Analog dual to a 2+1-dimensional holographic superconductor

We study an analog hydrodynamic model that mimics a 3+1 AdS planar BH spacetime dual to a 2+1-dimensional superconductor. We demonstrate that the AdS$_4$ bulk and its holographic dual could be realized in nature in an analog gravity model based on fluid dynamics. In particular we mimic the metric of an $O_2$ holographic superconductor and calculate the entanglement entropy of a conveniently designed subsystem at the boundary of the analog AdS$_4$ bulk.


Introduction
A pseudo-Riemannian geometry of spacetime can be mimicked by fluid dynamics in Minkowski spacetime. The basic idea is the emergence of an effective metric which describes the effective geometry for acoustic perturbations propagating in a fluid potential flow with u µ ∝ ∂ µ θ. The quantity c s is the adiabatic speed of sound, the conformal factor a is related to the equation of state of the fluid, and the background spacetime metric g µν is usually assumed Minkowski. The metric of the form (1) has been exploited in various contexts including emergent gravity [1,2], scalar theory of gravity [3], Einstein-aether gravity [4], acoustic geometry [5,6,7,8] and euclidean gravity [9,10,11]. The work presented here is motivated by recent development of anti-de Sitter/conformal field theory (AdS/CFT) dual theory of 2+1-dimensional superconductor [12,13,14,15,16,17,18,19,20,21] (for a review and additional references see [22]). The AdS/CFT duality in these models is based on a correspondence between gravitational theory and dynamics of quantum field theory on the boundary of asymptotically anti-de Sitter (AdS) spacetime. The gravity side can be well described by classical general relativity, while the dual field theory involves the dynamics with strong interaction. This correspondence is often referred to as "holography" since a higher dimensional gravity system is described by a lower dimensional field theory without gravity, which resembles optical holography.
A particularly important work in this context is the minimal model of a holographic superconductor by Bobev et al [19] with an Abelian gauge field embedded in the truncation of four-dimensional maximal gauged super-gravity. Besides, it is worth mentioning the work on d-wave superconductivity by Benini et al [16] in which interesting physical phenomena are demonstrated such as the formation of Fermi arcs.
The AdS 4 spacetime as a solution to Einstein's equations cannot actually exist in nature due to instability problems. However, it can inspire some configurations where the underlying general gravitational structure can be studied through analogue models. The aim of this paper is to demonstrate that AdS 4 and its holographic dual could be realized in nature in an analog gravity model based on hydrodynamics of a physical fluid. In particular we will mimic the bulk metric of the minimal model of a holographic superconductor consisting of the metric, a charged scalar with a non-trivial potential and an Abelian gauge field embedded in the truncation of four-dimensional maximal gauged super-gravity [19]. This model was recently studied in the context of holographic entanglement entropy [20,21]. The entanglement entropy is an important tool for keeping track of symmetry breaking and phase transition in strong coupling systems. In the context of black-hole thermodynamics the entropy of a black hole is proportional to the area of the horizon in the same way as is the entanglement entropy proportional to the boundary area of between two subsystems of a quantum system.
Our first task is to derive an analog acoustic geometry which mimics a d + 1-dimensional asymptotic AdS geometry with a general planar black hole (BH). Furthermore, we will apply this to a 3+1-dimensional model and calculate the entanglement entropy for a particular geometry obtained as solution related to the holographic O 2 superconductor. The reason why we are specifically interested in the O 2 type is due to its pronounced first order phase transition at finite temperature.
It is important to stress that analog gravity in general is concerned by curved geometry per se without referring to sources of the gravitational field as in general relativity. More specifically, the fluid analog mimics the geometry only and says nothing about the source such as matter and other fields. There are no equations analog to Einstein's which, as in general relativity, would involve curvature tensor and stress tensor. However, even without Einstein's equations, the analog BH horizon entropy is realized via quantum entanglement of phonons. This is why the model studied in this paper and analog gravity in general can teach us something about black holes and related phenomena.
We divide the remainder of the paper into three sections and an two appendices. We start with section 2 in which we derive an analog metric for a d + 1-dimensional AdS planar BH hole of the form relevant for a holographic description of the superconductor. In the next section, Sec. 3, we apply our formalism to a 3+1-dimensional bulk related to the minimal model of the 2+1-dimensional holographic superconductor. For a particular geometry related to the O 2 superconductor we calculate the entanglement entropy. Concluding remarks are given in section 4. In appendix A we outline a derivation of the relativistic acoustic metric and in appendix B we derive the effective speed of sound in a fluid with an external pressure.

Analog planar black hole
Geometric structures in the form of a planar BH may have interesting applications in condensed matter physics [23]. In this section we construct a model of an analog planar BH hole in a general asymptotic AdS d+1 . A similar model for d = 4 was discussed in detail by Hossenfelder [24,25] and recently in [26,27]. We will discuss in more detail the case d = 3 which is of particular interest for 2+1-dimensional superconductor [19,20,22]. In our approach we will consider a nonisentropic fluid flow which yields the desired analog metric.
We start from a general form of the AdS planar BH metric in an arbitrary number of space-like dimensions d where is the curvature radius of AdS d+1 and For d = 3 we will relate the functions χ and γ to the truncated Lagrangian of the fourdimensional N = 8 super-gravity [28] studied by Bobev et al [19] in the context of holographic superconductivity. In order to have an asymptotic AdS for z → 0 we can always rescale the time coordinate so that, without loss of generality, we may assume Next, the dimensionless functions χ and γ can be thought of as functions of the dimensionless variable z/z h , where z = z h is the location of the horizon. In other words γ(z h ) = 0 (5) and γ has no zeros on the interval 0 < z < z h . Then, the horizon temperature is This temperature measured in some chosen fixed units, e.g., in units of −1 is ambiguous because the geometry (2) is invariant under rescaling Thus, the metric (2) has a rescaled horizon z h /α with the corresponding rescaled horizon temperatureT However, the temperature T expressed in units of 1/z h is unique, i.e., the quantity T z h is invariant under the rescaling (7). Therefore, in the following we will express the temperature and other dimensionfull physical quantities in units of some power of z h . Now we seek a fluid analog model which would mimic the induced metric of the form (2). The basic idea is to find a suitable coordinate transformation t →t, z →z such that the new metric takes the form of the relativistic acoustic metric (92) derived in appendix A with g µν replaced by the Minkowski metric η µν Here n and w denote the particle number density and specific enthalpy, respectively, and an arbitrary mass scale m is introduced to make G µν dimensionless. The specific enthalpy is defined as usual where p and ρ denote the pressure and energy density, respectively. The quantity c s is the so-called "adiabatic" speed of sound defined by where | s denotes that the specific entropy, i.e., entropy per particle s = S/N , is kept fixed. The second equality in (11) follows from the thermodynamic law dw = T ds + 1 n dp.
Following Hossenfelder [24] we transform the metric (2) by making use of a coordinate transformation where the functions z(z) and h(z) are determined by the requirement that the transformed metric takes the form (9). By simple algebraic manipulations the line element (2) can be recast into a convenient form where we have set and an abbreviationγ = e −χ γ.
From (4) and (5) it follows Comparing (14) with the acoustic metric (9) we identify c s as the speed of sound and the non-vanishing components of the velocity vector ut and uz in transformed coordinates as These equations implyγ ≤ c 2 s ≤ 1.
Next, by applying the potential-flow equation (see appendix A) we derive closed expressions for w, n, and c s in terms of the variable z. Since the metric is stationary, the velocity potential must be of the form where m is an arbitrary mass parameter which we can identify with the mass scale that appears in (9) and g(z) is a function ofz through z. Then, from (21) and (22) it follows and the function g in (22) must satisfy The particle number density can be obtained from the condition that the conformal factor in (2) must be equal to that of (9), i.e., we require As m is arbitrary it is natural to choose so using this and (23) we find In this way, both w and n are expressed as functions of z and c s . However, c s is not independent since by the definition (11) Using (28) with (23) and (27) we obtain a differential equation for c s with solution The integration constant must satisfy the constraint 1 ≥ K ≥ 0 as a consequence of the condition 0 ≤ c 2 s ≤ 1. Dimensionless physical quantities such as w, c s and the components of the fluid velocity field are functions of z/z h and are invariant under the rescaling (7).
Plugging (30) into (23) and (25) one obtains w and n as functions of z. Note that explicit functional forms of z(z), γ(z), and g(z) can be obtained by making use of (30) and integrating respectively (15), (16), and (24). However, the precise forms of these functions are not really needed for obtaining a closed expression for the analog metric.
It is of particular interest to discuss the above solution in the asymptotic limit, i.e., in the limit z → 0. Motivated by the asymptotic behavior of the O 2 holographic superconductor with d = 3 (see section 3.1)γ in the following we assume for general d with c d < 0. Then, it may be easily shown that in the limit z → 0 the sound speed squared tends to a constant c 2 s → d/(d + 4) < 1. However, in this limitγ → 1 so from equations (19) it follows that the limit z → 0 cannot be reached since we must have c 2 s ≥γ. This puts the constraint as to how close to the boundary is our analog metric applicable. Our analog model breaks down at a point z = z min which is the maximal root of the equation c 2 s =γ. For the minimal value of K, K min = 0, this equation reads In the case of a Schwarzschild AdS planar black hole, i.e., for χ = 0 and γ = 1 − (z/z h ) d , the integration in (29) can be easily performed yielding The condition c 2 s − γ = 0 now reads For example, for d = 4, the root z min is given by and for d = 3 we find numerically Hence, the simple prescription for an analog model is only valid from the point z min up to the location of the horizon at z h . In principle we could place the boundary of our model at z min and cut off the section of AdS from z = 0 to z min as it has been done in the Randall-Sundrum model [29,30]. However, as we aim to make a connection with CFT at the boundary of AdS and calculate the boundary entanglement entropy at z = 0, we would like to extend our model all the way down to the AdS boundary at z = 0. As we demonstrate in appendix B, such an an extension can be achieved by manipulating the equation of state by adding an external pressure. For a fluid with an external pressure of the form where α is a function of z, one finds the effective speed of sound Depending on the functional form ofγ we can choose α to make the quantityc 2 s satisfy equation (20) in the interval 0 ≤ z ≤ z h . For example, ifγ behaves as in (32) near z = 0, we can choose to obtain c 2 s ≥γ in the entire interval 0 ≤ z ≤ z h and lim z→0c 2 s = 1.
3 Analog bulk for the holographic superconductor Here we consider a concrete example of the analog metric of the form (2) for d = 3 related to the holographic superconductor. Instead of solving the field equations we will implement the already known solutions [19,20,21] into our analogue setup. Based on the known results we will construct approximate analytic expressions for γ corresponding to a chosen horizon temperature. With this we can calculate the entanglement entropy and by comparison with the results of Refs. [20,21] we can also find an analytic expression for χ(z). The analog geometry which we have derived in general form can be used to mimic these analytic expressions.

Holographic superconductor
Here we briefly review the minimal model of a holographic superconductor following Bobev et al [19]. We consider the minimal model of a holographic superconductor realized by an SO(3) × SO(3) invariant truncation of four-dimensional N = 8 gauged super-gravity [28]. The truncated action is where L involves two real dimensionless scalar fields λ and ϕ coupled to an Abelian gauge field A µ and gravity. The Lagrangian can be written as with potential The gauge coupling g sets the scale of AdS 4 via the relation P 2 = −6 [18] with scalar potential evaluated at a critical point. For the critical point λ = 0 related to SO(8) global symmetry [19,28] we obtain the relation g 2 2 = 1. The spacetime metric can be parameterized as where the functions γ and χ are to be determined by solving the field equations with appropriate boundary conditions. As we have noted in section 2, the value χ 0 ≡ χ(0) can be set to zero by rescaling the time coordinate. The field equations are derived in Ref. [19] for the gauge choice ϕ = 0 and A µ = (ψ(z), 0, 0, 0) and solved for two types of superconductors depending on the choice of boundary conditions, with non-trivial gauge fields and scalar condensates below some critical value of the temperature. Here and from here on we use the dimensionless variablez = z/ . As functions of temperature, the condensates O 1 and O 2 exhibit the second and first order phase transitions, respectively. The typical behavior of the condensates as functions of temperature is shown in Figs. 1 and 2 of Ref. [19]. The quantity ρ c which was chosen to set the units in these figures appears as a coefficient in the expansion ψ = µ − ρ c z + . . . near the AdS boundary. Physically, µ and ρ c are appropriately normalized chemical potential and charge density, respectively. From the field equations one can derive the following asymptotic expansions near z = 0: As we have mentioned, χ 0 can be set to 0 and the other coefficients in the expansion are related to physical quantities as follows: For λ = χ = 0 there are no condensates and the solution is just the Reisner-Nordstrom (RN) AdS 4 planar BH with and The charge squared Q 2 ranges between 0 and 3 where Q 2 = 0 corresponds to a Schwarzschild AdS 4 planar BH and and Q 2 = 3 to the maximal RN AdS 4 planar BH.

Entanglement entropy
Here we present the calculation of the holographic entanglement entropy in the analogue model discussed in section 3.1. Before we proceed to do that let us first discuss basic notions related to the entanglement entropy in general. Supose we have a quantum system with the density of states matrix ρ = |Ψ Ψ|. If we divide the total system into two subsystems A and B we define the reduced density matrix for the subsystem A by taking a partial trace over the subsystem B. i.e., ρ A = tr B |Ψ Ψ|. Then, the entanglement entropy defined as is the entropy for an observer who can access information only from the subsystem A and can receive no information from B. The subsystem B is analogous to the interior of a black hole horizon for an observer outside of the horizon. However, it is often not easy to compute the entanglement entropy, in particular in field theory in 3+1 or higher dimensions. A convenient description of the entanglement entropy is derived in a d + 1-dimensional field theory. It has been shown that the leading term of the entanglement entropy can be expressed as the area law [31,32] S A = a Area(∂A) where ∂A is the boundary of A, is an ultraviolet cutoff or the minimal length in the theory, and a is a constant which depends on the system. It is not accidental that this area law is of the same form as the Bekenstein-Hawking entropy of black holes in 3+1 dimensions which is proportional to the area of the event horizon, with the constants d = 3, a = 1/4, and equal to the Planck length. It is of particular relevance here that the entropy-area relation arises in the context of AdS/CFT duality. AdS/CFT, or gauge/gravity duality, is a correspondence between string theories in asymptotically anti-de Sitter bulk spacetimes and certain conformal field theories living on the holograhic boundary [33].
According to AdS/CFT, the entanglement entropy being basically tied to the gravity in the bulk, should reflect fundamental features of the boundary gauge theory. In this regard we will study the so called holographic entaglement entropy in 3+1 dimensions in the context of holographic superconductivity. There is a subtle difference between the usual entanglement entropy and holographic entanglement entropy: although both obey the area low, in the case of holographic entanglement entropy, as we will shortly demonstrate, for a fixed twodimensional subsystem on the holographic boundary the area depends on the geometry in the bulk. In particular, we expect that the holographic entanglement entropy in our model should exhibit the phase transition discussed in the previous section as demonstrated by the temperature dependence of the superconductor condensates [19].
The holographic entanglement entropy S in a 2+1-dimensional boundary CFT for a subsystem A that has an arbitrary one-dimensional boundary ∂A is defined by the following area law [34,35,36] where Σ is the two-dimensional static minimal surface in AdS 4 with boundary ∂A and Pl is the Planck length. As we are dealing with an analog geometry we will assume that there exist a minimal length, typically of the order of the atomic separation, below which the bulk description of the fluid fails. This length is referred to in the condensed matter literature as the coherence length, where the meaning of the word "coherence" is different from that in optics. Since it describes the distance over which the wave function of a BE condensate tends to its bulk value when subjected to a localized perturbation, it is also referred to as the healing length [37]. In analog gravity systems, a healing length hl plays the role of the Planck length [38,39,40,41,42] and for a BE gas is typically of order hl 1/(mc s ) where m is the boson mass. Hence, to calculate the entanglement entropy we use (56) with the Planck length Pl replaced by the healing length hl . Furthermore, we will identify the arbitrary scale with hl . Next we apply the prescription (56) to the geometry suggested in Refs. [20,34] illustrated in Fig. 1 and calculate the entropy S as a function of the strip width d for a fixed temperature.
Consider the bulk metric (2) with d = 3 and a surface Σ defined by the equation where z(x) is a function of x such that Σ extends into the bulk and is bounded by the perimeter of A as illustrated in Fig. 1. The induced metric σ ij on Σ defines the line element The area of Σ can be viewed as a functional where L and d are respectively the length and width of the strip, and Next we calculate the maximal area of Σ. Clearly, a maximum of Area corresponds to a minimum of I and variation of I yields the equation of motion for z. Instead of solving the equation of motion we will use the Hamiltonian approach. We define the conjugate momentum π = ∂L ∂z (61) and construct the Hamiltonian It can be easily shown that the equation of motion is satisfied if and only if the Hamiltonian is a constant of motion. In particular, at the bottom of the surface z = z * we have z = 0 and the Hamiltonian is equal to 2 /z 2 * . In this way we obtain the equation from which we can express z as Inserting this into (59) and changing the integration variable from x to z with dx = dz/z we obtain the area of the extremal surface Dividing this by 4 2 we obtain the entanglement entropy expressed as an integral over z S = Area The location of the bottom z * of the extremal surface is related to the strip width The integral in (66) is divergent near z = 0 and can be regularized by adding and subtracting a counter-term The entropy is then expressed as where the finite part reads Next we calculate the entanglement entropy using the bulk profile corresponding to an O 2 superconductor at fixed temperature. The reason why we specifically address the O 2 type is that the O 2 superconductor exhibits a first order phase transition which manifests itself as a discontinuity depicted in Fig. 2 of Ref. [19]. To calculate S fin we use a polynomial function We plot this function in Fig. 2. This choice is motivated by the superconductor bulk metric profile plotted in Fig. 7(b) of Ref. [21] for a fixed horizon temperature T = 0.61 × 10 −2 √ ρ c where ρ c is the charge density of the O 2 superconductor (see section 3.1). The function (71) is an analytic approximation to the bulk metric found by numerically solving the field equations of the holographic superconductor.
