissipation arises from the leading gradient terms in the energy momentum tensor. In the rest frame of the fluid these corrections have the same form as in non-relativistic fluids. We have

$$\begin{equation} \delta T^{ij}= -\eta \left(\partial^i u^j+\partial^j u^i - \frac{2}{3}\delta^{ij}\partial\cdot u \right) - \zeta \delta^{ij} \partial \cdot u , \end{equation} \tag{ 44 }$$

where η and ζ are the shear and bulk viscosity.

The application of hydrodynamics to relativistic heavy-ion collisions goes back to the work of Landau [262] and Bjorken [249]. Bjorken discussed a simple scaling solution of the equations of fluid dynamics that corresponds to the space–time picture discussed in section 3.3. This solution provides a natural starting point for more detailed studies in the ultra-relativistic domain. In the Bjorken solution the initial entropy density is independent of rapidity, and the subsequent evolution is invariant under boosts along the z-axis. The evolution in proper time is the same for all comoving observers. The flow velocity is

$$\begin{equation} u_\mu = \gamma(1,0,0,v_z)= (t/\tau,0,0,z/\tau), \end{equation} \tag{ 45 }$$

where $\gamma =\sqrt {1-v_z^2}$ is the boost factor and $\tau = \sqrt {t^2-z^2}$ is the proper time. The velocity field in equation (45) solves the relativistic Euler equation (43). In particular, there is no longitudinal acceleration. The remaining hydrodynamic variables are determined by entropy conservation. Equation (42) gives

$$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}\tau}\left[\tau s(\tau) \right] = 0 \end{equation} \tag{ 46 }$$

and s(τ) = s₀τ₀/τ. For an ideal relativistic gas s ∼ T³ and T ∼ 1/τ^1/3. We saw in the previous section that the initial entropy density is constrained by the final state multiplicity. Typical parameters at RHIC are τ₀ ≃ 0.6–1.6 fm and T₀ ≃ 300–425 MeV. We note that the initial temperature is significantly larger than the critical temperature for the QCD phase transition. The temperature drops as a function of τ and eventually the system becomes too dilute for the hydrodynamic evolution to make sense. At this point, the hydrodynamic description is matched to kinetic theory, generally using the formalism described by Cooper and Frye in the early 1970s [263], and the spectra of produced particles are computed.

In order to quantitatively describe the observed particle distributions several improvements of the simple Bjorken model are necessary. Firstly, one has to include the transverse expansion of the system [265]. Secondly, one has to include deviations from boost invariance in the longitudinal directions. Rapidity distributions at RHIC most likely result from physics somewhere in between the Bjorken scenario, which assumes boost invariance, and the Landau picture, which assumes complete stopping of the initial nuclei [266]. One also has to include realistic equations of state and take into account the geometry of the initial state. Results for v₂(p_T) obtained from a calculation in ideal hydrodynamics are shown in figure 16. This calculation focuses on the central rapidity regime and maintains the assumption of boost invariance. The figure shows the result for a given centrality class, corresponding to a specific range of impact parameters. Hydrodynamic calculations show that the elliptic flow response v₂ is approximately linear in the spatial anisotropy 〈y² − x²〉/〈x² + y²〉 of the initial state. We observe that ideal hydrodynamics provides an excellent fit to the RHIC data for , depending on the particle species. This includes the observed hierarchy in the p_T dependence of v₂ for different species at low p_T. The mass splitting of v₂(p_T) reflects an approximate transverse energy E_T = (p²_T + m²) scaling of the particle spectra in hydrodynamics.

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Having established a baseline description of the spectra using ideal hydrodynamics, we can now discuss the role of dissipative effects. We begin with the effect of shear and bulk viscosity on the Bjorken solution. The scaling flow given in equation (45) is a solution of the relativistic Navier–Stokes equation. If the transverse expansion of the system is neglected, viscosity does not affect the flow profile but it does generate entropy. We find that

$$\begin{equation} \frac{1}{s}\frac{\mathrm{d}s}{\mathrm{d}\tau} = -\frac{1}{\tau} \left( 1 - \frac{\frac{4}{3}\eta+\zeta}{sT\tau}\right). \end{equation} \tag{ 47 }$$

The applicability of the Navier–Stokes equation requires that the viscous correction be small [234],

$$\begin{equation} \frac{\eta}{s}+\frac{3}{4}\frac{\zeta}{s} \ll \frac{3}{4}(T\tau). \end{equation} \tag{ 48 }$$

For the Bjorken solution, Tτ ∼ τ^2/3 grows with time, and this condition is most restrictive during the early stages of the evolution. Using τ₀ = 1 fm and T₀ = 300 MeV gives η/s < 0.6. This result implies that hydrodynamics cannot be used in relativistic heavy-ions collisions unless the QGP is strongly coupled and the shear viscosity is small.

It is instructive to study in more detail the viscous contribution to the stress tensor. Neglecting bulk viscosity the stresses in the central rapidity slice are given by

$$\begin{equation} T_{zz} = P -\frac{4}{3}\frac{\eta}{\tau},\quad T_{xx}=T_{yy} = P + \frac{2}{3}\frac{\eta}{\tau} . \end{equation} \tag{ 49 }$$

This means that shear viscosity decreases the longitudinal pressure and increases the transverse one. In the Bjorken scenario there is no acceleration, but if pressure gradients are taken into account shear viscosity will tend to increase radial flow. At finite impact parameter shear viscosity reduces the pressure along the x-direction and increases the pressure in the y-direction. As a consequence there is less acceleration in the x-direction, and elliptic flow is suppressed.

Viscosity modifies the stress tensor, and via the matching to kinetic theory at freeze-out, this modification changes the distribution functions f_p of the produced particles. In [267] a simple quadratic ansatz for the leading correction δf to the distribution function was proposed,

$$\begin{equation} \delta f_p = \frac{1}{2T^3} \frac{\eta}{s}f_0(1\pm f_0) p_\alpha p_\beta \partial^{\langle \alpha} u^{\beta\rangle} , \end{equation} \tag{ 50 }$$

where f₀ is the Bose–Einstein/Fermi–Dirac distribution and ∂^〈αu^β〉 is the symmetric traceless tensor that appears in equation (44). This ansatz is a very good approximation to the result of a more involved calculation using kinetic theory [6]. The modified distribution function leads to a modification of the single-particle spectrum. For a simple Bjorken expansion and at large p_T, we find that

$$\begin{equation} \frac{\delta (\mathrm{d}N)}{\mathrm{d}N_0} = \frac{1}{3\tau_fT_f}\frac{\eta}{s} \left( \frac{p_{\mathrm{T}}}{T_f} \right)^2 , \end{equation} \tag{ 51 }$$

where dN₀ is the number of particles produced in ideal hydrodynamics, δ(dN) is the dissipative correction and τ_f is the freeze-out time. There is an analogous formula for the second Fourier moment of the spectrum, related to v₂ [267]. We observe that the dissipative correction to the spectrum is controlled by the same parameter η/(sτT) that appeared in the entropy equation. We also note that the viscous term grows with p_T. These results are in agreement with experiment: deviations from ideal hydrodynamics grow with p_T, and they are larger in smaller systems (which freeze out earlier). More detailed analyses can be found in the contributions by Snellings [187] and Nagle et al [189]. A conservative bound for η/s at RHIC is η/s < 0.4, but the best fits tend to give values that are even smaller, η/s ≃ 0.1–0.2.

Another way to directly probe the density of gluons present in the initial state is to scatter fast partons, formed in the initial collisions, which can be detected in the final state as hadronic jets. This idea was proposed as long ago as 1982 by Bjorken, placed on a firmer theoretical footing in the early 1990s, and finally discovered experimentally by RHIC experiments in the early part of the 2000s [268]. Figure 17 shows the nuclear suppression factor, defined as

$$\begin{equation} R_{\mathrm{AA}} = \frac{1}{N_{\mathrm{coll}}} \frac{\mathrm{d}N/\mathrm{d}p_{\mathrm{T}}^{\mathrm{AA}}}{\mathrm{d}N/\mathrm{d}p_{\mathrm{T}}^{\mathrm{pp}}}, \end{equation} \tag{ 52 }$$

i.e. the yield of a particle species in heavy-ion collisions, typically measured as a function of its transverse momentum, divided by a similar yield in proton–proton collisions, divided by the number of binary collisions calculated from a Glauber model. This ratio essentially tests the hypothesis that each individual binary collision has an equal probability to induce a hard process identical to that found in proton–proton collisions. The latter acts as a reference system in which hard processes can usually be calculated perturbatively [269]. As can be seen in figure 17, only the production of direct photons appears to be unmodified, and even then only below 12–13 GeV, above which isospin effects are expected [270]. Conversely, light hadrons such as π⁰ and η are suppressed by about a factor of five above 5 GeV and perhaps slowly rise at high p_T. Electrons from charmed hadron decays are unsuppressed at low p_T but quickly fall to a level similar to that found for light hadrons, as discussed in more detail in the next section.

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Considerable effort has been devoted to the development of a perturbative QCD-based formalism that describes the energy loss of fast partons moving through a hot, dense medium. The main transport parameter that appears in the theory is the transverse momentum diffusion constant $\hat {q}$ [274],

$$\begin{equation} \hat{q} = \rho \int \mathrm{d}q_{\mathrm{T}}^2 \, q_{\mathrm{T}}^2 \, \frac{\mathrm{d}\sigma}{\mathrm{d}q_{\mathrm{T}}^2}, \end{equation}\noindent \tag{ 53 }$$

which determines the mean transverse momentum kick of the fast parton per unit length traveled. Here, dσ/dq²_T is the differential cross section for the scattering of the parton off the constituents of the dense medium, and ρ is the density of the medium. Recent work has focused on non-perturbative definitions of $\hat {q}$ that make no reference to the structure of the dense medium; see [184] for a review. The energy loss per unit length scales as $\mathrm {d}E/\mathrm {d}z\sim \hat {q}L$ for short path lengths (L < L_c) and as $\mathrm {d}E/\mathrm {d}z\sim \sqrt {\hat {q}E}$ for long path lengths. The characteristic length scale is $L_{\mathrm {c}}\sim \sqrt {E/\hat {q}}$.

Extracting $\hat {q}$ from experimental data on jet quenching at RHIC has proven to be difficult. Based on an analysis of R_AA using a Monte-Carlo implementation of the theory of perturbative energy loss [274, 275], the PHENIX collaboration has reported that $\hat {q}=13.2^{+2.1}_{-3.2}\,{\mathrm {GeV}}^2\,{\mathrm {fm}}^{-1}$ (the 2σ errors are $\hat {q}=13.2^{+6.3}_{-5.2}\,{\mathrm {GeV}}^2\,{\mathrm {fm}}^{-1}$) [276]. This number is large compared to expectations for a perturbative QCD plasma, $\hat {q}\simeq 1$–2 GeV² fm⁻¹ [277]. However, there are significant uncertainties associated with different implementations of energy loss. Bass et al [278] consider three different methods and find that equally good descriptions for R_AA can be obtained for values of $\hat {q}$ ranging from $\hat {q}=2.5$ to 10 GeV² fm⁻¹.

The LHC offers significantly larger jet production cross sections, a much increased p_T range and the possibility for detailed studies of not only leading particles but also identified jets and jet shapes. First results from the LHC are shown in figures 18 and 19. Figure 18 shows the observable R_AA introduced above. The low-p_T behavior agrees with the results at RHIC, despite the much larger collision energy. At high p_T the suppression is smaller, R_AA ∼ 0.5, but there is no hint of R_AA approaching 1. This is seen even more dramatically in figure 19, which shows the suppression factor R_CP for identified jets and not just high-p_T hadrons. The ratio R_CP is defined relative to the yield in very peripheral nucleus–nucleus collisions. We observe that R_CP is approximately equal to 0.5 for jets with transverse energies as high as 300 GeV. These results have important implications for the dependence of energy loss on the energy density of the medium, but the theoretical analysis of the LHC data is still very much in progress. A discussion of the first jet data from the LHC can be found in the contribution by Steinberg to this focus issue [186].

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There are some important connections between $\hat {q}$ and other transport parameters. In a quasiparticle description the cross section in equation (53) is the same cross section that governs momentum diffusion of approximately thermal particles, i.e. shear viscosity. This implies that $\eta /s\propto T^3/\hat {q}$, where the constant of proportionality is roughly 1. In the context of kinetic theory, a small shear viscosity therefore implies strong jet quenching [279]. In detail, the situation is more complicated. One can show that the viscous correction to v₂(p_T) at moderate p_T is proportional to $1/\sqrt {\hat {q}}$ and not $1/\hat {q}$ [280]. Also, elliptic flow at large p_T is mainly a probe of the path length dependence of energy loss [281]. In the next section we will see that there are important connections between energy loss of light quarks and energy loss of heavy quarks.

An important diagnostic of the properties of the QGP is the drag force on a heavy quark initially produced in a hard collision during the pre-thermal-equilibrium stage of the collision. We will see below that the drag force can be related to the heavy quark diffusion constant. Combining experimental constraints on heavy quark diffusion and shear viscosity, which is related to momentum diffusion, provides additional information about transport properties of the nearly perfect fluid created in heavy-ion collisions.

The diffusion of heavy quarks in a QCD plasma was first studied by Svetitsky [288]. In the following we will follow the arguments presented by Teaney and Moore [286]. Consider a small density n_Q of heavy quarks inside a QGP. In a heavy-ion collision, heavy charm and bottom quarks are produced in hard collisions between quarks and gluons during the pre-equilibrium stage of the reaction. If there is enough time, and if the interaction is sufficiently strong, then the heavy quarks can reach thermal equilibrium²⁵. In this case the time evolution of the density can be described by a diffusion equation

$$\begin{equation} \frac{\partial n_{\mathrm{Q}}}{\partial t}= D{\boldsymbol{\nabla}}^2 n_{\mathrm{Q}}, \end{equation}\noindent \tag{ 54 }$$

where D is the diffusion constant. We can relate D to the drag force on the quark by using a stochastic (Langevin) equation:

$$\begin{equation} \frac{\mathrm{d}{\bf p}}{\mathrm{d}t} = -\eta_{\mathrm{D}} {\bf p} +\xi (t), \quad \langle {\xi_{i}}(t) \xi_{j}(t') \rangle = \kappa\delta_{ij}\delta(t-t'). \end{equation}\noindent \tag{ 55 }$$

Here, p is the momentum of the particle, η_D is the drag coefficient and ξ(t) is a stochastic force. In a kinetic theory picture, the stochastic force models collisions with quarks and gluons in the plasma. The coefficient κ is related to the mean square momentum change per unit time, 3κ = 〈(Δp)²〉/(Δt). The Langevin equation can be integrated to determine the mean squared momentum. In the long time limit (t ≫ η⁻¹_D) the particle thermalizes and we expect that 〈p²〉 = 3mT. This requirement leads to the Einstein relation

$$\begin{equation}\eta_{\mathrm{D}}=\frac{\kappa}{2mT} .\end{equation} \noindent \tag{ 56 }$$

The relation between η_D and the diffusion constant can be determined from the mean square displacement. At late times 〈[Δx(t)]²〉 = 6D|t| and

$$\begin{equation} D= \frac{T}{m\eta_{\mathrm{D}}}=\frac{2T^2}{\kappa}. \end{equation}\noindent \tag{ 57 }$$

The diffusion constant for heavy quarks in a QGP can be determined by computing the mean square momentum transfer per unit time. At weak coupling and for approximately thermal heavy quarks, the diffusion constant is dominated by heavy quark scattering on light quarks and gluons, qQ → qQ and gQ → gQ. These processes are similar to the processes that determine the shear viscosity, except that heavy quarks move slowly and the interaction only involves the color Coulomb interactions, whereas shear viscosity is sensitive to both electric and magnetic interactions. The leading order result in QCD with three light flavors is [286, 288]

$$\begin{equation} D=\frac{6\pi}{g^4T\log(2T/m_{\mathrm{D}})}. \end{equation} \noindent \tag{ 58 }$$

Comparing this result with equation (31), we observe that heavy quark and momentum diffusion are related to each other. In the relevant range of coupling constants and keeping terms beyond the leading logarithm, one finds that DT ≃ 6(η/s). We note that the heavy quark relaxation time η⁻¹_D contains an extra factor m_Q/T compared to the hydrodynamic relaxation time η/(sT). For charm quarks at T = 200 MeV this factor is m_c/T ≃ 7. This implies that even if hydrodynamic behavior is reached very quickly, η/(sT) ∼ 0.2 fm, charm quarks have barely enough time to equilibrate, m_cD/T ∼ 8 fm.

This simple relation between DT and η/s is broken in the strong coupling limit of the SUSY Yang–Mills plasma. Using holographic duality, one finds that [289–291]

$$\begin{equation} D = \frac{2}{\pi T}\frac{1}{\sqrt{\lambda}}. \end{equation} \tag{ 59 }$$

This result implies that in the strong coupling limit there is no bound on the diffusion constant and that even very heavy quarks can possibly equilibrate. It is not clear how λ should be chosen in order to compare equation (59) to experiments in QCD. Gubser has advocated a value λ ≃ 6π and concludes that D ≃ 1/(2πT) [292].

Experimental information on the diffusion constant comes from the observation of single electrons from charm and bottom decays, as shown in figure 20 [282]. There are data on both the nuclear modification factor R_AA = dN_Au+Au/(〈T_AA〉dσ_p+p), where dN_Au+Au is the differential yield in Au + Au, 〈T_AA〉 is the nuclear overlap and dσ_p+p is the differential cross section in p + p, as well as on the elliptic flow parameter v₂. The nuclear modification factor is sensitive to energy loss, which in turn is governed by the drag force. Drag also implies that the elliptic flow of light quarks and gluons induces a non-vanishing v₂ parameter for heavy quarks. Different models for the drag force are shown as dashed and dash-dotted lines in figure 20. We observe that these models can account qualitatively for the behavior of R_AA but fail to reproduce the observed flow. This indicates that charm quarks show some degree of equilibration. A Langevin simulation with (2πT)D = (4–6) provides a qualitative description of both R_AA and v₂. It is interesting to note that this result, combined with the estimate DT ≃ 6(η/s), gives (4π)η/s ≃ 1.3–2.0, which is close to the result obtained from the elliptic flow of light particles [282].

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There are a number of uncertainties in this analysis that will be addressed in the future. Experiments have not yet been able to determine the flavor of the heavy quark, and there are significant theoretical uncertainties in predictions of the charm and bottom spectra. Future experiments will be able to identify the flavor of the heavy quark. Two contributions in this focus issue address transport properties of heavy quarks. Meyer [293] presents a lattice study of the Euclidean chromo-electric field correlation function. This correlation function is related by a Kubo formula to the diffusion constant. Rapp and Riek [294] perform a non-perturbative T-matrix analysis of the charm quark diffusion constant and the charmonium spectral function. These and other studies, together with ongoing efforts to measure the spectra of identified heavy quarks, will shed light on the relation between the different transport properties of the plasma such as diffusion, energy loss and shear viscosity and help to characterize the degree to which the initial state in a heavy-ion collision thermalizes.

Why should a QFT in d-dimensions have anything to do with gravity in d + 1, or vice versa? To give some intuition we describe two heuristic arguments that motivate such a holographic correspondence. We first consider a system that includes gravity, i.e., objects falling into a black hole, and argue that it should admit an equivalent description without gravity in one lower dimension. We then examine a strictly non-gravitational system, a QFT on a lattice, and argue that it should be related to an equivalent theory with gravity in one higher dimension.

4.1.1. Starting with gravity: falling into a black hole

Imagine standing far from a black hole³⁰ in (d + 1) dimensions and sending a robotic probe on a straight line trajectory directly toward the black hole. To follow the trajectory of the robot, we have attached to it a strobe light which flashes once a second. According to the robot, it will itself reach the horizon in finite proper time, as measured for example by a clock onboard the robot, and will thus have flashed its strobe only a finite number of times before crossing the horizon into the black hole interior.

However, according to you, the distant observer, the story looks very different. As the robot falls into the gravitational well of the black hole, the light emitted by the flashing strobe must do work to escape the gravitational potential, and is thus red-shifted. By the constancy of the speed of light, this means that the flashes arrive to you at ever greater intervals. As the robot approaches the horizon, the observed red-shift in fact diverges, so that the final flash emitted at the moment of crossing the horizon takes an infinite time to arrive at the distant observer; light from flashes after the robot has crossed the horizon will never make it out to you. Thus, what you actually see is not the robot falling into the black hole, but rather the robot approaching the black hole and gradually slowing down and compressing into a thin membrane on the surface of the black hole. Indeed, to you, it appears as if the robot has stopped falling toward the massive black hole entirely, as if it had stopped responding to gravity at all (figure 21).

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Now imagine sending in a swarm of robots in an isotropic shell around the black hole, so as to increase the mass of the black hole by a small amount corresponding to the matter and energy in the collapsing shell. Again, according to you, the distant observer, the shell will never actually appear to fall into the black hole. Rather, you will see it approach the black hole, slow down and effectively stop, forming a non-gravitating membrane which stretches over the true horizon. As each new bit of matter reaches this stretched horizon, it generates a disturbance in the membrane which spreads in waves through the stretched horizon as if through a fluid. Remarkably, all of the observable behavior of this system can be precisely captured by a non-gravitational viscous hydrodynamical model for a fluid on the stretched horizon which lives in only d rather than (d + 1) space–time dimensions.

If we describe this same process from the perspective of one of the robotic probes, the physics looks very different. In this frame, nothing special happens as the robots approach and pass the horizon. In contrast, the swarm of robots continues to fall toward the black hole, interacting with each other and with the gravity of the black hole in the full volume of (d + 1) dimensions. These dynamics can be modeled in a straighforward manner with GR in (d + 1) dimensions coupled to the matter and energy of the robots and the black hole.

This leaves us with two very different descriptions of our probes as they approach the horizon, one involving the gravitational dynamics of probes falling into a (d + 1)-dimensional black hole and the other involving the hydrodynamic response to our probes of a d-dimensional fluid. Both descriptions accurately capture the behavior of our probes as viewed by two different observers. But there cannot be two facts of the matter about the physics of our probes before they cross the horizon: two observers cannot accurately observe contradictory events unfold³¹. The gravitational dynamics of a (d + 1)-dimensional black hole must thus be, in some deeply non-local way, equivalent to the hydrodynamics of a d-dimensional fluid.

4.1.2. Starting without gravity: taking the renormalization group literally

Rather than start with gravity, let us start with a familiar non-gravitational field theory. Consider a system on a lattice with lattice spacing a and Hamiltonian,

$$\begin{equation} H = \sum_{x,i}J_{i}(x){\cal O}^{i}(x). \end{equation}\noindent \tag{ 60 }$$

Here, x labels the sites in the lattice, i labels the various operators ${\cal O}^{i}(x)$ defined at each site and the J_i(x) are coupling constants/sources for the operators³² ${\cal O}^{i}$. Note that the sources will, in general, depend on both space and time; we take x to stand for both as is the typical space–time notation in relativity. Given this setup, what we generally want to compute is the physics of the ground state and the low-energy excitations over this ground state as a function of the microscopic coupling constants at the lattice scale. In general, however, exactly diagonalizing the Hamiltonian is intractably difficult.

Kadanoff and Wilson taught us a beautiful approach to this problem: the RG [324–326].

The basic move is to iteratively coarse-grain the lattice, making the lattice spacing larger with each step, so that at each step a single site represents the average of multiple sites in the previous lattice. We then tune the couplings so as to preserve the physics of the ground state and the low-energy excitations over it, replacing the fixed coupling J_i(x) with a scale-dependent coupling J_i(x,u), where u denotes the length scale at which we probe the system. By iterating this procedure, we eventually arrive at an effective description of long-wavelength modes of the system. As explained by Kadanoff and Wilson, the resulting flow of the couplings with scale u can be encoded in a β-function which is, remarkably, local in energy scale,

$$\begin{equation} {u} {\partial\over\partial {u}} J_{i}(x,{u}) = \beta_{i}(J_{j}(x,{u}),{u}). \end{equation}\noindent \tag{ 61 }$$

When the beta function can be determined, for example in perturbation theory, this renormalization group approach is extremely powerful. Indeed, even finding the fixed points of the RG flow, corresponding to scale-invariant or conformal points governed by CFTs, can be enormously revealing, so much so that we often define scale-dependent QFTs with non-trivial RG flows by starting with a well-understood CFT and turning on a relevant deformation to generate the desired RG flow.

However, in many complex, strongly coupled systems, including many strongly correlated systems governed by interesting conformal fixed points, the β-functions cannot be straightforwardly derived. It is thus tempting to search for a reorganization of the RG which determines the correct RG flow of the couplings without requiring an explicit calculation of the β-functions.

To this end, consider the following recasting of the Kadanoff–Wilson picture (figure 22). As before, consider our series of coarse-grained lattices with coarse-grained Hamiltonians and suitably tuned couplings, J_i(x,u). Now imagine arranging each coarse-graining of the lattice into a stack ordered by scale, so that at each subsequent level of the stack the lattice spacing grows, ${u} =\{a, 2a, 4a,\dots \}$. By construction, motion down the stack reproduces RG flow in the original lattice theory. In this one-higher-dimensional hyper-lattice, however, the couplings at each scale, J_i(x,u), look a lot like fluctuating fields in a one-higher-dimensional lattice, varying both in space–time and in the scale direction (figure 23).

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This raises an interesting question. We know that the true solution J_i(x,u) solves the β-function equations for our original lattice theory, but we do not know what the correct β-functions for our lattice model are. Might there be some simple equation of motion on this one-higher-dimensional lattice whose solution is also J_i(x,u)? More generally, might there be a conventional local field theory³³ whose dynamics encode the full, and unknown, β-functions of the original lattice theory?

Consider the full RG stack of lattices labeled by a new RG coordinate, r, which runs from the original UV lattice cutoff, r = a, to the deep infrared (IR), r → ∞. We seek a simple QFT defined on this one-higher-dimensional space whose (d + 1)-dimensional bulk fields, Φ_i(x,r), are in one-to-one correspondence with the couplings, J_i(x), of the underlying lattice theory, with the values of the bulk fields at the UV end of the stack, r = a, determined by the microscopic couplings,

$$\begin{equation} \Phi_{i}(x,a)= J_{i}(x). \end{equation} \tag{ 62 }$$

As a consequence, the bulk field Φ(x,r) must have the same charges, tensor structure and other quantum numbers as the corresponding coupling³⁴, J_i(x). Thus with each scalar operator ${\cal O}(x)$ is associated a bulk scalar field Φ(x,r) such that $\Phi (x,a){\cal O}(x)$ is a scalar operator we can add to our Hamiltonian, with each current operator ${\cal J}^{\mu }(x)$ is associated a bulk vector field, A_μ(x,r), such that $A_{\mu }(x,a){\cal J}^{\mu }(x)$ is a scalar and so forth. In particular, with the canonical stress-tensor T^μν(x) is associated a canonical spin-2 field in the bulk, g_μν(x,r).

What can the Lagrangian of this bulk QFT be? It is not the original lattice Hamiltonian—the fields of the bulk QFT encode the dynamics of the couplings in RG space and not of the operators on the original lattice. A priori, the natural thing to do is to write down the most general effective field theory allowed by the fields and symmetries of the system. However, a simple argument greatly constrains the possibilities. Any QFT defined by perturbation from a fixed point CFT as discussed above includes among its operators a canonical stress-energy tensor, T_μν, arising as the Noether current corresponding to the space–time symmetry group. The bulk QFT should thus include a corresponding canonical spin-2 field, g_μν. But by the Weinberg and Weinberg–Witten theorems³⁵ and assuming the existence of a Lorentz-invariant continuum limit, any spin-2 field either must decouple at low energy, which would imply that sources of momentum or energy in the original lattice theory do not affect the system, or must couple universally according to the equivalence principle. In other words, the bulk field theory must either be topological or be a theory of gravity that reduces, at long wavelength and low energy, to GR.

If all this can be done consistently, we would appear to have two distinct descriptions of our system: the original lattice theory with lattice spacing a and sources J_i(x) which satisfy first-order β-function equations; and a gravitational description living in a one-higher-dimensional bulk whose fields, Φ_i(x,r), are in one-to-one correspondence with the sources J_i(x) but which satisfy conventional second-order local field equations. In other words, we have two ways to compute correlation functions in our system: either we may work in the lattice theory and solve the QFT problem; or we may work in the gravitational description and solve a GR problem.

In order for these two descriptions to have any chance of being equivalent³⁶, the bulk theory must satisfy two among many rather unusual properties, both of which turn out to be basic properties of black holes. Firstly, the entropy as computed in the two descriptions must be the same. In the original lattice QFT, the entropy will generically be extensive, with the entropy S_QFT of the d-dimensional QFT in a region R_d scaling linearly with its spatial volume, S_QFT(R_d)∝V (R_d). As a result, the entropy as measured in the one-greater-dimensional bulk gravity, S_GR, must be sub-extensive, scaling as the area bounding a region, S_GR(R_d+1)∝A(R_d+1)∝V (∂R_d+1). This is a basic feature of any theory of gravity³⁷: the entropy in a volume is bounded by the entropy of a black hole that fits inside that volume [330]. Since the entropy of a black hole is proportional to the surface area of its horizon, $S_{\mathrm {BH}}={1\over 4}A_{\mathrm {H}}$, this tells us that the entropy of any region in a theory of gravity is bounded by its surface area, $S_{\mathrm {GR}}(R_{d})\leqslant {1\over 4}A(R_{d})$. Indeed, the fact that the entropy of a black hole can be entirely associated with its surface, and that as a result the entropy in gravity is necessarily sub-extensive, is the origin of the term holography [331, 332]³⁸.

Secondly, the β-function equations in the boundary lattice theory are first order in the scale r, while the gravitational equations of motion in the bulk are second order in r. Physically, this means that specifying the sources in the lattice QFT at one scale completely determines the coupling at all other scales, i.e. completely fixes the RG trajectory. By contrast, in the gravitational system we have two solutions to our second-order equation, so we must further specify the derivative of the couplings with scale to uniquely fix the RG trajectory. How can these be equivalent? Somehow it must be the case that the gravitational system automatically selects one of the two solutions of the bulk equations as physical.

It is again the physics of black holes which explains how this can happen. Imagine solving a simple wave equation in the neighborhood of a black hole horizon. In principle, there are two solutions to the second-order wave equation. However, since things can fall into a black hole horizon but nothing can escape, it is natural to organize the solutions near the horizon into in-going and out-going waves. Moreover, when working in the Euclidean signature as is done, for example, in computing thermodynamic data, only the in-going solution is regular at the horizon; the out-going solution is always irregular at the horizon. Thus, while the gravitational field equations are indeed second order, the presence of a black hole horizon effectively adds a second boundary condition, so that we again need only specify a single boundary condition to completely determine the solution. Since the location of the horizon encodes the thermodynamic variables (temperature, entropy) of the system, the physical meaning of this second boundary condition is to specify the state of the QFT.

These heuristics suggest a connection between QFT in d space–time dimensions and quantum gravity in d + 1 space–time dimensions, with the fields in the gravitational system exactly paired with sources in the dual QFT and with the extra spatial coordinate in the gravitational bulk playing the role of an RG scale. The next sections will describe a precise and computationally effective theory of this connection known as holographic duality and review some of the lessons gleaned from its study to date.

4.1.3. Coda: the surprise of locality

Holographic duality [302–304] is a precise equivalence between certain d-dimensional quantum field theories and (d + 1)-dimensional gravitational theories which provides a sharp realization of the heuristics described in section 4.1. In particular, all the basic features noted above will reappear below: the fields in the bulk correspond to the couplings in the QFT; the RG flow of the QFT is encoded in the radial evolution of the gravitational theory along an extra dimension; and black hole horizons play a key role. Of particular importance is the precise geometry of the emergent space–time, as well as the precise relationship between observables in the QFT and those of the gravitational dual. We will explore a few simple examples along the way.

For clarity of presentation, we will focus our discussion on the simplest possible QFTs from the point of view of the RG, i.e. Lorentz-invariant CFTs which are fixed points of the RG. Since the RG maps to radial evolution in the gravitational dual, QFTs corresponding to RG fixed points must be dual to geometries which are translationally invariant along the emergent radial direction up to overall rescalings; adding Lorentz invariance then entirely fixes the dual geometry to be AdS, whose geometry we will now describe⁴⁰.

4.2.1. Gravity and matter in anti-de Sitter space

AdS_d+1 is a homogeneous, isotropic space–time with d + 1 space–time dimensions and constant negative curvature whose metric can be conveniently written as⁴¹

$$\begin{equation} \mathrm{d}s^{2} = {L^{2}\over r^{2}}\left[-\mathrm{d}t^{2}+\mathrm{d}\vec{x}^{2} +\mathrm{d}r^{2} \right], \end{equation} \tag{ 63 }$$

where $\vec {x}$ are (d − 1) spatial coordinates, t is a time-like coordinate and r is a final 'radial' spatial coordinate. The constant parameter L, known as the AdS-radius, sets the radius of curvature, with the Ricci curvature scalar given by the constant

$$\begin{equation} R=-{d(d+1)\over L^{2}}. \end{equation} \tag{ 64 }$$

The space is thus weakly curved when L is large and strongly curved when L is small. At fixed values of the radial coordinate r_*, the metric reduces to the d-dimensional Minkowski metric on flat space–time rescaled by L²/r²_*,

$$\begin{equation} \mathrm{d}s^{2}_{r_{*}} = {L^{2} \over r_{*}^{2}}\left[-\mathrm{d}t^{2}+\mathrm{d}\vec{x}^{2}\right]. \end{equation} \tag{ 65 }$$

It is thus useful to think of AdS as a stack of slices of flat space, with r controlling the physical scale on that slice. The slice at r → 0 is referred to as the boundary. Note that the volume element of the boundary slice at r → 0 diverges, while the proper distance to the boundary also diverges. Thus the boundary at coordinate r → 0 should be understood as the set of points at spatial infinity⁴².

Importantly, the full geometry is invariant under not just translations, rotations and Lorentz boosts along each slice, but also under general scaling transformations

$$\begin{equation} (r,\vec{x},t) \to (\alpha r,\alpha \vec{x},\alpha t) \end{equation} \tag{ 66 }$$

as well as rotations and boosts that mix r with the other dimensions. The full isometry group of AdS_d+1 is, in fact, identical to the conformal group in d space–time dimensions⁴³.

It is often useful to think of AdS as a harmonic trap for gravity. More precisely, the constant negative curvature of AdS acts as a harmonic trap such that if you sit inside AdS at an arbitrary point and throw a ball, it will inevitably fall back to you in finite time. Indeed, if you fire out a photon, that photon will run away to the boundary at r = 0 and return, again in finite observed time⁴⁴. And yet while nothing, not even gravity, gets out of the AdS box, the space–time remains completely homogeneous and isotropic. AdS thus acts as a peculiarly graceful IR regulator for gravity which breaks none of the symmetries of flat space.

All of this implies an intimate connection between the radial coordinate in the bulk of AdS and spatial scales along the boundary: probing short distances (or high energies) along the boundary corresponds to probing the bulk only near the boundary at r = 0, while probing long distances (or low energies) along the boundary corresponds to data deep in the interior of AdS. Roughly, then, we can think of the region near the boundary of AdS as associated with the UV physics of the boundary, and of the deep interior of AdS as associated with the IR physics of the boundary.

To visualize this association, consider a fixed tension rope whose ends are glued to the boundary at r = 0. Due to the constant negative curvature of AdS, the bulk of the rope is drawn into the bulk of the AdS space–time, forming an arc drooping away from the boundary. If the ends of the rope are held close together—probing short distances or high energy in the four-dimensional (4D) boundary—the rope barely hangs into the bulk (figure 24). If we instead pull the ends of the rope far apart—probing long distances or low energy in the 4D boundary—the rope dips much further into the bulk. In this way, probing the IR of the boundary corresponds to probing the deep interior of the AdS, r → ∞, while probing the UV of the boundary corresponds to focusing on the near-boundary region of the geometry, r → 0. Note that the near-boundary cutoff at r = a ≪ 1 thus acts as a UV regulator, while the horizon at r = r_H provides a natural IR regulator.

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Importantly, the AdS is a solution to the equations of motion of a generic Wilsonian action for the metric

$$\begin{equation} I_{\mathrm{Gravity}} = {1\over 16\pi G_{\mathrm{N}}} \int \mathrm{d}^{d+1}\!x\,\sqrt{-g}\left( -2\Lambda \,+ R \,+c_{2} R^{2} \,+c_{3} R^{3} +\cdots \right). \end{equation}\noindent \tag{ 67 }$$

Here, G_N is the Newton constant, g is the determinant of the space–time metric, g = det(g_μν), R is the Ricci curvature scalar built out of two derivatives of the metric, R ∼ ∂∂g, Λ is a cosmological constant (also known as the tension of the vacuum), and the ···  represent all other scalars one can build out of the metric and its derivatives. The R term in the action plays the role of a two-derivative kinetic term for the metric, g_μν. GR is defined by retaining only the lowest dimension kinetic term, R, giving the Einstein–Hilbert action whose equation of motion is the Einstein equation,

$$\begin{equation} R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R = -\Lambda g_{\mu\nu}. \end{equation} \tag{ 68 }$$

The c_nRⁿ + ···  terms then represent higher-derivative corrections to GR. The reader can verify that the AdS metric solves the Einstein equation (68) with the AdS radius L determined by the cosmological constant, Λ, as $\Lambda =-{\mathrm {d}(d-2)\over 2L^{2}}$. One can then show [336] that AdS continues to be a solution even when we include generic higher-curvature corrections: all that changes is the precise relationship between Λ and L.

Importantly, the Ricci scalar in AdS scales as $R\sim {1\over L^{2}}$ (cf (64)). As a result, we may neglect higher-curvature corrections in AdS when L² is large compared to the appropriate powers of the dimensionful couplings c_n. For example, if the higher-derivative operators are generated by quantum gravity effects, the dimensional couplings are controlled by the Planck length, ℓ_p. Quantum corrections to the metric can thus be neglected in AdS when L is large compared to the Planck length, ${L\over \ell _{\mathrm {p}}}\gg 1$. Similarly, GR receives corrections from string theory even at the classical level. Such corrections become important at a characteristic length scale known as the string scale, ℓ_s. In AdS, these stringy corrections can be neglected so long as ${L\over \ell _{\mathrm {s}}}\gg 1$. Our generic Wilsonian action for quantum gravity in AdS is thus well approximated by classical GR when the AdS radius is large compared to all scales in the problem,

$$\begin{equation} {L\over\ell_{\mathrm{p}}}\gg1,\quad{L\over\ell_{\mathrm{s}}}\gg1,\dots, \end{equation} \tag{ 69 }$$

where the ... denotes other sources of corrections to GR which may kick in at other length scales.

Black holes in AdS.

If pure empty AdS is a ground state for gravity, finite-temperature states correspond to black holes inside AdS. The simplest such asymptotically AdS black hole is the AdS–Schwarzchild black brane,

$$\begin{equation} \mathrm{d}s^{2} = {L^{2}\over r^{2}} \left[-f(r)\,\mathrm{d}t^{2}+\mathrm{d}\vec{x}^{2} +{1\over f(r)}\mathrm{d}r^{2}\right], \end{equation}\noindent \tag{ 70 }$$

with emblackening factor

$$\begin{equation} f(r) = 1-{r^{d}\over r_{\mathrm{H}}^{d}}.\end{equation}\noindent \tag{ 71 }$$

Near the asymptotic boundary at r → 0, f → 1, so this metric is asymptotically AdS_d+1. At r = r_H, however, f → 0, signaling the presence of a black hole horizon. This horizon in turn shields us from a physical singularity at r → ∞ by ensuring that nothing which is inside the horizon, and thus sensitive to the singularity, can ever escape to influence events in the rest of the space–time. Since the entire solution is translationally invariant in the (d − 1) spatial directions, $\vec {x}$, this black brane is not a compact object, but rather extended in all directions other than r.

Importantly, a classical black hole is a thermodynamic object with a definite temperature, energy and entropy, as shown by Bekenstein and Hawking (see [330] and references therein). If this were not the case, we could violate the third law of thermodynamics by throwing all of our waste heat into a black hole and thus build perfectly efficient Carnot engines. Black holes in AdS similarly carry definite temperatures, energies and entropies, although the details are a bit different from the flat space case [337]. In the case at hand, the Hawking temperature T, energy density and entropy density s of the black brane are then

$$\begin{equation} T = {d\over 4\pi \,r_{\mathrm H}}, \quad \epsilon = {d-1\over16\pi\,r_{\mathrm H}^{d}} \left({L\over\ell_{\mathrm{p}}}\right)^{\!d-1}, \quad s ={1\over 4\, r_{\mathrm H}^{\,d-1}} \left({L\over\ell_{\mathrm{p}}}\right)^{\!d-1}, \end{equation}\noindent \tag{ 72 }$$

where ℓ_p is the Planck length defined by G_N = ℓ^d−1_p. The ratio ${L\over \ell _{\mathrm {p}}}$ thus measures the AdS scale in Planck units.

Physically, these thermodynamic relations are telling us that if we increase by throwing additional mass or energy into the black brane, the horizon swells outward toward the asymptotic boundary (r_H → 0) and the black brane heats up. The specific heat of this black brane is thus positive. This means that black holes in AdS can come to equilibrium. Again, this can be traced to AdS playing the role of a harmonic trap for gravity: any radiation that the black hole evaporates away will, in finite time, fall back into the horizon⁴⁵.

Charged black holes in AdS.

More generally, we can add charge to our black brane by adding a non-trivial Maxwell field

$$\begin{equation} \mathrm{d}s^{2} = L^{2}{-f(r)\mathrm{d}t^{2}+\mathrm{d}\vec{x}^{2} +\mathrm{{1\over f(r)}d}r^{2} \over r^{2}} , \quad A = A_{t}(r) \mathrm{d}t,\end{equation}\noindent \tag{ 73 }$$

with emblackening factor

$$\begin{equation} f=1-M \,r^{d} + Q^{2} \,r^{2(d-1)}, \end{equation} \tag{ 74 }$$

and electromagnetic scalar potential⁴⁶

$$\begin{equation} A_{t}(r) = \mu \,\left(1-\left({r\over r_{\mathrm{H}}}\right)^{d-2}\right), \end{equation} \tag{ 75 }$$

where $\mu = {2Q\over {\cal C}} \,{r_{\mathrm {H}}^{~d-2}}$ and ${\cal C}=\sqrt {2(d-2)\over d-1}$.

This metric and gauge field together extremize the Einstein–Maxwell action⁴⁷,

$$\begin{equation} I_{\mathrm{ME}} = {1\over 16\pi G_{\mathrm{N}}}\int \mathrm{d}^{d}x\,\sqrt{-g}\left[-2\Lambda +R - {L^{2}\over 4e^{2}} F^{2}\right], \end{equation}\noindent \tag{ 76 }$$

again with $\Lambda ={-d(d-2)\over 2 L^{2}}$. In this geometry, the horizon lies at the radial position r = r_H implicitly defined as the value of r where f(r) vanishes. M and Q then determine the Hawking temperature of the horizon,

$$\begin{equation} T = {d\over 4\pi r_{\mathrm H} }\left(1-{d\!-\!2\over d}Q^{2}r_{\mathrm H}^{2d-2}\right), \end{equation}\noindent \tag{ 77 }$$

as well as its energy, entropy and charge densities,

$$\begin{equation} \fl \epsilon = M\, {d-1\over16\pi} \left({L\over\ell_{\mathrm{p}}}\right)^{\!d-1}\quad s ={1\over 4\, r_{\mathrm H}^{\,d-1}} \left({L\over\ell_{\mathrm{p}}}\right)^{\!d-1}, \quad \rho =Q\,{d-1\over8\pi{\cal C}} \left({L\over\ell_{\mathrm{p}}}\right)^{\!d-1}. \end{equation} \noindent \tag{ 78 }$$

It is then straightforward to check that these variables satisfy the first law of thermodynamics, d = Tds + μ dρ, with μ playing the role of the chemical potential.

A curious feature of these charged black holes is that the zero-temperature limit has finite entropy density. To see this, note that we can tune Q to make T vanish, $Q\to Q_{*}=\sqrt {d\over d-2}{1\over r_{\mathrm H}^{d-1}}$. In this limit, r_H, and thus the entropy density, remains non-zero. This finite zero-temperature entropy represents an enormous degeneracy of the ground state of this black hole⁴⁸ and correspondingly a large number of potential instabilities of these black hole solutions; these turn out to play an interesting role in holographic descriptions of quantum phase transitions. For our present purposes, let it suffice to say that this is an interesting feature of these black branes which will crop up from time to time in the following.

Matter in AdS: the effective action as functional of boundary conditions.

The fact that AdS behaves like a homogeneous harmonic trap has important consequences when we consider matter fields in AdS. Consider, for simplicity, a scalar field Φ with mass m² in the background of an uncharged AdS black brane (70) with classical action

$$\begin{equation} I_{\Phi} \propto \int \!\mathrm{d}^{d+1}\!x\, \sqrt{-g}\left(-\frac{1}{2}\left(\partial\Phi\right)^{2}-{m^{2}\over2}\Phi^{2} +\cdots\right), \end{equation} \tag{ 79 }$$

where the ···  denotes interactions with other matter fields which we will, for the moment, neglect. Working for a moment with a simple plane wave with momentum k in the boundary dimensions, Φ(x,r) = Φ(r)e^ik·x, the wave equation following from the above action becomes

$$\begin{equation} r^{2}f \,\Phi''(r) -r\left[rf'\!-\!(d\!-\!1)f\right]\Phi'(r) -\left[k^{2}r^{2}\!+\!m^{2}L^{2}\right]\Phi(r) = 0. \end{equation} \tag{ 80 }$$

This equation has two simple degenerate points, one at the boundary where r = 0 and the other at the horizon where f = 0. Let us study a general solution in the neighborhood of each.

Near the boundary at r → 0, where f → 1, the general solution of this equation reduces to

$$\begin{equation} \Phi \sim \phi_{d-\Delta}(k) \,r^{d-\Delta} + \phi_{\Delta}(k)\, r^{\Delta} + \cdots, \end{equation} \tag{ 81 }$$

where the scaling dimension Δ is determined by⁴⁹ Δ(Δ − d) = m²L². So long as m² ⩾ − d²/4L², the scaling dimension is real. This Breitenloner–Freedman (BF) bound tells us that a small negative m² does not lead to an instability in AdS as it would in flat space—instead, the would-be instability is lifted by the harmonic potential generated by the AdS curvature. Note that in the flat space limit, L → ∞, this window of 'allowed tachyons' disappears. So long as the BF bound is satisfied, specifying a solution reduces to specifying the two radial integration constants, ϕ_d−Δ and ϕ_Δ.

A key fact about this asymptotic behavior is that, so long as $m^{2}L^{2}>1-{d^{2}\over 4}$, the mode ϕ_d−Δ is non-normalizable according to the natural norm on a constant-time spatial slice, Σ_t,

$$\begin{equation} \left(\Phi_{1},\Phi_{2}\right) = -\mathrm{i}\int_{\Sigma_{t}}\mathrm{d}z\, \mathrm{d}\vec{x} \sqrt{-g}g^{tt} \left(\Phi_{1}^{*}\partial_{t}\Phi_{2}-\Phi_{2}^{*}\partial_{t}\Phi_{1}\right). \end{equation} \tag{ 82 }$$

Varying this non-normalizable mode thus corresponds to a large (divergent) change in the action. We must thus fix the non-normalizable mode to make the variational problem in the bulk well posed, for example by specifying a boundary condition for the bulk field Φ near the boundary, r → 0. This leaves us with a one-parameter family of solutions labeled by the remaining integration constant, ϕ_Δ, which can be chosen arbitrarily.

We thus draw a general conclusion for studying matter fields in AdS: in order for the least-action principle to be well defined in AdS, we must fix the values of the non-normalizable modes of all fields. This is typically done by fixing boundary conditions for the non-normalizable modes of all fields at the AdS boundary, r → 0.

Near the horizon at r_H, where f ∼ f_H(1 − r) → 0, the equation of motion again degenerates. Here, however, the physics is rather different. Physically, this equation has two kinds of solution: one which is regular, so that the degenerating fΦ'' term is negligible; and one which is irregular at the horizon, so that the fΦ'' term is not negligible. But why are half the solutions irregular? The reason was alluded to above: near the horizon, all waves can be expressed as a superposition of in-going and out-going waves. To see that this is precisely what is going on, let us study our scalar equation at non-zero frequency. Recalling that the magnitude of the d-momentum k in the metric (70) is $k^{2} = -{r^{2}\over f} \omega ^{2}+r^{2}\vec {k}^{2}$, the dominant terms in the scalar equation near the horizon are

$$\begin{equation} f\,\partial_{r}(f\,\partial_{r} \Phi) +{\omega^{2}}\Phi \sim 0 . \end{equation} \tag{ 83 }$$

It is useful to switch to the near-horizon coordinate ρ defined by f∂_r = ∂_ρ, i.e. e^−ρ ∼ (1 − r)^1/f_H, in terms of which the horizon lies at ρ → ∞. In terms of this coordinate, the general solution can be expressed as a superposition of in-going and out-going waves as

$$\begin{equation} \Phi = \Phi_{\mathrm{in}}(\rho)\,\mathrm{e}^{-\mathrm{i}\omega(t-\rho)} + \Phi_{\mathrm{out}}(\rho)\,\mathrm{e}^{-\mathrm{i}\omega(t+\rho)} .\end{equation} \tag{ 84 }$$

For future reference, note that the in(out)-going solutions satisfy the first-order constraint at r_H,

$$\begin{equation} f\,\partial_{r} \Phi_{\mathrm{in}} =\mathrm{i}\omega\Phi_{\mathrm{in}},\quad f\,\partial_{r} \Phi_{\mathrm{out}} =-\mathrm{i}\omega\Phi_{\mathrm{out}}. \end{equation} \tag{ 85 }$$

Note that, in terms of the original radial coordinate, r, the phases of both of these time-dependent solutions accumulate near the horizon. One can check that, in the limit of zero frequency, the in-going wave is regular at the horizon while the out-going wave is irregular, matching our zero-frequency analysis above. Note, too, that this suggests a natural prescription for analytic continuation to Euclidean time: we should choose the continuation so that the in-going wave is regular at the Euclidean horizon, e^iωρ → e^−ω_Eρ, i.e. we should define ω = iω_E, where ω_E > 0.

Again, this turns out to be a general result: in the presence of a black hole horizon, the Laplacian in any tensor representation takes the form $\square \sim r^{2}f \partial _{r}^{2}+{r^{2}\omega ^{2}\over f}+\dots$, so we must impose a boundary condition on our solutions such that they correspond to regular, in-falling modes at the horizon.

Thus, while our equation of motion is second order, there is only one linearly independent solution which is regular everywhere in the bulk and in-going at the horizon. Our gravity problem in AdS, in the presence of a black hole horizon, becomes effectively a first-order problem⁵⁰. Explicitly, fixing the non-normalizable mode ϕ_d−Δ at the boundary and imposing in-falling boundary conditions at the horizon completely determines our solution for Φ throughout the bulk of AdS and as a result determines ϕ_Δ. However, in order to actually compute ϕ_Δ given ϕ_d−Δ, we must solve our problem not just at the boundary but all the way through the bulk to the horizon, where we impose the appropriate boundary conditions. The relation between ϕ_d−Δ and ϕ_Δ is thus also determined by IR physics and not just UV physics.

It is illuminating to see this work in detail in an analytically solvable example. Consider again our scalar field Φ but now in pure AdS, i.e. with no black hole. This corresponds to studying our dual QFT at zero temperature and zero density. Expanding in plane waves, e^{−i(ωt−kx)}, the general solution to the bulk equation of motion can be found in closed form

$$\begin{equation} \Phi(t,x,r) = \left[\phi_{\mathrm{reg}}\, r^{d\over2}\,K_{\nu}\left(\kappa r\right)+\phi_{\mathrm{irreg}}\, r^{d\over2}\,I_{\nu}\left(\kappa r\right)\right]\mathrm{e}^{-\mathrm{i}(\omega t -kx)}, \end{equation}\noindent \tag{ 86 }$$

where $\nu =\Delta -{d\over 2}$, $\kappa =\sqrt {\omega ^{2}+\vec {k}^{2}}$, K_ν and I_ν are modified Bessel functions and ϕ_reg and ϕ_irreg are the two integration constants. Since I_ν(κr) ∼ e^κr near the 'AdS horizon' at r → ∞, regularity requires that ϕ_irreg = 0. We thus have just a single integration constant, ϕ_reg, which fixes the overall normalization of the in-falling mode in Φ. Near the boundary, the two asymptotic integration constants ϕ_d−Δ and ϕ_Δ in (81) are not independent. We can see how they are related by setting ϕ_irreg = 0 in (86) and expanding the solution near the boundary. After a little algebra, this gives

$$\begin{equation} \phi_{\Delta} = {\Gamma({d\over2}-\Delta)\over2^{\Delta-{d\over2}}\Gamma(\Delta-{d\over2})} \left(\omega^{2}+k^{2}\right)^{\Delta-{d\over2}} \phi_{d-\Delta}. \end{equation} \tag{ 87 }$$

Thus, once we impose regularity on our bulk solutions, we need only one boundary condition in the UV at r → 0 to fully specify a solution of the second-order bulk equations of motion. Meanwhile, we know that we need to fix the non-normalizable mode ϕ_d−Δ at the boundary so that our variational problem is well posed. Once we impose regularity in the bulk, i.e. in-falling at the horizon, the value of ϕ_Δ is completely determined.

This teaches us an important lesson. Suppose that we want to study the partition function for our gravitational theory in AdS, integrating over all bulk fields, Φ(x,r). To make the variational problem well posed, we must specify a boundary condition for each of the bulk fields at the boundary of AdS, as discussed above,

$$\begin{equation} \phi_{d-\Delta}(x) =\lim_{r\to0}r^{\Delta-d}\Phi(x,r), \end{equation} \tag{ 88 }$$

where the factor of r^Δ−d should be thought of as the appropriate wavefunction renormalization of the bulk field Φ(x,r). This means that the partition function in AdS, and thus the effective action Γ_AdS = −ln[Z_AdS], is a functional of the boundary conditions for all bulk fields,

$$\begin{equation} Z_{\mathrm{AdS}}[\phi_{d-\Delta}(x)]\equiv Z_{\mathrm{AdS}}[\Phi[\phi_{d-\Delta}(x)]]. \end{equation}\noindent \tag{ 89 }$$

What one means by 'a theory of gravity in AdS'.

In discussing holography, we will regularly refer to classical gravity, quantum gravity and perturbative gravity in AdS, so it is worth taking a moment to define our terms here.

By quantum gravity in AdS we mean a complete quantum description of the gravitational system, i.e. string theory in AdS. In general, string theory is a rich and complicated quantum theory which has no simple perturbative classical description. However, in some cases closed strings have a tractable semiclassical low-energy expansion involving a metric, g_μν, minimally coupled to a host of dynamical matter fields, {Φ_i}, governed by an effective action of the form

$$\begin{equation} I_{\mathrm{GR}} = {1\over 16\pi G_{\mathrm{N}}} \int \mathrm{d}^{d+1}\!x\,\sqrt{-g}\left( -2\Lambda + R + \cdots + {\cal L}_{\mathrm{matter}}(\Phi_{i}) \right). \end{equation} \noindent \tag{ 90 }$$

Here ${\cal L}_{\mathrm {matter}}(\Phi _{i})$ is the Lagrangian density for the matter fields and R is the Ricci curvature scalar, which is roughly two derivatives of the metric. $\sqrt {g}R$ plays the role of a kinetic term for the metric in GR. The ···  then represents an infinite tower of higher-derivative terms involving the metric, for example R², R⁴, etc, which represent closed string corrections to GR. Since these higher-derivative terms are also higher-dimension irrelevant operators, they must appear with dimensionful couplings. In general, these couplings are controlled by two independent length scales: ℓ_p, the Planck length, which controls quantum corrections to the dynamics of classical strings; and ℓ_s, which controls classical 'stringy' corrections to the dynamics of point particles.

When expanding the theory around AdS with AdS-length L (and thus with Ricci curvature R ∼ L⁻²), we thus have two dimensionless parameters controlling the various possible corrections to GR: ${\ell _{\mathrm {p}}\over L}$, which tells us whether quantum corrections are important; and ${\ell _{\mathrm {s}}\over L}$, which tells us whether stringy corrections are important. Note that ensuring that the curvature is weak in Planck units, ${\ell _{\mathrm {p}}\over L}\ll 1$, does ensure that the system can be treated classically, but does not tell us that the gravity must be exactly GR: the curvature of our AdS may still be large compared to the string scale, so higher-curvature corrections to the Lagrangian of GR may not be negligible. For pure classical GR to be a good approximation, two independent conditions must thus hold,

$$\begin{equation} {\ell_{\mathrm{p}}\over L}\ll1 \quad {\mathrm{and}} \quad {\ell_{\mathrm{s}}\over L}\ll1, \end{equation} \tag{ 91 }$$

i.e. the curvature must be small compared both to the Planck scale and to the string scale.

4.2.2. Field theories in flat space

Our next job is to define the d-dimensional QFT we want to study. A canonical way to do so begins by specifying a list of local operators ${\cal O}_{i}$ labeled by their Lorentz structure, their charges q_i and their scaling dimensions Δ_i, all defined at some UV fixed point. To generate an RG flow of interest, we perturb away from this fixed point by turning on appropriate sources J_i. The basic observables of the theory are then correlation functions of products of local operators

$$\begin{equation} \langle{\cal O}_{1}(x_{1}) \dots {\cal O}_{n}(x_{n})\rangle. \end{equation} \tag{ 92 }$$

These can be conveniently encoded in terms of the quantum generating functional,

$$\begin{equation} Z_{\mathrm{QFT}}[J_{i}(x)] = \langle \mathrm{e}^{\int \mathrm{d}x^{d} J_{i}(x){\cal O}_{i}(x)}\rangle, \end{equation} \tag{ 93 }$$

where the J_i(x) represent a set of sources and couplings for the operators ${\cal O}_{i}$. Note that the scaling dimensions and tensor structure of the sources J_i are completely determined by those of ${\cal O}_{i}$. This allows us to express correlation functions as derivatives of the partition function

$$\begin{equation} \langle{\cal O}_{1}(x_{1}) \dots {\cal O}_{n}(x_{n})\rangle ={\delta^{n} \ln Z_{\mathrm{QFT}}[J(x)] \over \delta J_{1}(x_{1})\dots \delta J_{n}(x_{n})} {\Big |}_{J_{i}=0} , \end{equation} \tag{ 94 }$$

where the restriction |_Ji = 0 means that evaluate the final result with all remaining sources turned off. Solving the theory then requires one to compute the partition function, Z_QFT[J_i(x)].

Canonical example: SU(N) Yang–Mills at large N.

The classic example of a field theory with a well-understood and controlled holographic dual, to which we will appeal below, is a special version of the SU(N) Yang–Mills gauge theory in flat 4D space–time called the ${\cal N}=4$ theory. Here ${\cal N}$ refers to the amount of supersymmetry enjoyed by the theory; ${\cal N}=4$ is the most one can have in 4D without including gravity. For our purposes, the role of supersymmetry is nothing more than a way of turning off quantum mechanics without totally trivializing the theory: with ${\cal N}=4$ supersymmetry, some quantities do not receive quantum corrections beyond one loop, and so can be computed at weak coupling and reliably extrapolated to strong coupling. In particular, the b-function of the theory can be computed exactly and is identically zero for all values of the coupling⁵¹.

The basic ingredients of the ${\cal N}=4$ theory are a gauge group, G = SU(N), a gauge field, A_μ(x), transforming in the adjoint of G, six scalars ϕ^I(x) also transforming in the adjoint of G and further enjoying a global SO(6) symmetry, and a large set of fermions. We can safely neglect the fermions for now, as their only role is to ensure supersymmetry. The Lagrangian is

$$\begin{equation} {\cal L}_{\mathrm{YM}} = -{1\over 4g^{2}}\left({\mathrm{Tr }}\, F^{2} + {\mathrm{Tr }}\, |D\phi_{I}|^{2} + \cdots\right), \end{equation} \tag{ 95 }$$

where g²_YM is the gauge coupling, D_μ is the gauge-covariant derivative, ···  denotes fermionic terms we neglect and all fields are canonically normalized with an overall factor of $-{1\over 4g^{2}}$ in front of their kinetic terms.

What are the good operators of this theory? We may suppose that any good gauge-invariant combination of the fundamental fields is a perfectly good observable. However, since we have a global symmetry, it is useful to write this list in terms of SO(6) irreducible representations. For example, the following three Lorentz scalars,

$$\begin{equation} {\cal O} = \sum_{I}{\mathrm{Tr }}\, \phi_{I}^{2},\quad {\cal O}_{IJ} = {\mathrm{Tr }}\, \phi_{(I}\phi_{J)},\quad {\cal O}^{\mu} = {\mathrm{Tr }}\, {\cal J}^{\mu}, \end{equation} \tag{ 96 }$$

transform as a scalar, a symmetric two-tensor and a vector, respectively, under the global SO(6), with the latter corresponding to a charge current operator. In the weak-coupling limit, their scaling dimensions Δ_i are just their engineering dimensions which can be read off the Lagrangian⁵².

Now, as pointed out by 't Hooft [341] (see also [342]), anytime you have a gauge theory of N × N matrices, there is a natural way to organize the Feynman diagrams in terms of underlying genus-g surfaces⁵³. Explicitly, since every field comes with two gauge indices, and since all indices in a gauge invariant observable must be contracted, every Feynman diagram can be written as a ribbon diagram in which each propagator is a ribbon with a gauge running along each side of the ribbon. One can then show that any given Feynman diagram, with any number of underlying loops, can be drawn without the index lines crossing only on a genus-g Riemann surface, where the specific genus g depends on precisely how the index lines are contracted to make the given Feynman diagram. Explicitly, a diagram with genus g = 0 can be drawn on a piece of paper with no lines crossing, a diagram of genus g = 1 must be drawn on a torus to have no lines cross, etc.

A remarkable result is that if we reorganize the loop expansion in g²_YM as a double expansion in N and λ = g²_YMN, then every observable can be expressed as a power series expansion in N where the powers of N for a given diagram is determined only by the genus g on which that diagram can be drawn with no crossings. For example, the free energy ${\cal F}=\ln Z$ takes the form

$$\begin{equation} {\cal F} = N^{2} f_{0}({\lambda}) + f_{1}({\lambda}) +{1\over N^{2}} f_{2}({\lambda}) + \cdots = N^{2}\sum_{g}{f_{g}({\lambda})\over N^{2g}}, \end{equation} \tag{ 97 }$$

where f_g(λ) is the sum of all diagrams arising at genus g and is a function of only λ, independent of N. A similar result is obtained for every possible observable, with the only difference being the overall factor of N,

$$\begin{equation} {\cal A} = N^{m} \left({\cal A}_{0}({\lambda}) + {1\over N^{2}}{\cal A}_{1}({\lambda}) +{1\over N^{4}} {\cal A}_{2}({\lambda}) + \cdots\right) = N^{m}\sum_{g}{{\cal A}_{g}({\lambda})\over N^{2g}}. \end{equation} \tag{ 98 }$$

This double expansion implies a remarkable simplification at large N: if N ≫ 1, the only term that matters in the genus expansion is the leading planar term; any diagram that cannot be drawn on a piece of paper without crossings, while possibly large in numerical value, is dwarfed by the much larger terms coming from the planar diagrams.

Another way of expressing this simplification at large N is on terms of large N factorization. Consider a set of single-trace operators⁵⁴ of the form ${\cal O}_i ={\mathrm {Tr }}\,(\dots )$. At large N, correlation functions of single trace operators factorize

$$\begin{equation} \langle{\cal O}_1(x_1){\cal O}_2(x_2)\rangle = \langle{\cal O}_1(x_1)\rangle \langle{\cal O}_2(x_2)\rangle + O\left({1\over N^2}\right). \end{equation} \tag{ 99 }$$

This follows because the disconnected parts of the diagrams necessarily involve more closed index loops than connected parts and thus additional powers of N. This does not imply that the large N theory is free, for while these correlators do factorize, the individual one-point functions remain non-trivial. For example, the anomalous dimensions of all operators are controlled by the 't Hooft coupling, λ.

We thus see that large N gauge theories have two natural coupling constants: $1\over N$, which controls the genus expansion and factorization; and λ, which controls the perturbative corrections to each term in the genus expansion and the anomalous dimensions of large N factorized operators. Suggestively, this is precisely the structure of closed string perturbation theory, where every amplitude is expressed as a sum of contributions from various genera, with the loop-counting parameter given by the string coupling g_s and the amplitude of a given genus determined by an auxiliary QFT whose interactions are determined by the string tension α' via the relation

$$\begin{equation} {\cal A} = g_{\mathrm{s}}^{n} \left({\cal A}_{0}(\alpha') + {g_{\mathrm{s}}^{2}}{\cal A}_{1}(\alpha') +{g_{\mathrm{s}}^{4}} {\cal A}_{2}(\alpha') + \cdots\right) = g_{\mathrm{s}}^{n}\sum_{g} g_{\mathrm{s}}^{2g}{\cal A}_{g}(\alpha'). \end{equation}\noindent \tag{ 100 }$$

This analogy led various people to speculate that large N gauge theories should be captured by some theory of closed strings. This turned out to be wrong, but only just barely: at large N, these theories are dual to closed string theories, but the closed strings live in one higher dimension!

As an aside, while not every theory is a gauge theory of N × N matrices, it is useful to keep in mind this example when thinking about more general examples. What N² is really measuring is the number of degrees of freedom per unit volume, or per lattice site if we put the theory on a lattice. Similarly, λ is measuring the strength of the dominant interactions when the number of degrees of freedom grows large. Many theories that are not simple gauge theories, nonetheless, have a useful notion of N and λ. Exactly when one obtains such a controlled double expansion remains, in general, a key open question in holography.

4.2.3. The holographic dictionary

The basic claim of holography is that every d-dimensional QFT defined as above can be exactly reorganized into a (d + 1)-dimensional quantum theory of gravity and matter propagating in AdS_d+1. The precise relationship between these two theories is known as the holographic dictionary, a sketch of which is outlined in table 1. The rest of this subsection will develop various entries in the dictionary, including a few canonical examples of computations performed via the holographic dictionary.

Operators and fields.

The first entry in the holographic dictionary relates the operators of the QFT to the matter fields in the bulk gravity: with every local operator ${\cal O}_{i}(x)$ with quantum numbers (or charges) q_i and scaling dimension Δ_i in our QFT is associated a field Φ_i(x,r) in AdS which carries the Lorentz structure and quantum numbers required to act as a coupling for ${\cal O}_{i}(x)$ in the Hamiltonian. Note that the dimension of the operator, Δ_i, is determined by the mass of the bulk field, as we saw above in (81).⁵⁵ This means that the mass and other couplings of the bulk gravity theory generally do not correspond to interactions of the boundary theory in any simple way, but rather determine which boundary QFT we are studying.

An equivalence of partition functions.

Given this map, the central claim of holographic duality [302–304] can be succinctly expressed as

$$\begin{equation} Z_{\mathrm{QFT}}[J_{i}]=Z_{\mathrm{QG}}[\Phi[J_{i}]], \end{equation} \tag{ 101 }$$

where Z_QFT[J_i] is the partition function of the QFT as a function of the sources J_i for each operator ${\cal O}_{i}$, while Z_QG[Φ[J_i]] is the quantum partition function of the gravitational theory described in section 4.2.1 computed in an AdS space–time background⁵⁶. As discussed, the quantum gravity partition function must be evaluated on field configurations Φ_i which asymptote at the boundary to the sources J_i of the QFT, hence the notation Φ[J]. As in the heuristic derivation above, the bulk fields are precisely the coupling constants of the QFT promoted to dynamical fields on the full RG-extended space–time in which the RG scale becomes a physical coordinate.

An important ingredient in this recipe is a proper definition of the boundary value of the bulk field. As we saw above for a bulk scalar Φ of mass m, dual to a scalar operator ${\cal O}$ of dimension Δ given by Δ(d − Δ) = m²L², the boundary value of the bulk field is in general divergent. To specify the boundary condition, we must rescale the bulk field by an appropriate wavefunction renormalization which picks off the leading divergence of the bulk field near the boundary

$$\begin{equation} J(x)\equiv \phi_{d-\Delta}(x) =\lim_{r\to0}r^{\Delta-d}\Phi(x,r) . \end{equation} \tag{ 102 }$$

We will return to this renormalization when we discuss the computation of one- and two-point functions.

Strong coupling and perturbative gravity.

In general, both sides of equation (101) are complicated and computationally intractable objects. However, when one side or the other can be evaluated semi-classically, we can use equation (101) to generate a controlled strong-coupling expansion via the dual description. Precisely whether, and when, such a limit is obtained is a delicate question that must be studied in detail on a case-by-case basis. To get a sense of the general structure, let us consider the canonical example, the large N limit of the 4D ${\cal N}=4$ gauge theory with gauge group G = SU(N) and 't Hooft coupling λ. The dual theory turns out to be a closed string theory on AdS₅ with AdS-radius L. The holographic dictionary then tells us that

$$\begin{equation} N^{2} = \left({L\over \ell_{\mathrm{p}}}\right)^{d-1},\quad {\lambda} = \left({L\over\ell_{\mathrm{s}}}\right)^{d}, \end{equation} \tag{ 103 }$$

where ℓ_p is the Planck length determining the scale at which quantum effects occur in the bulk, and ℓ_s is the string length controlling the scale of stringy higher-curvature corrections to the bulk gravitational action, as discussed in section 4.2.1. The first relation in equation (103) thus tells us that the bulk description becomes classical (L ≫ ℓ_p) only in the large-N limit⁵⁷ and that the gravitational description is deeply quantum mechanical when N is small. In other words, the loop counting parameter in the bulk is $1\over N$.

With an eye toward the general class of holographic theories, it is worthwhile unpacking equation (103) in more detail. In the case at hand, i.e. SU(N) SYM, N² measures the number of degrees of freedom per point in the QFT. In particular, all extensive thermodynamic quantities (entropy, energy, etc) must scale as N². In this light, what equation (103) shows is that the quantum corrections in the bulk are negligible when the number of degrees of freedom per point in the dual QFT is large, and vice versa. Indeed, in a host of controlled models in string theory, the appropriate version of the first equation in equation (103) takes the form, $N^{b}= \left ({L/ \ell _{\mathrm {p}}}\right )^{d-1}$, where N is a conserved charge and b is some real parameter [343]. Similarly, it is not always possible to interpret λ as a 't Hooft parameter in the dual QFT. More generally, the role of λ is played by the typical scale of anomalous dimensions in the QFT [333]. We can understand this as follows: if the QFT is weakly interacting, the effects of renormalization will generally be weak and most observables will be well approximated by their classical cousins, i.e. quantum anomalous dimensions will be small; if interactions are strong, quantum effects will generally drive anomalous dimensions to be large.

While N ≫ 1 ensures that the Planck scale is negligible, so that the gravitational interactions can be treated with classical effective field theory, it does not tell us that the system is well approximated by pure GR. Indeed, in principle, there will be a host of higher-curvature corrections to the classical GR action suppressed by powers of $\ell _{\mathrm {s}}\over L$. To ensure that these corrections are in fact negligible, i.e. that L ≫ ℓ_s, we must furthermore take the 't Hooft coupling to be large, λ ≫ 1. Conversely, when the QFT is weakly coupled, λ ≪ 1, the bulk geometry is sufficiently strongly curved that GR is swamped by higher curvature corrections to the stringy effective action.

The main lesson here is that when the QFT is perturbative, the bulk gravity is out of control, and that when the QFT is strongly interacting and has a large entropy density, the dual gravitational description is simple semi-classical gravity. To see why such a weakly curved limit is interesting, let us consider our canonical example at large N, N ≫ 1, and with strong 't Hooft coupling, λ ≫ 1. Let us also analytically continue to imaginary time, ω = iω_E with ω_E > 0. In this limit, the gravity sector reduces to classical Euclidean GR. In terms of our fundamental relation, the rhs can thus be expressed as a sum over saddles. Focusing on the dominant saddle, we obtain

$$\begin{equation} Z_{\mathrm{QFT}}[J] ~\simeq~ \mathrm{e}^{-I_{\mathrm{GR}}[\Phi[J]]}, \end{equation} \tag{ 104 }$$

where

$$\begin{equation} I_{\mathrm{GR}}[\Phi[J]] = {1\over 16\pi G_{\mathrm{N}}} \int \!\mathrm{d}^{d+1}\!x\, \sqrt{-g}\left(R-2\Lambda +{\cal L}_{m}\left(\Phi\right)\right) \end{equation} \tag{ 105 }$$

is the classical gravitational action expanded about the dominant classical saddle, subject again to the condition that the normalizable modes of all matter fields are determined by the sources of the dual QFT. Equation (105) thus defines a simple classical functional of the sources from which we can compute quantum correlation functions of the dual QFT via the above expressions as

$$\begin{equation} \langle{\cal O}_{1}(x_{1}) \dots {\cal O}_{n}(x_{n})\rangle={\delta^{n} I_{\mathrm{GR}}[\Phi[J_{i}]] \over \delta J_{1}(x_{1})\dots \delta J_{n}(x_{n})}{\Big |}_{J_{i}=0}. \end{equation} \tag{ 106 }$$

The upshot is that, when such a dual semi-classical limit exists and is reliable, we can compute the quantum correlation functions of our QFT by finding solutions to the classical field equations of the dual gravitational system, evaluating the action on-shell as a functional of the boundary values of the fields, and taking appropriate derivatives.

Example: one- and two-point functions for a scalar operator.

As an example of this machinery at work, let us compute the one- and two-point functions for a boundary operator ${\cal O}$ dual to a free bulk scalar field ϕ. For simplicity, we will work in Euclidean momentum space throughout.

For the one-point function, the basic form of the computation is straightforward. Intuitively, our prescription (106) tells us that the one-point function corresponds to a variation of the classical gravitational action with respect to the boundary value of the bulk field. But a variation of the action with respect to the boundary value of the field is precisely the conjugate momentum, Π, of the field Φ in the direction normal to the boundary,

$$\begin{equation} \Pi = -\sqrt{-g}g^{rr}\partial_{r}\Phi. \end{equation} \tag{ 107 }$$

Thus we should expect to find, with $k\equiv \left (\omega _{\mathrm {E}},\vec {k}\right )$,

$$\begin{equation} \langle{\cal O}(k)\rangle={\delta I_{\mathrm{GR}}[\Phi[J]] \over \delta J(k)}\, \sim \,\lim_{r\to0}\Pi(k,r). \end{equation} \tag{ 108 }$$

This is almost correct. The trouble is that the on-shell classical action, I_GR[ϕ], and the bulk field, Φ, both generically diverge as we approach the boundary, cf equation (81). To absorb these divergences we must (a) regulate all quantities by evaluating them not at the boundary at r = 0 but at r =  ≪ 1, (b) rescale the bulk field by an overall wavefunction normalization as in (88) and (c) add boundary counterterms to the action evaluated at r =  so that the renormalized on-shell action remains finite as we remove the cutoff. In retrospect, this should not be surprising: the radial coordinate r is playing the role of a lattice cutoff, so we must work with properly renormalized quantities to avoid confusion. This holographic renormalization [335, 344–348] is simply the bulk realization of the UV renormalization of the boundary QFT.

In fact, we have already seen this effect above. Recall that the relationship between the amplitude of the non-normalizable mode ϕ_d−Δ and the boundary value of the bulk field Φ also required a subtle wavefunction renormalization (88), so that the source is given by

$$\begin{equation} J(k)=\phi_{d-\Delta}(k) =\lim_{r\to0}r^{\Delta-d}\Phi(k,r) . \end{equation} \tag{ 109 }$$

Upon performing a similar renormalization of the bulk Euclidean action, one finds that the correct expression for the one-point function in terms of the renormalized radial momentum is⁵⁸

$$\begin{equation} \langle{\cal O}(k)\rangle = \lim_{r\to0}r^{d-\Delta}\Pi(k,r). \end{equation} \tag{ 110 }$$

Together with (107) and (81), this gives

$$\begin{equation} \langle{\cal O}(k)\rangle ={2\Delta-d\over L}\phi_{\Delta}(k), \end{equation} \tag{ 111 }$$

where ϕ_Δ is the coefficient of the sub-leading term in (81). Thus, just as the non-normalizable mode of the bulk field ϕ_d−Δ determines the source, J, for the boundary operator ${\cal O}_{\Delta }$, the subleading normalizable term ϕ_Δ determines the response, $\langle {\cal O}\rangle$.

This makes computing the linear-response Green functions easier. In linear response theory, an infinitesimal source J(x) generates a response which is linearly proportional to the source, with the ratio defining the linear-response Green function

$$\begin{equation} G_{\mathrm{E}}(k)\equiv{\langle{\cal O}(k)\rangle \over J(k)}= \lim_{r\to0}r^{2(\Delta-d)}{\Pi(k,r)\over\Phi(k,r)}. \end{equation}\noindent \tag{ 112 }$$

Using our above expressions for the source, J, and response, $\langle {\cal O}\rangle$, we thus find that

$$\begin{equation} G_{\mathrm{E}}(k)={2\Delta-d\over L}{\phi_{\Delta}(k)\over\phi_{d-\Delta}(k)}. \end{equation}\noindent \tag{ 113 }$$

Note that we could also have derived this result by taking two derivatives of the partition function.

Computing the one- and two-point functions thus follows from determining ϕ_Δ in terms of ϕ_d−Δ. As we have seen, in AdS these two modes are not independent: requiring the bulk solution to be regular at the Euclidean horizon imposes a relation which we can use to determine the response ϕ_Δ in terms of the source ϕ_d−Δ. In practice, finding the precise relation involves solving the bulk equation of motion subject to the boundary conditions, i.e. to solving a set of second-order elliptic partial differential equations. For simple examples this can be done analytically; more generally, one is forced to use some form of matched asymptotic expansion or numerical integration to determine ϕ_Δ in terms of ϕ_d−Δ.

The key point is that the computation of the quantum correlation function of the boundary QFT has been reduced to solving a set of classical partial differential equations with fixed boundary conditions.

An example where this can be done analytically is our example of a massive scalar in pure AdS, for which we have in fact already determined this relation in momentum space, equation (87). Plugging this in and again working in momentum space gives, for the Euclidean Green function,

$$\begin{equation} G_{\mathrm{E}}(k) = {2\Delta-d\over L}{\Gamma\left({d\over2}-\Delta\right)\over2^{\Delta-{d\over2}}\Gamma\left(\Delta-{d\over2}\right)} \left(k^{2}\right)^{\Delta-{d\over2}}. \end{equation}\noindent \tag{ 114 }$$

This is exactly the form for the two-point function of a scalar operator with scaling dimension Δ in a CFT.

Real-time response and retarded Green functions.

So far we have focused on Euclidean correlation functions. In fact, these holographic techniques can be readily extended to intrinsically Lorentzian computations needed for real-time response.

The simplest thing to do would be to simply compute the Euclidean Green function and analytically continue. In practice, this is very often not tractable. For example, we will sometimes be interested in particles moving on light-like trajectories which are not easily described in Euclidean continuation. Instead, one can construct an intrinsically real-time holographic prescription, as was first proposed by Son and Starinets [352] by essentially analytically continuing the Euclidean prescription. Their results have since been justified by developing a full holographic Schwinger–Keldysh formalism [353–355]. Here we will forego the formalities. We assume that such a justification can be made and proceed to the prescription.

The prescription requires repeating our Euclidean prescription step by step in the real-time Lorentzian geometry. The key difference is that we must choose an appropriate boundary condition at the horizon. The appropriate choice depends on which Green function we wish to compute⁵⁹. Intuitively, one might expect that the retarded Green function, G_R, should correspond to imposing causal in-falling boundary conditions at the horizon, while the advanced Green function, G_A, should involve acausal out-going boundary conditions. This turns out to be precisely correct. Once we have a solution satisfying the appropriate boundary conditions, we again compute the properly renormalized on-shell Lorentzian action, identify source and response from the asymptotic behavior of the solution near the boundary and compute the Green function via a linear response. The final result takes an analogous form to equation (113),

$$\begin{equation} G_{\mathrm{R}}(\omega,\vec{k})=\lim_{r\to0}r^{2(\Delta-d)}{\Pi_{r,\mathrm{in}}(\omega,\vec{k},r)\over\Phi_{\mathrm{in}}(\omega,\vec{k},r)}={2\Delta-d\over L}{\phi_{\Delta,\mathrm{in}}(\omega,\vec{k})\over\phi_{d-\Delta,\mathrm{in}}(\omega,\vec{k})}, \end{equation}\noindent \tag{ 115 }$$

where ω is the Lorentzian frequency and ϕ_Δ, ϕ_d−Δ are found by solving the linearized Lorentzian equations in the bulk with in-falling boundary conditions at the horizon. The advanced Green function is computed analogously to out-going boundary conditions enforced at the horizon.

Cautionary notes on the semi–classical limit.

While it is true that we have simplified our job by focusing on limits where the gravitational partition function Z_GR can be computed by saddle point approximation, this is not the same as expanding Z_QFT semi-classically; in the present saddle, the classical description is of classical gravity in one higher dimension. Thus, rather than studying a semi-classical limit of the QFT governed by quasiparticle perturbation theory, we are focusing on a very different emergent limit in which quasiparticles are replaced by geometry.

Note, too, a practical consistency condition: it cannot be the case that both Z_QFT and Z_QG are simultaneously perturbative, for if that were the case we would learn that every classical field theory can be reorganized into classical gravity—but this is explicitly forbidden by the Weinberg–Witten theorem. It is thus clear that this holographic duality had to be a strong–weak duality, with one side becoming strongly quantum mechanical whenever the dual becomes semi-classical.

Finally, we could just as easily have inverted our logic and studied a limit in which the QFT partition function became semi-classical. While this is unlikely to teach us much about the QFT, since perturbation theory is already well understood, it may teach us a great deal about the strongly quantum gravitational dual. In particular, this approach provides the only non-perturbative definition of 4D quantum gravity currently known, and can be used to quickly and convincingly argue for the preservation of unitarity by the formation and evaporation of black holes.

Thermodynamics encoded by black holes.

The final entry in the holographic dictionary sketched in table 1 relates the thermodynamic state and ensemble in which we place our QFT to the precise geometry in which we study the bulk gravity [356]. When the QFT is exactly conformal, the bulk geometry is pure AdS. Recalling that the UV physics of the boundary is associated with the asymptotic near-boundary region of AdS. Deforming the theory away from this conformal UV fixed point to generate a QFT with non-trivial RG flow corresponds to studying a geometry which is asymptotically AdS near the boundary but flows away from pure AdS as we run into the bulk. For example, turning on a finite temperature, T, and chemical potentials, μ_i, in this CFT corresponds to studying asymptotically AdS black hole space–times with Hawking temperature T_H = T and charges Q_i = μ_i. More generally, the thermodynamic data of the QFT are entirely encoded in the thermodynamics of the black hole in the dual geometry (see table 1 for the mapping between the thermodynamic parameters of the QFT and the quantum numbers of the dual black hole).

The classic example of the power of these thermodynamic relations is the computation of the thermodynamic properties of a large-N 4D Yang–Mills gauge theory at the strong 't Hooft coupling. The holographic dual of this system is a planar AdS black brane, as discussed in section 4.2.1. In the large-N, large-λ limit, the gravitational description reduces to GR and the black brane entropy becomes

$$\begin{equation} S_{\mathrm{BH}} = {A_{\mathrm{H}}\over 4G_{\mathrm{N}}}, \end{equation} \tag{ 116 }$$

where A_H is the area of the horizon. Since the system is translationally invariant, this diverges, so it is convenient to consider instead the entropy per unit area in the field theory

$$\begin{equation} s = {S_{\mathrm{BH}}\over A_{\partial}} = {A_{\mathrm{H}}\over 4 G_{\mathrm{N}}A_{\partial}}, \end{equation} \tag{ 117 }$$

where A_∂ is the area of the AdS boundary. Plugging in factors, this gives a strong-coupling computation of the entropy density and energy density as

$$\begin{equation} s_{\mathrm{strong}} = {\pi^{2}\over2}N^{2}T^{3}, \quad \epsilon_{\mathrm{strong}} = {3\pi^{2}\over8}N^{2}T^{4}. \end{equation} \tag{ 118 }$$

These strong coupling results are extremely close to the results in the free theory as computed in perturbation theory:

$$\begin{equation} s_{\mathrm{free}} = {2\pi^{2}\over3}N^{2}T^{3}, \quad \epsilon_{\mathrm{free}} = {\pi^{2}\over2}N^{2}T^{4}. \end{equation} \tag{ 119 }$$

Remarkably, despite running from weak to strong coupling, λ = 0 to λ → ∞, all that happened to the thermodynamics is a mild renormalization by a factor of $3\over 4$:

$$\begin{equation} {s_{\mathrm{strong}}\over s_{\mathrm{free}}} = {\epsilon_{\mathrm{strong}}\over \epsilon_{\mathrm{free}}} = {3\over4}. \end{equation} \tag{ 120 }$$

Studying the full λ-dependence of this ratio reveals that the energy and entropy densities flow smoothly and monotonically as we run from weak to strong coupling. Analogous results are obtained for a wide variety of gauge groups, matter contents and even dimensions, with the ratio ${s_{\mathrm {strong}}\over s_{\mathrm {free}}}$ typically within 10% or so of the ${{\cal N}}=4$ value, $3\over 4$.

This holographic result already tells us an important fact: thermodynamic quantities such as the free energy and the entropy density are not good probes of the strength of interaction of a quantum liquid. Similar results apply to other thermodynamic quantities, for which the strong/weak ratio is typically ${\cal O}(1)$ as well⁶⁰.

4.2.4. A bit of history

Applied holography is the application of holographic dualities to construct computationally tractable toy models of behavior which has proven difficult to model with traditional tools. Key to this endeavor is the fact that various features of the physics which appear universal, or even trivial, on one side of the duality may be significant, or simply obscure, in the dual. Most current work in applied holography has used universal features of gravitational models, and in particular, the universality of black hole horizon physics, to make new discoveries about QFTs, although using universal features of weakly coupled field theories to learn about extreme phases of gravity is of interest as well. Given the aim of this focus issue, we will emphasize the first direction in what follows.

The simplest application of holography involves studying QFTs for which the precise gravitational dual is explicitly known. In such cases we can use the perturbative QFT to compute any quantities of interest at weak coupling, λ → 0, and then use the weakly coupled gravitational dual to compute the corresponding quantities at strong coupling, λ → ∞. This strategy has very limited applications because the precise duals of most QFTs are not yet known. At the moment we only know the precise duality for a special set of non-generic quantum field theories, although they are fairly close cousins of the theories studied in particle physics. For example, we know the precise gravitational dual of the ${\cal N}=4$ theory discussed in detail above and have used it to learn a great deal about the structure and strong-coupling physics of this intricate and exotic theory (which we emphasize does not appear to be realized anywhere in nature). By contrast, we do not know the dual of the Hubbard model.

This is not to say progress has not been made, but just that it is difficult. An example of this point is the recent conjecture of a duality between the 1D Ising model and pure Einstein gravity in AdS₃. For this relation to hold the gravitational side must be treated not only quantum mechanically, but fully non-perturbatively, summing over an infinite number of topologically inequivalent locally AdS₃ space–times, including a series of quantum corrections around each classical saddle, something that can only be done in closed form in the absolutely simplest possible gravitational systems, namely pure Einstein gravity in 2 + 1 dimensions. This conjecture underscores the difficulty of constructing the precise holographic dual of a garden-variety QFT.

If the precise dual is not known, a second approach to applied holography is to make use of a known exact duality with a class of QFTs that are structurally similar to the QFT of interest. Universal properties of these sibling QFTs can then be inferred to apply to the original QFT, too. An example of this strategy is the construction by Kachru et al of holographic dimer models [357, 358]. In this construction, the degrees of freedom of interest to the dimer problem form a small subset of the full set of holographic degrees of freedom. At low energy, this subset forms a relatively generic dimer model. By exploiting the holographic description, much can be derived about the low-energy physics, including a novel phase transition between Fermi-liquid and non-Fermi liquid behaviors. The most powerful example of this approach so far has been the study of strongly coupled quark–gluon liquids in large N toy models of QCD, for which a menagerie of models have been considered, each emphasizing and reproducing various features of the observed phenomena. As we shall see in detail in sections 4.3.1 and 4.3.2, this approach has been used to motivate the anomalously small viscosity of the RHIC fireball, to explain why the rapid thermalization of the RHIC droplet is not so surprising, and to estimate the jet quenching parameter.

However, perhaps the most interesting use of holography is to provide an entirely new way of defining consistent QFTs. Traditionally, QFTs in dimensions greater than 2 are defined by specifying some Lagrangian or Hamiltonian governing the interactions of a set of well-defined quasiparticles. However, many systems, including some of the most theoretically and experimentally interesting systems, manifestly do not have any well-defined quasiparticles upon which to base a standard QFT. From this point of view, holography looks interesting because it provides a recipe for computing quantum amplitudes that manifestly satisfy the conditions of a good QFT (local, causal, etc) but which is defined without any appeal to a quasiparticle picture or any sort of conventional perturbation theory. Holography thus provides an entirely new way of constructing consistent models of many-body physics, replacing quasiparticles with geometry as the central organizing principle.

4.3.1. Viscous hydrodynamics from gravity

Suppose we are given a QFT at finite temperature and density and want to know whether it is weakly or strongly interacting. A natural observable to query is the shear viscosity, η, which measures the efficiency of momentum transport across a velocity gradient. Explicitly, the shear viscosity η of a fluid determines the frictional force F induced on a plate of area A moving at a velocity v a distance L over a fixed surface,

$$\begin{equation} F = \eta {A v \over L}. \end{equation}\noindent \tag{ 121 }$$

This relation follows from the hydrodynamic stress tensor given in equation (44). As discussed in section 3.2 shear viscosity has dimensions ℏn, where n is a density. The natural thermodynamic quantity to which one can compare the viscosity is thus the entropy density, s, with dimensions of k_Bn. η/s thus gives a simple dimensionless measure of the efficiency of transverse momentum transport.

From kinetic theory, the transport of momentum proceeds by scattering quasiparticles between layers of the fluid, and is thus proportional to the mean free path between scattering events and fluid quasiparticles. Since the mean free path decreases as the scattering cross section increases, the shear viscosity should be inversely related and thus strongly sensitive to the strength of interactions. However, if we drive up the strength of interaction until the mean free path becomes of the order of the Compton wavelength of the would-be quasiparticle, the kinetic-theory relation between interaction and viscosity should no longer apply. Meanwhile, in this post-quasiparticle setting, the mean free path is much smaller than any typical velocity gradient, so a hydrodynamic description should set in. Thus, as the interactions become strong, we expect the shear viscosity to become small, and furthermore expect coupling dependence of the viscosity to qualitatively change, since the kinetic behavior must eventually cease. But what values should η/s take and how will it vary as the coupling grows large? A priori, the answers to these questions would appear to be strongly system dependent.

To address this question, let us compute η/s holographically⁶¹. This will give us a controlled strong-coupling estimate of η/s. However, before proceeding, let us pause to consider, from the holographic perspective, why there must be any viscosity in the first place. Recall that the holographic dual of a strongly coupled relativistic large N gauge theory at finite temperature is a black hole with non-zero mass. But anytime we have a black hole, we will have dissipation: if we scatter a wave off the black hole, some fraction of its amplitude, and thus of the momentum it carried, will be absorbed into the black hole. So the fact that there is always a generic source of dissipation is very much built into holographic duality.

A useful way to compute η/s begins with the Kubo formula for η,

$$\begin{equation} \eta = -\lim_{\omega,k\to0} {{\mathrm{Im}} G_{\mathrm{R}}(\omega,k)\over \omega}, \end{equation} \tag{ 122 }$$

which relates η to the retarded Green function G_R for shear modes of the stress tensor,

$$\begin{equation} G_{\mathrm{R}}(t,x) = \langle T_{xy}(t,x)T_{xy}(0,0)\rangle\,\Theta(t). \end{equation} \tag{ 123 }$$

Note that we take ω → 0 only after sending k → 0. The Kubo formula follows from matching linear response theory to the expected low-frequency, low-momentum limit of the correlation function in hydrodynamics; see, for example, [317]. To obtain η we need to compute G_R(t,x). In general, this is difficult in a strongly coupled QFT. However, holographically this calculation turns out to be relatively straightforward. According to the holographic dictionary, the operator T_xy is dual to the corresponding shear mode of the bulk metric, g_xy. We thus need the response of the classical gravitational action, I_AdS[g], to variations of this mode, δg_xy = h_xy. So long as we are expanding around pure AdS, h_xy is governed by a simple massless scalar action of the form in equation (79), but with the normalization of the kinetic term determined by the normalization of the gravitational kinetic term, ${1\over 16\pi G_{\mathrm {N}}}$,

$$\begin{equation} I_{\phi} = - {1\over 16\pi G_{\mathrm{N}}}\int \!\mathrm{d}^{d+1}\!x\, \sqrt{g}\, \,\frac{1}{2}\left(\partial\phi\right)^{2}. \end{equation} \tag{ 124 }$$

Since m = 0, we have Δ(d − Δ) = 0, so the dimension of our dual operator is Δ = d. Let us use this action and the formalism developed above to compute G_R. With Δ = d, the retarded Green function takes the form

$$\begin{equation} G_{\mathrm{R}}(k) = \lim_{r\to0}{\Pi_{\mathrm{in}}(\omega,\vec{k},r)\over\Phi_{\mathrm{in}}(\omega,\vec{k},r)}. \end{equation} \tag{ 125 }$$

It is readily checked (cf (114)) that the real part of the ratio above scales, after sending $\vec {k}\to 0$, as ω³, and thus vanishes in our limit. We can thus replace ImG_R(ω,k) with ${1\over {\rm i}}$ G_R(ω,k) to obtain

$$\begin{equation} \eta = -\lim_{\omega,k\to0} \lim_{r\to0}{\Pi_{\mathrm{in}}(\omega,\vec{k},r)\over \mathrm{i}\omega\Phi_{\mathrm{in}}(\omega,\vec{k},r)}. \end{equation} \noindent \tag{ 126 }$$

At this point, something remarkable happens. In the limit that $\omega ,\vec {k}\to 0$, the radial evolution of Π and ωΦ trivialize. This can be seen by inspection of the definition of Π and the Hamiltonian equation of motion, $\partial _{r}\left (\omega \Phi \right )\sim \omega \Pi$ and ∂_rΠ ∼ k^μk^ν∂_μ∂_nΦ. Thus, in the limit $\omega ,\vec {k}\to 0$, both Π and ωΦ become independent of r.

We can thus evaluate the ratio appearing in our Kubo formula at any radial coordinate we like. A convenient choice is the horizon, where the solution must satisfy in-falling boundary conditions. From our earlier discussion of the in-falling solutions (85), which implied that f∂_rΦ_in = iωΦ_in at the horizon, and using the definition $\Pi _{\mathrm {in}} = -{1\over 16\pi G_{\mathrm {N}}}\sqrt {-g}g^{rr}\partial _{r}\Phi _{\mathrm {in}}$, we have that

$$\begin{equation} \lim_{r\to r_{\mathrm{H}}} \Pi_{\mathrm{in}} = -{1\over 16\pi G_{\mathrm{N}}r_{\mathrm{H}}^{3}} {\mathrm{i}}\omega\Phi_{\mathrm{in}}(r_{\mathrm{H}}). \end{equation} \tag{ 127 }$$

The factor of r⁻³_H can be nicely interpreted as the area of the horizon in boundary units, ${A_{\mathrm {H}}\over A_{\partial }}$. Plugging all of this into our Kubo formula, equation (126) then becomes

$$\begin{equation} \eta = {A_{\mathrm{H}}\over 16\pi G_{\mathrm{N}} A_{\partial}}, \end{equation} \tag{ 128 }$$

where A_H is the area of the horizon and A_∂ is the area of the boundary. From equation (117), ${A_{\mathrm {H}}\over A_{\partial }}=4 G_{\mathrm {N}} s$. Putting these together, we see that the viscosity-to-entropy-density ratio takes the simple form now known as the KSS form after the authors of [11] (however see also, e.g., [235, 236, 301]),

$$\begin{equation} {\eta\over s} = {1\over 4\pi}, \end{equation}\noindent \tag{ 129 }$$

which is orders of magnitude smaller than that of typical liquids. In fact, the liquids that are believed to come close to this level of perfection are the QGP at RHIC and fermions at unitarity, as shown in figure 2.

Many features of this result are remarkable. Firstly, as motivated heuristically above, η/s is indeed a good diagnostic of strong coupling in any many-body system. Near weak coupling, QFT perturbation theory gives

$$\begin{equation} {\eta\over s}\sim {A\over\lambda ^2\ln(\sqrt{{\lambda}})}, \end{equation} \noindent \tag{ 130 }$$

where the constant of proportionality is system dependent. In particular, weakly coupled QFTs generally have ${\eta \over s}\gg 1$. Near strong coupling, by contrast, a more careful gravitational analysis gives

$$\begin{equation} {\eta\over s}={1\over4\pi} +O\left({1\over\lambda^{3/2}}\right), \end{equation} \noindent \tag{ 131 }$$

almost independently of the details of the system. Thus, in any such system, the behavior of $\eta \over s$ changes dramatically as we run from weak to strong coupling.

Secondly, equation (129) is remarkably generic in holographic systems. Indeed, in any QFT which is holographically dual to GR, whether the gauge group is SO(N) or SP(2N) or something more complicated, whether enjoying maximal supersymmetry or no supersymmetry, we obtain precisely ${\eta \over s} = {1\over 4\pi }$. To see why this result is so general, note that the radial evolution dropped out of our computation of η, so that all that we needed was the local physics near the horizon. The rest of the holographic description (including the UV physics near the boundary) was moot. Meanwhile, s is explicitly proportional to the horizon area, so again it is completely determined by the physics of the horizon. The generality of this holographic result then follows from the exceptional universality of the physics of horizons in GR.

Observations such as these led KSS to make the conjecture that $\eta \geqslant {1\over 4\pi }$ is a universal lower bound on the shear viscosity of any quantum many-body system, with saturation occurring only for those strongly coupled theories which are holographically dual to pure Einstein gravity. We now understand that this result is not universal. Explicitly, equation (129) holds when the space–time geometry is AdS, the gravitational action is Einstein–Hilbert and the fluid is time independent, homogeneous and isotropic. However, when higher-derivative terms are present in the gravitational action (corresponding to finite-N and finite-λ effects in the dual QFT) or when the geometry is deformed away from AdS such that the radial evolution does not factor out (as occurs, for example, when the geometry is time dependent, spatially disordered or sufficiently anisotropic), $\eta \over s$ can depart from the KSS value, and indeed can be lower than $1\over 4\pi$.

The cleanest example of such an effect involves [10, 360, 361] adding the leading irrelevant operator to the bulk gravitational action, the 5D Gauss–Bonnet term, a quadratic scalar built out of the curvature tensors,

$$\begin{equation} I_{\mathrm{GB}} = {1\over 16\pi G_{\mathrm{N}}} \int \mathrm{d}^{d+1}\!x\,\sqrt{-g}\left( -2\Lambda + R + \lambda_{\mathrm{GB}}R^{2}_{\mathrm{GB}} \right). \end{equation} \noindent \tag{ 132 }$$

Such terms arise in known string theories where we know and have control over both sides of the duality⁶². As it turns out, the holographic analysis including λ_GB ≠ 0 is straightforward. All that materially changes is the normalization of the action for the scalar mode h_xy, which picks an extra factor of (1–4λ_GB), yielding

$$\begin{equation} {\eta\over s} = {1-4\lambda_{\mathrm{GB}}\over 4\pi}. \end{equation} \tag{ 133 }$$

When λ_GB > 0, the KSS conjecture is violated⁶³.

It is worth understanding why this happens. The value of η/s is controlled by the normalization of the kinetic term in the scalar action for our tensor mode h_xy. Typically, such coefficients are best understood as coupling constants, since scaling them away to make the kinetic term canonical shifts the coefficient into the interactions⁶⁴. When we make the coupling stronger, as in the case of Gauss–Bonnet with λ_GB > 0, the viscosity goes down. This is reminiscent of the dependence of viscosity on coupling in kinetic theory: increasing the strength of quasiparticle scattering inhibits transport and decreases the shear viscosity. At strong coupling, there are no longer any well-defined quasiparticles to which to apply kinetic theory; instead, the effective degrees of freedom mediating low-energy momentum transport are weakly coupled gravitons in the holographic space–time.

Interestingly, while this evidence rules out the KSS conjecture, there exist compelling arguments for a weakened version of the KSS bound. Physically, the arguments for such a refined bound involve demonstrating that violations of these bounds would lead, in a wide class of theories, either to violations of causality in the field theory [10, 362] or to violations of the positivity of energy [363]; in fact, these two pathologies are intimately related. In the case of Gauss–Bonnet gravity which is holographically dual to 4D QFTs with ${\cal N}=1$ supersymmetry, both arguments lead to the same constraint, ${-7\over 36}\leqslant \lambda _{\mathrm {GB}}\leqslant {9\over 100}$, leading to the refined lower bound

$$\begin{equation} {\eta\over s}\geqslant {1\over4\pi}{16\over25}. \end{equation} \noindent \tag{ 134 }$$

In more general models, similar bounds obtain, but the precise values differ. The meaning and reliability of such refined bounds remain a matter of active debate. For example, it has been shown [364, 365] that sufficiently contrived examples may be concocted which violate any such bound and in fact push ${\eta \over s}$ arbitrarily close to zero, and even negative! However, it is not at all clear that these models are themselves well defined; indeed, one can immediately show [366] that all such models with ${\eta \over s}$ negative or vanishing are pathological, with causality and self-consistency enforcing a minimal value for ${\eta \over s}$. Meanwhile, whether any such bound should be expected when we give up time independence, homogeneity or isotropy is a topic of much current research. To summarize, there are well-defined lower bounds on $\eta \over s$ in various classes of theories, but each such bound has been violated by considering a more general class of models. It remains possible that there is a simple, completely universal lower bound on $\eta \over s$, but the evidence for such a bound is increasingly tenuous.

Worrying about universality of a particular bound, however, misses the point. The lasting import of equation (129) is that it establishes a link between ultra-low shear viscosity and strongly interacting many-body systems. Said differently, if one finds a fluid with an extremely small shear viscosity, one has very strong evidence that it enjoys strong quantum correlations. Conversely, extreme quantum fluids should be expected to have ultra-low viscosity of the order of $1\over 4\pi$. Whatever the final story about a hard lower limit, this connection is an important lesson from holography.

4.3.2. Diffusion, jet quenching and dynamics in the holographic QGP

A related transport coefficient which can be fruitfully studied holographically is the diffusion constant for a heavy quark in a strongly coupled holographic QGP. It is an interesting example for us for several reasons. Firstly, the structure of the holographic computation is very different from the computations described above, and in fact owes a great deal to the string-theoretic origin of holographic duality. Secondly, the result is very different from the simple universality of the viscosity computation. Specifically, the diffusion constant for a heavy quark scales as

$$\begin{equation} D={2\over\pi T}{1\over\sqrt{{\lambda}}} \end{equation} \tag{ 135 }$$

and is thus very strongly sensitive to the 't Hooft coupling, λ. In particular, this suggests that even very heavy quarks should rapidly equilibrate in the QGP, in sharp contrast to perturbative estimates. Note that this also differs sharply from the perturbative YM result⁶⁵

$$\begin{equation} D={6\over2\pi T} \left({1\over 2\alpha_{\mathrm{s}}}\right)^2 , \end{equation} \noindent \tag{ 136 }$$

for while equations (135) and (136) look similar, the latter is only valid for α_s ≪ 1, while the former is only valid for λ ≫ 1.

The basic structure of the holographic computation is described in [289, 291]; see [286] for a perturbative analysis. Suppose that we place a heavy quark in the QGP and treat it in a Born–Oppenheimer approximation as a slow, heavy, point-like variable which sources the gauge field. As we run the RG, this charge is conserved. Holographically, this means that we should have a point-like defect not just on the UV boundary, but also on every slice of the holographic bulk. The heavy quark on the boundary is thus the endpoint of a string in the bulk which hangs down from the boundary into the AdS horizon, hanging along a straight line according to the gravitational tug of AdS. The string can be thought of as the holographic image of the color flux of the fundamental quark in the 3 + 1 boundary. Explicitly, the dynamics of the string are determined by a simple least action principle, where the action is the area of the surface in space–time swept out by the string in AdS. For a static string, the extremal string is simply the string hanging straight down from the boundary quark.

To measure the diffusion constant, we need to give our quark a kick. The easiest way to do this is to drag the quark through the QGP with a fixed velocity and then determine the drag force. In equilibrium, we must have

$$\begin{equation} f=-\eta_{\mathrm{D}} p, \end{equation}\noindent \tag{ 137 }$$

where η_D is the drag coefficient. Holographically, the string hanging off the quark into the bulk will now dangle behind the moving quark, imparting a drag force to the quark. Using the action principle we just defined, the drag force is computed to be

$$\begin{equation} f = -{\pi\over2}\sqrt{{\lambda}}T^{2}{v\over\sqrt{1-v^{2}}}.\end{equation}\noindent \tag{ 138 }$$

For heavy quarks, for which the thermal corrections to the kinetic mass are negligible, this gives

$$\begin{equation} \eta_{\mathrm{D}} ={\pi\over2m}\sqrt{{\lambda}}T^{2} .\end{equation} \noindent \tag{ 139 }$$

Using this in the relation D = T/η_Dm derived in equation (57) yields equation (135).

It is worth commenting on how this computation is similar to the viscosity calculation and how it differs. Perhaps the most important similarity is that in both cases we computed deeply quantum-mechanical transport quantities in our strongly coupled QGP by solving simple, classical differential equations in the dual gravitational system. This is a basic hallmark of holographic duality. Secondly, the resulting computation was surprisingly independent of the details of the holographic setup, depending only on the gravitational dynamics of a string hanging into AdS from a moving point on the boundary. This suggests that a heavy fundamental in any strongly coupled QGP should have a similar qualitative behavior, namely efficient diffusion and rapid equilibration. By contrast, the final results differ in significant ways. For example, while η/s was remarkably insensitive to the coupling, D depends sensitively and is thus a sensitive probe of precisely how strongly coupled the theory is.

A similar computation leads to a holographic estimate of the jet-quenching parameter, $\hat {q}$, introduced in section 3.6. Holographically, the role of the QGP fireball is played by the dual black hole, while the hard jet can be modeled as a heavy quark at very high energy, and thus moving on an effectively light-like trajectory. Our job is thus to compute the transverse momentum gained by a light-like quark moving through the QGP. This can be reduced [367–369] to computing a light-like Wilson loop in the QGP. Holographically, the expectation value of this Wilson loop can again be computed by studying the surface swept out by the trailing string, determined as before by extremizing the string action. Since the Wilson loop is light-like, the computation must be performed in Lorentzian signature. The resulting real-time holographic analysis, while somewhat intricate, leads to a simple prediction for the jet-quenching parameter in ${\cal N}=4$ SYM [369, 370],

$$\begin{equation} \hat{q}_{{\cal N}=4} = {\pi^{3/2}\Gamma\left({3\over4}\right)\over \Gamma\left({5\over4}\right)}\sqrt{{\lambda}}\,T^{3}. \end{equation} \tag{ 140 }$$

Recalling that λ = g²N_c and g² = 4πα_s, and plugging in T ∼ 300 MeV and $\alpha _{\mathrm {s}}=\frac {1}{2}$, which are reasonable estimates for the values at RHIC, this gives $\hat {q}_{{\cal N}=4}\sim 4.5\,{\mathrm {GeV}}^{2}\,{\mathrm {fm}}^{-1}$, which is comparable with the results from RHIC; see section 3.6. Note that this holographic result leads to a semi-strongly coupled treatment of energy loss, treating transverse momentum diffusion in strong coupling via a holographic estimate of $\hat {q}$, while the actual energy loss (longitudinal drag) is done in perturbation theory, as discussed in section 3.6. For fully strongly coupled treatments, see, e.g., [371, 372].

An important lesson of this holographic computation, however, is not the specific value of $\hat {q}$ we find but rather how $\hat {q}$ depends on the physical parameters of the system. In particular, it turns out that in any holographic QGP dual to pure AdS, and thus governed by a dual CFT, one finds that [370]

$$\begin{equation} { \hat{q}_{\mathrm{CFT}} \over \hat{q}_{{\cal N}=4} } = \sqrt{ s_{\mathrm{CFT}} \over s_{{\cal N}=4} }, \end{equation} \tag{ 141 }$$

where s is the entropy density of the QGP. The energy lost by a hard parton plowing through the QGP is thus not proportional to the entropy density and so cannot be well modeled by scattering off a gas of persistent quasiparticles.

This holographic approach to studying the physics of a generic SU(N) QGP to glean insight into the dynamics of e.g. the physical QGP at RHIC has been followed in many directions, including several of the papers in this focus issue. For example, Hubeny [373] uses a simple but remarkably illuminating gravitational model for an accelerated heavy quark in large-N QCD to study the collimation of synchrotron radiation emitted by a circling quark. Like the diffusion and jet-quenching computations sketched above, Hubeny's calculation involves tracing the path of the bulk string which trails behind the accelerating quark into the bulk of AdS. The result of the acceleration is that this bulk string becomes coiled, leading to the radiation emitted by the string remaining tightly collimated. Curiously, this result at first appears to be in tension with standard holographic intuition. Hubeny resolves this apparent conflict through a careful study of the gravitational backreaction induced by the coiled string.

4.3.3. Gravitational engineering and holographic superconductors

An interesting example of defining the field theory by its holographic dual is the holographic superconductor (figure 25). The idea, first outlined by Gubser [374] and developed in detail by Hartnoll et al [375, 376], goes roughly as follows. Suppose we have a QFT which enters a superconducting phase at low temperatures by developing a non-zero condensate for some scalar operator, ${\cal O}$, spontaneously breaking a global U(1) symmetry. What would that look like in the holographic context? By appeal to the holographic dictionary in table 1, the global current in the boundary is dual to a gauge field in the bulk, while the charged scalar order parameter in the boundary is dual to a scalar field in the bulk, which is charged under the bulk gauge field. Spontaneously breaking (or Higgsing) the global U(1) symmetry on the boundary by giving a vacuum expectation value to ${\cal O}$ is then dual to spontaneously breaking the bulk gauge U(1) via a non-trivial profile for the bulk scalar field, Φ(r).

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Our goal is thus to build a gravitational system which is dual to some QFT which exhibits this symmetry breaking pattern—not a specific QFT, but any QFT with this structure. By adjusting the gravitational theory we will then be able to explore a whole space of strongly coupled QFTs with low-energy superfluid phases. Procedurally, we will take the gravitational system as a definition of the dual QFT, computing the correlation functions, etc, of the QFT entirely by appeal to the holographic dual.

The holographic dictionary thus tells us that, whatever else exists in the gravitational dual of a superconducting QFT, we will need a vector field A_μ dual to the boundary current ${\cal J}_{m}$ and a scalar field Φ dual to the scalar order parameter ${\cal O}$. The dictionary, however, does not tell us the Lagrangian, so we must choose it. A minimal first guess is

$$\begin{equation} I_{\mathrm{grav}} = -{1\over 16\pi G_{\mathrm{N}}}\int\sqrt{g}\left(-2\Lambda +R -{1\over e^{2}}\left[{1\over4}F^{2}+|D\Phi|^{2}+m^{2}|\Phi|^{2}\right]\right). \end{equation}\noindent \tag{ 142 }$$

We can further simplify our lives by considering the probe limit, g → ∞, in which case the matter field stress tensor (which scales as $1\over e^{2}$) is negligible so that the Einstein equation, and thus the metric, is not modified from its original AdS form.

Our task now is to find non-trivial solutions of the bulk equations of motion which follow from our trial action expanded around an AdS black brane geometry, so as to put the system at finite temperature. Non-trivial means that the scalar field Φ acquires a non-vanishing profile which spontaneously breaks the gauge symmetry in the bulk. From our dictionary, this corresponds to the boundary operator ${\cal O}$ acquiring a vacuum expectation value which spontaneously breaks the global symmetry on the boundary. Once we find such a solution, we can compute correlation functions in the resulting system by evaluating the bulk action on-shell as a functional of the non-normalizable modes and taking derivatives with respect to those modes, as discussed in section 4.2.3.

For simplicity, let us focus on homogeneous configurations of the form A = A_t(r)dt and Φ = Φ(r). The bulk equations of motion in our AdS black brane take the form

$$\begin{equation} \fl r^{2}f A_{t}'' - 2|\Phi|^{2}A_{t} = 0,\quad r^{2}f\, \Phi'' +(rf'\!-\!2f)r\Phi'+ \left({r^{2}A_{t}^{2}\over f}-m^{2}\right)\Phi = 0. \end{equation} \tag{ 143 }$$

Near the boundary, the general solution to the bulk field equations takes the form

$$\begin{equation} A_{t} \sim \mu \,+ \rho\,r + \cdots,\quad \Phi \sim \phi_{d-\Delta} r^{d-\Delta} + \phi_{\Delta} r^{\Delta} + \cdots, \end{equation} \tag{ 144 }$$

where m²L² = Δ(Δ − d). A convenient choice for the mass is m²L² = −2, which gives Δ = 1 or 2. Let us focus on Δ = 2. The holographic dictionary tells us to interpret μ as the source of the ${\cal J}_{t}$ component of the current, so μ plays the role of the chemical potential for the boundary charge density. ρ, being the subleading piece of A_t near the boundary, then represents the expectation value of the operator conjugate to μ, $\rho =\langle {\cal J}_{t}\rangle$, so we can identify ρ as the charge density induced by μ. Similarly, ϕ_d−Δ, as the leading term in Φ near the boundary, is interpreted as the source for the boundary operator ${\cal O}$, and ϕ_Δ as the vacuum expectation value, $\phi _{\Delta }=\langle {\cal O}\rangle$.

To construct a spontaneous condensate, we should turn off the source, ϕ_d−Δ = 0, fix a reference value for the chemical potential, μ = μ_o, and look for a solution of the bulk equations of motion which have, below some critical temperature T_c, a non-zero value for $\langle {\cal O}\rangle =\phi _{\Delta }$. This requires that we solve equations (143) subject to the boundary conditions that Φ(0) = 0 (no source) and A_t(0) = μ_o. Generally, this must be done numerically, as equations (143) are nonlinearly coupled. By evaluating the on-shell action and using the holographic dictionary in table 1, we can compute physical quantities. For example, we find the critical temperature by searching for the minimum T of the background black brane such that a non-trivial solution with these boundary conditions exists. We find the ac conductivities by looking at linear response for fluctuations of A_x linearized around a non-trivial solution for A_t and Φ, etc. The precise behavior we find depends on the details of the gravitational model chosen. By adding additional fields and interactions, we can use these techniques to engineer a wide range of dynamical phenomena in the dual QFT.

The point to emphasize here is that while we are explicitly computing correlation functions in a strongly coupled QFT, and indeed studying the detailed physics of transport, the QFT under investigation is defined purely through its gravitational dual. We thus call these superconductors holographic superconductors, and more generally refer to such QFTs as holographic QFTs.

Importantly, these models can be used to study much more than just the translationally invariant ground states of these strongly interacting superfluids. For example, they provide a simple and powerful framework for studying spatially ordered phases and solitons. For example, the paper by Keränen et al in this focus issue [164] uses solitons in these holographic superconductors as precision probes of the superfluid, including, in particular, the excitation spectrum [378, 379] around a dark soliton at unitarity in the BCS–BEC crossover [162, 165–167]. As the solution of this problem reduces to a detailed study of coupled PDEs, considerable progress can be made with common computational techniques toward a concrete prediction observable in the present experiments on ultracold quantum gases.

Notably, this holographic mechanism for spontaneously breaking a symmetry can be applied to any boundary QFT with symmetry breaking. For example, the paper by Basu et al in this focus issue [380] uses the holographic superconductor paradigm to study color superconductivity in holographic models of the QGP. As good effective field theorists, Basu et al begin by writing down a gravitational effective theory with the minimal ingredients to mock up a color superconductor and then examine what constraints they must impose on the parameters of this effective description to reproduce the expected physics of a true color superconductor.

4.3.4. Non-relativistic holography and cold atoms

Strongly correlated quantum liquids of cold atoms provide an excellent target for applied holography. From a holographic point of view, cold atoms at unitarity are very similar to ${\cal N}=4$ models of the RHIC fireball: they provide examples of strongly correlated liquids at finite temperature and chemical potential that are governed by a conformal field theory. The key difference is that these systems are non-relativistic, with dispersion relations scaling as⁶⁶

$$\begin{equation} \omega\sim k^{z}, \end{equation}\noindent \tag{ 145 }$$

so that the RG fixed points that govern their dynamics are non-relativistic conformal field theories (NRCFTs) [381–383]. Furthermore, in the case z = 2 arising in cold fermions at unitarity, the symmetry algebra is enlarged [383] and includes a central number operator, $\hat {N}$, such that every operator has, in addition to a dimension, Δ, a conserved particle number eigenvalue, N.

Holographically, the non-relativistic scaling has a dramatic effect. In the case of ${\cal N}=4$ theory, relativistic conformal invariance was enough to determine the dual space–time, with the conformal symmetry group of the QFT, SO(4,2), mapping to the isometry group of the dual space–time; this fixes the geometry dual to the ${\cal N}=4$ theory to be AdS₅. More generally, the fact that relativistic QFTs flow to Lorentz-invariant RG fixed points in the UV implies that their holographic duals should asymptotically approach pure AdS near the boundary. If our QFT does not possess Lorentz invariance, the dual space–time cannot be simple AdS. To build the holographic dual of an NRCFT, then, we must start from the beginning and identify the right space–time.

The simplest geometry with these properties is the Lifschitz metric [308]

$$\begin{equation} \mathrm{d}s^{2} = L^{2}\left(-{\mathrm{d}t^{2} \over r^{2z}} + {\mathrm{d}\vec{x}^{2} +\mathrm{d}r^{2} \over r^{2}}\right). \end{equation}\noindent \tag{ 146 }$$

In this geometry, every slice at a fixed r is again a copy of flat space, but the scaling of the time-like coordinate and the spatial coordinates under a shift in r is inhomogeneous, with scaling ω ∼ k^z. At z = 1, this reduces precisely to the AdS case, a good check. However, at z = 2 there is no enhancement of the isometry group of the geometry, so it is natural to look for another geometry which realizes the full enhanced z = 2 non-relativistic conformal symmetry group. Following this logic leads to the Schrödinger metric [306, 307]

$$\begin{equation} \mathrm{d}s^{2} = L^{2}\left(-{\mathrm{d}t^{2} \over r^{2z}} + {2\mathrm{d}t\,\mathrm{d}\xi + \mathrm{d}\vec{x}^{2} +\mathrm{d}r^{2} \over r^{2}}\right), \end{equation}\noindent \tag{ 147 }$$

where ξ is a new periodic variable. Again we find that scaling implies that ω ∼ k^z. Now, however, the isometry group of the manifold is in fact enhanced at z = 2. Moreover, since ξ is compact, we can expand every field in eigenmodes of $\hat {N}=\mathrm {i}\partial _{\xi }$, so that bulk fields, and thus all boundary operators to which they are dual, are labeled by both a dimension, Δ, and also a ξ-momentum, N. We thus have a geometry which enjoys both of the peculiar features of fermions at unitarity identified above.

Repeating the full construction in section 4.2 in this non-relativistic setting remains an open problem. However, considerable progress has been made in the context of both Lifschitz and Schrödinger systems, including the construction of charged black holes corresponding to NRCFTs at finite temperature and chemical potential [384–387] and the construction of some toy non-relativistic superfluids (see, e.g., [388] in this focus issue). Many of the basic universal results from the relativistic setting have been replicated. For example, in the simplest models we again find ${\eta \over s}={1\over 4\pi }$. Note that, while we have suppressed factors of c throughout, there are no such factors in this result for the viscosity ratio so we get a non-trivial prediction for non-relativistic systems. Furthermore, while $\eta \over s$ is a natural observable in relativistic QFTs even if we do not know about holography (it is the dimensionless momentum diffusion constant in linearized hydrodynamics), the fact that $\eta \over s$ is relevant in non-relativistic systems is an interesting feature of holography; the momentum diffusion constant in non-relativistic hydrodynamics is $\eta \over n$. In the context of Lifschitz scaling, considerably more progress has been made, including interesting results for models of strange metals [389].⁶⁷

For models of cold atoms, on the other hand, our non-relativistic holographic models remain rather distant from the systems studied in actual experiments. Indeed, it seems likely that fermions at unitarity, like QCD, do not have weakly coupled holographic duals. However, unlike QCD, where we have already built holographic toy models which capture a great deal of the strong-coupling phenomena, non-relativistic CFTs with dynamical exponent z = 2 and Schrödinger symmetry have proven difficult to model holographically. In particular, the thermodynamics and phase structure of current holographic models are not obviously related to any known systems, if not outright pathological [384–386]. For example, we expect the finite-density ground state of these theories to form a superfluid which spontaneously breaks the U(1) symmetry generated by the number operator, $\hat {N}$. Holographically, $\hat {N}$ is mapped to ∂_ξ, the generator of translations in the second holographic direction. Such a superfluid ground state must thus break translation invariance in the ξ-direction. Unfortunately, all our current holographic models of z = 2 Schrödinger CFTs feature manifest unbroken ξ-translation invariance.

This problem can be traced [390–392] to the fact that all our current models were derived from highly symmetric stringy systems which enforce this symmetry. As a result, these limitations do not appear to be intrinsic to the holographic framework but rather to the very specific models which have so far been studied. It would be of great interest to develop more realistic models (see, e.g., [392] for a recent work in this direction). A step in this direction is described in the paper by Adams and Wang in this focus issue [388], in which a superconducting ground state for such a non-relativistic holographic CFT is constructed by extending the 'holographic superconductor' strategy from AdS to these non-relativistic geometries. This leads to various interesting effects, including a surprising quantum phase transition as the density of the fluid is varied and a multicritical point where the transition to the superfluid state switches from second to first order. This does not solve the underlying problem, as the geometry remains unmodified, but is at least a proof of principle that such holographic ground states do exist.

4.3.5. Holographic non-Fermi liquids and critical phenomena

4.3.6. Far-from-equilibrium physics in strongly coupled quantum field theories

• (i)  
  What is the structure of the unitary Fermi gas in both two and three dimensions? In particular, is it possible to understand the properties of the system in terms of well-defined quasiparticles?At high temperature the gas is always weakly correlated, independent of the strength of the interaction. In the weak coupling regime we know that the quasiparticles near T_c are BCS-like gapped fermions, and that at very low temperature the quasiparticles are phonons. At unitarity it has been argued that the regime can be understood in terms of a pseudogap [35, 404, 405]. This means that there is a gap in the single-particle spectrum, but no long-range coherence. Pseudogap behavior in three dimensions has been observed using RF spectroscopy [406, 407], and it has also been seen in quantum Monte Carlo calculations [408]. Pseudogap behavior was also reported in experiments with two-dimensional systems [409]. However, it is not entirely clear to what extent thermodynamic and transport properties can be understood in terms of gapped quasiparticles. One quasiparticle that may turn out to be relevant is the polaron, a single spin-up particle immersed in a sea of spin-down fermions [410]. In highly polarized gases, RF spectroscopy has been used to measure the dispersion relation of polarons. It was observed that the polaron description can be extended to more spin balanced systems. In particular, a simple estimate based on a gas of polarons provides a very good estimate for the critical polarization at the Chandrasekhar–Clogston point, the first-order transition from an unpolarized superfluid to a polarized normal gas [411]. Recently, polarons have also been observed in two-dimensional Fermi gases [412, 413].
• (ii)  
  How do we make local measurements of transport coefficients such as shear and bulk viscosity, thermal conductivity and the spin diffusion constant?Measurements of elliptic flow and collective mode damping provide trap averaged values of the shear viscosity (see section 2.5) but there is no direct measurement of the local value of the shear viscosity η(n,T). The status is roughly analogous to the situation after the first experiments aimed at measuring the equation of state. These experiments only determined the total energy and entropy [66]. Local measurements were made possible by new ideas such as the Gibbs–Duhem method employed by the ENS group [68] and the compressibility thermometer introduced by the MIT group [36]. It is possible, in principle, to unfold the currently available data for trap averaged viscosities, but this method requires a very careful treatment of the dilute corona, and a more direct method would be desirable. One possible direction is the development of new experimental techniques that generate local shear flows and study their relaxation. Another option is to measure the frequency-dependent shear viscosity via spectroscopic methods as suggested in [414]. Similar questions apply to other transport properties, such as bulk viscosity, thermal conductivity and spin diffusion. Current methods are mostly sensitive to average transport coefficients, but more local measurements are needed.
• (iii)  
  Are there reliable approaches to transport theory beyond the Boltzmann equation?In the unitary limit, transport properties can be computed reliably using the Boltzmann equation in the limit T ≫ T_F. For T ∼ T_F the quasiparticles are strongly interacting, and the kinetic description in terms of binary scattering between atoms breaks down. Enss et al [129] have computed the shear viscosity using a T-matrix resummation. A similar approach is discussed by Guo et al [110] in this focus issue. These methods are quite successful in describing the thermodynamics, but there is no expansion parameter, and it is not clear if non-equilibrium properties are predicted as well as equilibrium features. In section 2.2 we briefly discuss the application of 1/N and expansions to equilibrium properties of the unitary gas, but except for the recent work of [415] these methods have not been extended to non-equilibrium properties. Similar remarks apply to other methods as well: there are detailed studies of equilibrium properties using the exact RG [416], but no corresponding studies of transport properties. Finally, there now exist very accurate quantum Monte Carlo calculations of the equation of state, but there is only a single, pioneering, study of transport properties [5].
• (iv)  
  Are there other universal quantum gases that can be created by manipulating the dimensionality, the spin structure, the type of interaction or the dispersion relation?The unitary gas exhibits a very high degree of universality: in the limit of infinite scattering length and zero effective range the many-body system is completely characterized by the fermion mass, the temperature and the chemical potential. The dependence on dimensionful combinations of these parameters is completely governed by dimensional analysis. Universality can be exploited to use the unitary gas as a model system for dilute neutron matter; see section 3.1. The question is whether this idea can be extended to more complicated systems, for example three-species gases that can serve as a model for the quark matter/nuclear matter transition (see section 2.7) or four-species gases that correspond to self-bound nuclear matter, which is a liquid of equal numbers of spin-up and -down protons and neutrons. Universality in systems with more than two degrees of freedom is more complicated, because the interaction depends also on at least one three-body parameter [417], and the system may require repulsive short-range forces in order to prevent collapse. New universal states can also be created by combining Bose and Fermi gases in different dimensions [418] or by manipulating the dispersion relation of the atoms. Steps in this direction were recently taken by Salger et al [419], who demonstrated linear dispersion and Klein tunneling of a Bose–Einstein condensate in a 1D optical lattice, and by Taruell et al [420], who created a two-dimensional optical lattice with the same honeycomb lattice structure as graphene and therefore obtained low-energy fermions with a linear dispersion relation.
• (v)  
  What is the physics of the crossover in the discrete context of optical lattices?Will the predictions of holographic duality of a lower limit on the ratio of viscosity to entropy hold true in the lattice context or not? What class of theories from holographic duality might map onto the Fermi resonance Hamiltonian or other effective two-channel models? How can we extend such models to treat the imbalanced Fermi gas? How will a polaron gas be described in the lattice context?
• (vi)  
  What will be the interplay between disorder and interactions for the unitary gas?The relative effects of disorder and interactions on both thermal and quantum phase transitions in quantum many-body physics are an open question generally (see, e.g., [421]). An essential feature of disorder is Anderson localization, in which scattering off of many small defects conspires to localize all states above a certain threshold in three dimensions. This concept is, in fact, a wave physics or single-particle quantum concept: it has been so far unclear where and how interactions destroy or allow for Anderson localization in a generic sense. Disorder was treated in this focus issue for ultracold Fermi gases specifically by Han and Sá de Melo [172]. Here, too, we find potential connections to holographic duality, as explored in recent forays into disordered systems by Adams and Yaida [422, 423]. Although these explorations have not treated lattice physics in particular, disorder is an essential feature of lattice systems arising in nature, and can be induced artificially in various ways for optical lattices [424]. Can holographic duality capture the underlying lattice causes of disorder? How will experiments in ultracold Fermi gases combine disordered optical lattices with unitary fermions at sufficiently low temperatures? Might studies in this direction help us gain some fundamental insight into the interplay between disorder and interactions in strongly correlated systems?

