Table of contents

The Lebwohl–Lasher model describes the isotropic–nematic transition in liquid crystals. In two dimensions, where its continuous symmetry cannot break spontaneously, it is investigated numerically since decades to verify, in particular, the conjecture of a topological transition leading to a nematic phase with quasi-long-range order. We use scale invariant scattering theory to exactly determine the renormalization group fixed points in the general case of N director components (RP^N−1 model), which yields the Lebwohl–Lasher model for N = 3. For N > 2 we show the absence of quasi-long-range order and the presence of a zero temperature critical point in the universality class of the O(N(N + 1)/2 − 1) model. For N = 2 the fixed point equations yield the Berezinskii–Kosterlitz–Thouless transition required by the correspondence RP¹ ∼ O(2).

This review is a primer on recently established geometric methods for observables in quantum field theories. The main emphasis is on amplituhedra, i.e. geometries encoding scattering amplitudes for a variety of theories. These pertain to a broader family of geometries called positive geometries, whose basics we review. We also describe other members of this family that are associated with different physical quantities and briefly consider the most recent developments related to positive geometries. Finally, we discuss the main open problems in the field. This is a Topical Review invited by Journal of Physics A: Mathematical and Theoretical.

As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This consists in a quasi-universal hierarchy of equations, partly unifying an integro-differential generalization of the Painlevé II hierarchy, the finite-time solutions of the Kardar–Parisi–Zhang equation, multi-critical fermions at finite temperature and a notable solution to the Zakharov–Shabat system associated to the largest real eigenvalue in the real Ginibre ensemble. As a byproduct, we obtain the explicit unique solution to the inverse scattering transform of the Zakharov–Shabat system in terms of a Fredholm determinant.

The pressure and flow statistics of Darcy flow through a three-dimensional random permeable medium are expressed as a path integral in a form suitable for evaluation by simulated annealing. There are several advantages to using simulated annealing for this problem: (i) any probability distribution can be used for the permeability, (ii) there is no need to invert the transmissibility matrix which, while not a factor for single-phase flow, offers distinct advantages for multiphase flow, and (iii) the action used for simulated annealing, whose extremum yields Darcy's law, is eminently suitable for coarse graining by integrating over the short-wavelength degrees of freedom. We show that the pressure and flow statistics obtained by simulated annealing are in excellent agreement with those obtained from standard finite-volume calculations.

We propose a simple Langevin equation as a generator for a noise process with Laplace-distributed values (pure exponential decays for both positive and negative values of the noise). We calculate explicit expressions for the correlation function, the noise intensity, and the correlation time of this noise process and formulate a scaled version of the generating Langevin equation such that correlation time and variance or correlation time and noise intensity for the desired noise process can be exactly prescribed. We then test the effect of the noise distribution on a classical escape problem: the Kramers rate of an overdamped particle out of the minimum of a cubic potential. We study the problem both for constant variance and constant intensity scalings and compare to an Ornstein–Uhlenbeck process with the same noise parameters. We demonstrate that specifically at weak fluctuations, the Laplace noise induces more frequent escapes than its Gaussian counterpart while at stronger noise the opposite effect is observed.

A heterostructure composed of two parallel homogeneous layers is studied in the limit as their widths l₁ and l₂, and the distance between them r shrinks to zero simultaneously. The problem is investigated in one dimension and the squeezing potential in the Schrödinger equation is given by the strengths V₁ and V₂ depending on the layer thickness. A whole class of functions V₁(l₁) and V₂(l₂) is specified by certain limit characteristics as l₁ and l₂ tend to zero. The squeezing limit of the scattering data a(k) and b(k) derived for the finite system is shown to exist only if some conditions on the system parameters V_j, l_j, j = 1, 2, and r take place. These conditions appear as a result of an appropriate cancellation of divergences. Two ways of this cancellation are carried out and the corresponding two resonance sets in the system parameter space are derived. On one of these sets, the existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac's delta function, contrary to the widespread opinion on the non-existence of bound states in δ'-like systems. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.

We present an analytic proof of the existence of phase transition in the large N limit of certain random noncommutative geometries. These geometries can be expressed as ensembles of Dirac operators. When they reduce to single matrix ensembles, one can apply the Coulomb gas method to find the empirical spectral distribution. We elaborate on the nature of the large N spectral distribution of the Dirac operator itself. Furthermore, we show that these models exhibit both a single and double cut region for certain values of the order parameter and find the exact value where the transition occurs.

We show that gauge-independent terms in the one-loop and multi-loops β-functions of the standard model can be exactly computed from the Wetterich functional renormalization of a matrix model. Our framework is associated to the finite spectral triple underlying the computation of the standard model Lagrangian from the spectral action of noncommutative geometry. This matrix-Yukawa duality for the β-function is a first hint towards understanding the renormalization of the noncommutative standard model conceptually, and provides a novel computational approach for multi-loop β-functions of particle physics models.

The Volterra lattice admits two non-Abelian analogs that preserve the integrability property. For each of them, the stationary equation for non-autonomous symmetries defines a constraint that is consistent with the lattice and leads to Painlevé-type equations. In the case of symmetries of low order, including the scaling and master-symmetry, this constraint can be reduced to second order equations. This gives rise to two non-Abelian generalizations for the discrete Painlevé equations and and for the continuous Painlevé equations P₃, and P₅.