In Fig. 3 we plot S fin as a function of d/2. For comparison we plot in the same figure the entanglement entropies of a Schwarzschild AdS planar BH hole and a maximal RN AdS planar BH which have the same asymptotic behavior near z = 0. The metric profiles are determined so that the cubic terms are the same as in the O 2 superconductor case. Hence we have for the Schwarzschild AdS planar BH and for the maximal RN AdS planar BH. The coefficient of the quartic term in (74) was fixed by virtue of (52) and requirement γ RN (z RN ) = 0, where z RN = (−4/c 3 ) 1/3 z h is the location of the RN BH horizon. To complete our model we still have to determine the function χ(z). To do this we need to set the scale z h in relation to the previous works [19,20,21]. We will make a comparison of the scales at a fixed temperature T = 0.61 × 10 −2 √ ρ c , where ρ c is the charge density of dimension of length −2 . In Ref. [19] ρ c is chosen to set the scale whereas in Refs. [20,21] the In our approach the scale is set by z h so we have to find a relation between our z h andρ c or ρ c . To this end we compare the transition point d tr /2 = 0.744z h (Fig. 3) with that of Chakraborty [21] d tr /2 = 2.56ρ Using this we can express the horizon temperature of our configuration depicted in Fig. 2 in which yields Next, we express χ as a function of z using the expression (49) from section 3.1 in which we set χ 0 = 0, λ 1 = 0, keep the z 5 term and neglect the higher order terms. Hence we write where the coefficient λ 2 can be fixed from Eq. (50) with the value of O 2 deduced from Fig.  2 of Ref. [19]. At T = 0.61 × 10 −2 √ ρ c we find λ 2 = 1.1462 and using (79)

Summary and conclusions
We have derived an analog acoustic geometry which mimics a d + 1-dimensional asymptotic AdS geometry with a planar Black hole. In 3+1 dimensions, this geometry has been exploited as a holographic model for the 2+1-dimensional superconductor. We have applied this general analog geometry to a 3+1-dimensional bulk and calculated the entanglement entropy for a particular geometry obtained as solution related to the holographic O 2 superconductor. We have demonstrated that the entanglement entropy in our analog model exhibits the usual first order phase transition which characterizes the O 2 superconductor.
In this way we have confirmed the basic idea that a 3+1 AdS bulk with a planar BH can be realized in nature as a hydrodynamic analog gravity model. Moreover, the analog bulk metric can be parameterized so that the coefficient in the asymptotic expansion in powers of z are such that the dual AdS/CFT boundary field theory corresponds to the type O 2 superconductor. A procedure similar to the one described in section 3.2 can easily be applied to the case of type O 1 superconductor.
It would be of considerable interest to construct a concrete fluid system in the laboratory which would satisfy the properties of the analog geometry described above. It is fare to say that at this stage we cannot provide a clear proposal of how to prepare an adequate laboratory setup. As we are concerned with fluid velocities close to the speed of light c and sound speed close to c, we would need an essentially relativistic fluid. So far the only known realistic experimental set up for a relativistic-fluid laboratory is provided by high-energy colliders. The study of analog gravity in high energy collisions may in general improve our understanding of the dynamics of general relativistic fluids [43,44]. Maybe, with the advance of accelerator technology, one day it will be possible, e.g., by choosing appropriate heavy ions and specially designed beam geometry to obtain the desired equation of state and expansion flow of the fluid.
Then, it may be easily shown that equation (89) can be recast into the form where the matrix G µν is the inverse of the acoustic metric tensor with determinant G. Here m is an arbitrary mass parameter introduced to make G µν dimensionless and c s is the speed of sound defined by (11). The effective mass squared is given by Hence, the linear perturbations χ propagate in the effective metric (92) and acquire an effective mass.
In an equivalent field-theoretical description [1,26,46] the fluid velocity u µ is derived from the scalar field as u µ = ∂ µ θ/ √ X, and n and c s are expressed in terms of the Lagrangian and its first and second derivatives with respect to the kinetic energy term X = g µν θ ,µ θ ,ν . Obviously, the quantity √ X in this picture is identified with the specific enthalpy w. Equation (91) with (92) and (11) coincides with that of Ref. [1] derived in field theory with a general Lagrangian of the form L = L(X, θ).

B Effective sound speed with external pressure
Consider a fluid with internal variables p, ρ, and n. Suppose we apply to the fluid an external pressure p ext so that the total pressure is The speed of sound is still defined by but the thermodynamic TdS equation (12) must include the external pressure, i.e., dW = T ds + 1 n dP, where W = P + ρ n = w + p ext n .
Then the sound speed is given by For an isentropic process from (96) it follows dP = ndW, dρ = W dn, Now we make the following ansatz where α = α(z) will be determined by the requirement that the speed of sound is well defined as z → 0. With this ansatz we find a modified expression for the sound speed