• (i)  
  Can we determine all the transport properties of the QGP?We would like to determine, with fully quantified experimental and theoretical uncertainties, the value of the shear viscosity to entropy density ratio η/s, the heavy quark diffusion constant D and the jet quenching parameter $\hat {q}$. This program is probably closest to completion in the case of η/s. In this case the basic method, second-order relativistic viscous hydrodynamics, is now fairly well understood. Final state corrections can be handled by coupling to a hadronic cascade. There is a large set of data, including the energy, impact parameter and system size dependence of elliptic flow, as well as the observation of higher harmonics that points to the validity of the hydrodynamic description. The internal consistency of hydrodynamics already implies a fairly strong bound . The main problem is that we have no direct method for establishing the initial conditions for the hydrodynamic description. This problem may be addressed by focusing on a larger set of observables, in particular higher harmonics of the flow [425, 426]. There also remain theoretical uncertainties regarding the coupling of viscous fluid dynamics to a Boltzmann description, the role of bulk viscosity and the equation of state. A discussion of the known uncertainties in η/s can be found in [427].The main strategy for determining the heavy quark diffusion constant is to use a Langevin description of the diffusion process where the background collective expansion is taken from hydrodynamics. The most important problem at present is the fact that there are only data for single leptons, and not for identified heavy quarks. This problem will be solved by improved detection capabilities at RHIC and LHC. The main theoretical issue is how to handle the interplay between elastic and inelastic scattering. For very heavy quarks elastic scattering is dominant, but for charm quarks inelastic processes may be important. This includes hadronization effects such as coalescence.In the case of jet energy loss there are still a number of theoretical issues that need to be resolved. The radiative energy loss of an energetic parton (E ≫ T) in a partonic medium can be expressed in terms of a single medium parameter, the transverse diffusion constant $\hat {q}$. It is not clear if this carries over to the spectra of jets or leading particles produced in the medium. Another source of uncertainty is that different implementations of the in-medium radiation in perturbative QCD lead to significantly different values of $\hat {q}$ when applied to the RHIC data [278]. There is a significant ongoing effort to understand these differences⁷⁰. On the experimental side, the LHC represents a large step toward higher jet energies E ∼ 100 GeV, and significantly improved statistics in the range already explored at RHIC. Both at the LHC and at RHIC, experimentalists are studying not only spectra of hard particles, but also identified jets, jet shapes and correlations between hard particles and the associated soft emission.
• (ii)  
  Can we understand early thermalization?The success of the hydrodynamic description implies that the system must thermalize early. The question is: how to quantify this statement experimentally, and how to understand early equilibration theoretically? The most reliable constraint on the thermalization time arises from the observation of elliptic flow. Elliptic flow is driven by the anisotropy of the initial state. Since ballistic expansion dilutes the initial anisotropy, local equilibration must happen early for elliptic flow to develop. A very conservative constraint, discussed in [182], is τ₀ < 2.5 fm/c. Typical values used in hydrodynamic fits are . Another hint of early thermalization comes from the observation of thermal photons with temperatures significantly larger than those seen in hadronic spectra. The PHENIX collaboration has recently reported a photon transverse momentum spectrum with a slope parameter T = 221 ± 19 ± 19 MeV [428]. The spectra can be reproduced in hydrodynamic calculations with thermalization times ranging from to , corresponding to initial temperatures T₀ ∼ 300–600 MeV. Establishing experimental constraints on the equilibration time is further complicated by the fact that not all observables require full thermalization. For example, for the development of radial and elliptic flow it is not necessary that longitudinal momentum distributions be fully equilibrated [429].Understanding thermalization theoretically is difficult even if the early evolution is governed by weak coupling. The first attempt to understand thermalization starting from an initial color-glass state using 2 → 2 and 2 → 3 scattering is the bottom-up scenario described in [430]. It was later realized that this picture is modified by collective effects that lead to plasma instabilities [431–433]; see also the contribution by Dusling in this focus issue [434]. In all these approaches it is hard to understand how thermalization at RHIC can occur on a time scale .Attempts to understand thermalization at strong coupling have focused on holographic duality; see section 4.3.6 and [435, 436] for recent overviews. Holographic duality provides an interesting geometric picture of thermalization: the initial state corresponds to two colliding shock waves, and thermalization is signaled by the formation of an event horizon. The hydrodynamic stage is described by the relaxation or ringdown of the black hole. In holographic duality one can achieve fast equilibration [312], but it is hard to make contact with asymptotic freedom and the well-established theory and phenomenology of nuclear deep inelastic scattering, and it is difficult to understand the observed energy and impact parameter dependence of the total multiplicity [437].
• (iii)  
  Does the QGP have a quasiparticle description?We would like to understand how the properties of the hot and dense matter produced at RHIC and the LHC are related to its structure, in particular whether the matter can indeed be described as a plasma made of quarks and gluons. Holographic duality seem to indicate that a strongly interacting quantum fluid near the viscosity bound does not have a quasiparticle description; see sections 3.2 and 4. In section 3.2 we discussed a number of observables that have been suggested as probes of the quasiparticle structure: fluctuations, the observed constituent quark scaling of flow, and the relation between the heavy quark diffusion constant and the shear viscosity. None of these observables provides a conclusive test of the quasiparticle picture, and new ideas would certainly be helpful. We also discussed the possibility to study quasiparticles by extracting the spectral functions of the correlators of conserved charges in lattice QCD, such as energy, momentum, flavor, etc. These studies, too, are somewhat indirect, because on the lattice we only have direct access to imaginary time correlation functions. This means that the spectral functions have to be extracted via analytic continuation. Analytic continuation of numerical data is difficult, but there has been significant progress over the last couple of years [438].
• (iv)  
  Can we experimentally locate the phase transition?There is a continuing effort to establish the presence of a critical endpoint in the QCD phase diagram. The basic idea is to look for the non-monotonic behavior of fluctuations or correlations as a function of the energy of the colliding system. The first data from the RHIC beam energy scan have recently been released [439]; see also the contribution by Mohanty [440] in this focus issue. The data contain new information about the beam energy dependence of hydrodynamic flow which will help to establish the onset of a nearly perfect fluidity. Gupta et al [214] have argued that data on fluctuations can be directly compared with lattice QCD and that it can be used to set the scale for the phase transition at zero chemical potential. There is not yet conclusive evidence for the existence of a critical point, but data for additional beam energies and with better statistics will become available in the near future. In heavy-ion collisions the growth of the correlation length is limited by finite size and finite time effects, and it is important to identify observables that provide the best sensitivity to critical behavior. Observables that have been considered include fluctuations of conserved charges, such as baryon number, other fluctuation observables, such as the pion-to-kaon ratio, the mean p_T, or elliptic flow, and higher moments such as skewness and curtosis [441]. There is a parallel effort to locate the critical endpoint in lattice QCD. This effort is complicated by the sign problem that affects lattice calculations for fermions at finite chemical potential [442].The discovery of a critical point would establish the presence of a phase transition between hadronic matter and the QGP. At small chemical potential this transition is only a crossover. One might hope that it would be possible to detect the crossover through hydrodynamic effects associated with the minimum in the speed of sound near the crossover temperature [443]. However, explicit calculations seem to show that there is no direct link between the presence of a minimum in the speed of sound and the beam energy dependence of multiplicity and flow [182].Finally, we would like to demonstrate the existence of quark matter at high density and low temperature. It is not known whether the cores of compact stars are sufficiently dense to contain quark matter or other exotic phases. The presence of quark matter can be potentially detected through its effect on the equation of state, which is reflected in the mass–radius relationship. There is a very active program to pin down this relationship using observations of compact stars [444, 445]. Once the presence of a high-density phase has been established, more detailed information regarding its properties can be derived using the cooling and spin-down behavior of compact stars.

• (i)  
  What does it take to derive a holographic duality?We have no first-principles derivation of holographic duality. This is perhaps not surprising, since it is a strong–weak duality, but it is frustrating. We do have a number of heuristic arguments, such as those that appeared in section 4.1. We also know how to derive a long but finite list of dual pairs as decoupling limits of particular string theories, e.g. the original construction, ${\cal N}=4$ Yang–Mills in 4D. But we do not know how to derive the duality directly within the bulk or boundary theories⁷¹.This is important for two main reasons. Firstly, we do not have a thorough understanding of the consistency conditions or regimes of validity of our holographic dual descriptions. We do have general rules of thumb, such as the scalings discussed in section 4.2.3, and we can check for self-consistency in specific examples. But particularly when we are defining a QFT via the holographic prescription, it is not always obvious what the precise regime of validity is and to what extent we should trust the chosen gravitational truncation.Secondly, if we want to generalize holographic duality, it would be very helpful to know why it is true in the more familiar examples. For instance, we would like to generalize to intrinsically anisotropic systems, to asymptotically flat or positive-curvature (dS) space–times or even to understand why this is not possible.
• (ii)  
  Given a QFT, what is the gravitational dual?Even if a general derivation is not possible, it would be useful to have a constructive map which allowed us to take an arbitrary QFT and identify the appropriate gravitational dual. For example, what is the exact dual of QCD, not QCD plus a collection of exotic fields, not of large N QCD, not of some toy model, but pure QCD? Similarly, with an eye toward ultracold atoms, what is the gravitational dual of the Hubbard model, or of the Kondo problem?Again, the fact that holography is a strong–weak duality should give us pause. As we have seen, when the gravitational system is weakly coupled, the dual QFT generically does not have any simple quasiparticle interpretation. If we want to define the QFT using conventional Lagrangian methods, the best we will, in general, be able to do is say that it is the strong-coupling limit of some explicit perturbative theory. In the models we understand best, the coupling that is getting strong is tightly constrained by symmetries, e.g. gauge symmetries, flavor symmetries or supersymmetry, so we know how to follow the RG flow well beyond the perturbative, quasiparticle regime. We do not know how to solve the more general case without these extraordinary symmetries and the constraints they imply.
• (iii)  
  Can we derive gravity from field theory?The holographic graviton looks very much like an emergent vector boson in more familiar condensed matter systems. It arises as an emergent dynamical gauge field which reorganizes the strongly coupled physics into a set of perturbative local fluctuations. However, in holography the graviton (a) lives in one higher dimension and (b) couples universally. Point (a) fits naturally with a basic feature of gravity: since gauge transformations are coordinate transformations, there can be no truly local degrees of freedom, so any emergent graviton must come along with an emergent space–time dimension with which to fix its gauge redundancy, much as any emergent vector boson must come along with an extra Hubbard–Stratanovich scalar to fix its gauge redundancy. This was, in fact, the origin of the term holography in gravitational physics. Point (b) is more spectacular from the point of view of emergence, but is in fact a necessary feature of any theory with a fluctuating spin-2 gauge field.It is thus tempting to wonder whether we can, in fact, derive the graviton in a general space–time as an emergent gauge boson in a strongly interacting QFT. Some progress has been made on this front (cf [446]), but a detailed understanding remains elusive.
• (iv)  
  Do we always need the complete holographic description?In many of the examples discussed above, and indeed almost generically in the holographic literature, most of the work is being done by a very small fraction of the total degrees of freedom in the holographic description. For example, in the computation of η/s in ${\cal N}=4$ SYM, whose holographic dual, string theory in AdS₅ ×  S⁵, includes an infinite number of dynamical massive fields, the only fields that matter in the computation are the 5D metric and a single scalar mode encoding one particular fluctuation of this metric. In practice, keeping track of the full holographic description often seems to be overkill⁷².This raises two natural questions. Firstly, when does it make sense to truncate the full holographic description to a small subset of modes? Secondly, might this truncated holographic model have a more straightforward QFT representation than the full holographic description? Said differently, might there be a 'semi-holographic' formalism defined entirely within field theory which allows us to build conventional QFTs that reproduce the interesting phenomenology of fully holographic QFTs without using higher dimensions?Recent work on holographic Fermi liquids, holographic quantum critical points and holographic quantum liquids (see [309] and especially the work of Nickel and Son in this focus issue [311]) demonstrates that at least some of the interesting holographic effects can indeed be realized in a purely quantum field theoretic formalism. On the other hand, for questions where the nonlinear dynamics of the geometry such as far-from-equilibrium physics, as opposed to kinematics or linear response, are important, it seems unlikely that any semi-holographic description can usefully capture the salient physics. Just how far a semi-holographic approach can be pushed, then, is an extremely interesting question.
• (v)  
  What predictions do holography make for generic strongly coupled systems?Once we let go of the idea that our models must be defined by traditional quasiparticle effective QFTs, we can start asking general questions about the strongly coupled QFTs illuminated by our holographic examples. What common properties do these theories share? What general lessons can we draw about their kinematics, dynamics, phase structures, critical behavior, equilibration, etc? For example, we already know to expect nearly inviscid hydrodynamic behavior at finite density and low temperature. What else should we expect in general? Perhaps most importantly, what lessons can we draw for systems we really care about?
• (vi)  
  What is the holographic dual of a non-relativistic QFT?As discussed in section 4.3.4, non-trivial progress has been made in the study of non-relativistic QFTs and many-body systems. This has led to some simple heuristic lessons, for example that we should expect the viscosity to entropy ratio to again be extremely low and that the Bertsch parameter⁷³ should be expected to be of the order of unity and remain relatively insensitive to the coupling constant near strong coupling. However, the state of current models is rather poor, with awkward thermodynamics and bizarre phase structure. What are we missing? Can we build a holographic dual in the same universality class as fermions at unitarity?
• (vii)  
  How (and why) is quantum information encoded in the bulk geometry?Holographically, information about the thermodynamic state of the boundary QFT is encoded in the detailed geometry and matter field backgrounds of the bulk theory. For example, the temperature and chemical potential are encoded in the mass and charge of the bulk black hole. Similarly, when the QFT is in a pure state, the bulk geometry should encode the quantum correlations of the boundary QFT at zero temperature. But how?One approach to this question is the Ryu–Takanayagi conjecture [448, 449], which states that the entanglement entropy associated with a region R in a holographic QFT is given by the area of the minimal-area surface in the bulk which ends at the boundary of the region R. Holographically, this conjecture matches the relation between the thermodynamic entropy of a black hole and the area of its horizon. Interestingly, it suggests an intimate connection between quantum and thermal entropy—and more generally, between quantum and thermal fluctuations—all of which are encoded by the physics of horizons in the holographic dual.While a general proof is still lacking, considerable evidence has been found for this conjecture, and conversely, numerous predictions have been made for the entanglement entropy of strongly coupled many-body systems by exploiting this conjecture (see, e.g., [450–455] and references therein). For example, in this focus issue, Albash and Johnson [456] have studied the evolution of the Ryu–Takanayagi entropy under a thermal quench, leading both to evidence supporting the conjecture and to predictions on the evolution of the entanglement entropy.
• (viii)  
  How does spatial inhomogeneity change the story?As we have seen, holography translates quantum computations in the boundary QFT (i.e. Feynman diagrams and path integrals) into classical GR calculations in the bulk (i.e. solving nonlinear partial differential equation boundary value problems). As such, it is clear why most research to date has focused on homogeneous, isotropic problems. However, on the quantum many-body side, we know that inhomogeneity can radically alter the dynamics, and that the precise way they do so provides a detailed probe of the physics driving those dynamics. The same is true of anisotropy. Moreover, much of what we know about inhomogeneity and anisotropy in QFT comes from some form of perturbation theory. It would thus be particularly interesting to study the dynamics of holographic systems in the presence of inhomogeneity and anisotropy. Will we find evidence for many-body localization? Do (2 + 1)D QFTs at strong coupling and in the hydrodynamic regime display a canonical turbulent cascade, or an inverse cascade as is usually, but not universally, expected in two spatial dimensions?

Many of the issues discussed in sections 5.1–5.3 are addressed in the contributions to this focus issue, and we hope that this summary, as well as the insights contained in these contributions, will spur further advances in research on the intriguing subject of strongly correlated quantum fluids.