We construct SU(2|1), d = 1 supersymmetric models based on the coupling of dynamical and semi-dynamical (spin) multiplets, where the interaction term of both multiplets is defined on the generalized chiral superspace. The dynamical multiplet is defined as a chiral multiplet (2, 4, 2), while the semi-dynamical multiplet is associated with a multiplet (4, 4, 0) of the mirror type.

In two-party, two-input and two-output measurement scenario only relevant Bell's inequality is the Clauser–Horne–Shimony–Holt (CHSH) form. They also provide the necessary and sufficient conditions (NSCs) for local realism. Any other form, such as, Clauser–Horne and Wigner forms reduce to the CHSH one. The standard Leggett–Garg inequalities, proposed for testing incompatibility between macrorealism and quantum theory, are often considered to be the temporal analog of CHSH inequalities. However, they do not provide the NSCs for macrorealism. There is thus scope of formulating new macrorealist inequalities inequivalent and stronger than the standard Leggett–Garg inequalities. In this paper, we propose two different classes of macrorealistic inequalities. A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory, and the other class of inequalities are inequivalent to the all the other formulations of Leggett–Garg inequalities both in macrorealist model and in quantum theory. The latter class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett–Garg inequalities fails to do so. We extend the formulations of inequivalent Leggett–Garg inequalities to the four-time and two-time measurement scenarios. Further, we provide a brief discussion about the alternative formulation of macrorealism known as the no-signaling in time (NSIT) conditions.

We establish a convex resource theory of non-Markovianity inducing information backflow under the constraint of small time intervals within the temporal evolution. We identify the free operations and a generalized bona-fide measure of non-Markovian information backflow. The framework satisfies the basic properties of a consistent resource theory. The proposed resource quantifier is lower bounded by the optimization free Rivas–Huelga–Plenio (RHP) measure of non-Markovianity. We next define the robustness of non-Markovianity and show that it can directly linked with the RHP measure of non-Markovianity through a lower bound. This enables a physical interpretation of the RHP measure. We further relate robustness of non-Markovianity with the quantum capacity of dephasing channels.

In this article we investigate driven dissipative quantum dynamics of an ensemble of two-level systems given by a Markovian master equation with collective and local dissipators. Exploiting the permutation symmetry in our model, we employ a phase space approach for the solution of this equation in terms of a diagonal representation with respect to certain generalized spin coherent states. Remarkably, this allows to interpolate between mean field theory and finite system size in a formalism independent of Hilbert-space dimension. Moreover, in certain parameter regimes, the evolution equation for the corresponding quasiprobability distribution resembles a Fokker–Planck equation, which can be efficiently solved by stochastic calculus. Then, the dynamics can be seen as classical in the sense that no entanglement between the two-level systems is generated. Our results expose, utilize and promote techniques pioneered in the context of laser theory, which can be powerful tools to investigate problems of current theoretical and experimental interest.

We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrödinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system's eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet et al (2019 New J. Phys.21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.

In this work the momentum spreading of a multidimensional hydrogenic system in highly excited (Rydberg) states is quantified by means of the Rényi and Shannon entropies of its momentum probability density. These quantities, which rest at the core of numerous fields from atomic and molecular physics to quantum technologies, are determined by means of a methodology based on the strong degree-asymptotics of a modified ${\mathfrak{L}}_{q}$-norm of the Gegenbauer polynomials which control the wavefunctions of these states in momentum space. The leading term of these entropic quantities is found from first principles, i.e. by means of the Coulomb potential parameters (space dimensionality, the nuclear charge) and the states's hyperquantum numbers, in a rigorous simple manner. It is shown that they fulfill a logarithmic growth scaling law with the principal hyperquantum number n which characterizes the Rydberg state.

The quest for extension of holographic correspondence to non-relativistic sectors naturally includes Schrödinger backgrounds and their field theory duals. In this paper we study the holography by probing the correspondence with pulsating strings. The case we consider is pulsating strings in five-dimensional Schrödinger space times five-torus T^1,1, which has as field theory dual a dipole CFT. First we find particular pulsating string solutions and then semi-classically quantize the theory. We obtain the wave function of the problem and thoroughly study the corrections to the energy, which by duality are supposed to give anomalous dimensions of certain operators in the dipole CFT.

We study the epidemic spreading on spatial networks where the probability that two nodes are connected decays with their distance as a power law. As the exponent of the distance dependence grows, model networks smoothly transition from the random network limit to the regular lattice limit. We show that despite keeping the average number of contacts constant, the increasing exponent hampers the epidemic spreading by making long-distance connections less frequent. The spreading dynamics is influenced by the distance-dependence exponent as well and changes from exponential growth to power-law growth. The observed power-law growth is compatible with recent analyses of empirical data on the spreading of COVID-19 in numerous countries.

Renewal processes generated by a power-law distribution of intervals with tail index less than unity are genuinely non-stationary. This issue is illustrated by a critical review of the recent paper by Barma, Majumdar and Mukamel (2019 J. Phys. A52 254001), devoted to the investigation of the properties of a specific one-dimensional equilibrium spin system with long-range interactions. We explain why discarding the non-stationarity of the process underlying the model leads to an incorrect expression of the critical spin–spin correlation function, even when the system, subjected to periodic boundary conditions, is translation invariant.

Godrèche, in Comment on 'Fluctuation dominated phase ordering at a mixed order transition' [2021 J. Phys. A: Math. Theor.54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor.52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper.