Classical Lattice Systems Research Articles

High-Temperature Cluster Expansion for Classical and Quantum Spin Lattice Systems With Multi-Body Interactions

High-Temperature Cluster Expansion for Classical and Quantum Spin Lattice Systems With Multi-Body Interactions

DLR–KMS correspondence on lattice spin systems

The Dobrushin–Lanford–Ruelle condition (Dobrushin in Theory Prob Appl 17:582–600, 1970. https://doi.org/10.1137/1115049; Lanford and Ruelle in Commun Math Phys 13:194–215, 1969. https://doi.org/10.1007/BF01645487) and the classical Kubo–Martin–Schwinger (KMS) condition (Gallavotti and Verboven in Nuov Cim B 28:274–286, 1975. https://doi.org/10.1007/BF02722820) are considered in the context of classical lattice systems. In particular, we prove that these conditions are equivalent for the case of a lattice spin system with values in a compact symplectic manifold by showing that infinite-volume Gibbs states are in bijection with KMS states.

Self-correction in Wegner's three-dimensional Ising lattice gauge theory

Motivated by the growing interest in self-correcting quantum memories, we study the feasibility of self-correction in classical lattice systems composed of bounded degrees of freedom with local interactions. We argue that self-correction, including a requirement of stability against external perturbation, cannot be realized in system with broken global symmetries such as the 2d Ising model, but that systems with local, i.e. gauge, symmetries have the required properties. Previous work gave a three-dimensional quantum system which realized a self-correcting classical memory. Here we show that a purely classical three dimensional system, Wegner's 3D Ising lattice gauge model, can also realize this self-correction despite having an extensive ground state degeneracy. We give a detailed numerical study to support the existence of a self-correcting phase in this system, even when the gauge symmetry is explicitly broken. More generally, our results obtained by studying the memory lifetime of the system are in quantitative agreement with the phase diagram obtained from conventional analysis of the system's specific heat, except that self-correction extends beyond the topological phase, past the lower critical temperature.

Geometrical frustration in nonlinear photonic lattices

Geometrical frustration in either classical or quantum lattice systems is a phenomenon in which the impossibility of satisfying simultaneously all constraints of the system Hamiltonian leads to complex structures with degeneracy of the ground state. We show that geometrical frustration can be observed in a simple setup consisting of classical light propagating along a regular array of evanescently coupled photonic nonlinear waveguides arranged in a honeycomb geometry. Here, geometrical frustration emerges as competition between the number of waveguides and the need to satisfy the stationary constraints of the system. The phenomenology that exhibits these photonic arrays mimics the behavior observed in graphene-related spin lattices. Our results show that two-dimensional arrays of coupled photonic waveguides can be used as a proxy to study geometry frustration in diverse lattice systems.

Holographic encoding of universality in corner spectra

In numerical simulations of classical and quantum lattice systems, 2d corner transfer matrices (CTMs) and 3d corner tensors (CTs) are a useful tool to compute approximate contractions of infinite-size tensor networks. In this paper we show how the numerical CTMs and CTs can be used, {\it additionally\/}, to extract universal information from their spectra. We provide examples of this for classical and quantum systems, in 1d, 2d and 3d. Our results provide, in particular, practical evidence for a wide variety of models of the correspondence between $d$-dimensional quantum and $(d+1)$-dimensional classical spin systems. We show also how corner properties can be used to pinpoint quantum phase transitions, topological or not, without the need for observables. Moreover, for a chiral topological PEPS we show by examples that corner tensors can be used to extract the entanglement spectrum of half a system, with the expected symmetries of the $SU(2)_k$ Wess-Zumino-Witten model describing its gapless edge for $k=1,2$. We also review the theory behind the quantum-classical correspondence for spin systems, and provide a new numerical scheme for quantum state renormalization in 2d using CTs. Our results show that bulk information of a lattice system is encoded holographically in efficiently-computable properties of its corners.

Integrability of a deterministic cellular automaton driven by stochastic boundaries

We propose an interacting many-body space–time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (rule 54 of Bobenko et al 1993 Commun. Math. Phys. 158 127) on a finite one-dimensional lattice , and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying the existence of a unique nonequilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and k-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. The feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is suggested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.

Sensitive Dependence of Gibbs Measures at Low Temperatures

The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. We observe that for some classical lattice systems there are interactions exhibiting a related phenomenon of sensitive dependence of Gibbs measures: An arbitrarily small perturbation of the interaction can produce significant changes in the low-temperature behavior of its Gibbs measures. For some one-dimensional XY models we exhibit sensitive dependence of Gibbs measures for a (nearest-neighbor) interaction given by a smooth function, and for perturbations that are small in the smooth category. We also exhibit sensitive dependence of Gibbs measures for an interaction on a classical lattice system with finite-state space. This interaction decreases exponentially as a function of the distance between sites; it is given by a Lipschitz continuous potential in the configuration space. The perturbations are small in the Lipschitz topology. As a by-product we solve some problems stated by Chazottes and Hochman.

On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction

We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial correlation functions.

Bionic Coulomb phase on the pyrochlore lattice

A class of three-dimensional classical lattice systems with macroscopic ground state degeneracies, most famously the spin ice system, are known to exhibit ``Coulomb'' phases wherein long wavelength correlations within the ground state manifold are described by an emergent Maxwell electrodynamics. We discuss a more elaborate example of this phenomenon---the four-state Potts model on the pyrochlore lattice---where the long wavelength description now involves three independent gauge fields, as we confirm via simulation. The excitations above the ground state manifold are bions, defects that are simultaneously charged under two of the three gauge fields, and they exhibit an entropic interaction dictated by these charges. We also show that the distribution of flux loops exhibits a scaling with loop length and system size previously identified as characteristic of Coulomb phases.

A cluster expansion approach to renormalization group transformations

The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work treats the rigorous definition of the RG map for classical Ising-type lattice systems in the infinite volume limit at high temperature. A cluster expansion is used to justify the existence of the partial derivatives of the renormalized interaction with respect to the original interaction. This expansion is derived from the formal expressions, but it is itself well-defined and convergent. Suppose in addition that the original interaction is finite-range and translation-invariant. We will show that the matrix of partial derivatives in this case displays an approximate band property. This in turn gives an upper bound for the RG linearization.

Spectral Properties of the Renormalization Group at Infinite Temperature

The renormalization group (RG) approach is largely responsible for the considerable success that has been achieved in developing a quantitative theory of phase transitions. Physical properties emerge from spectral properties of the linearization of the RG map at a fixed point. This article considers RG for classical Ising-type lattice systems. The linearization acts on an infinite-dimensional Banach space of interactions. At a trivial fixed point (zero interaction), the spectral properties of the RG linearization can be worked out explicitly, without any approximation. The results are for the RG maps corresponding to decimation and majority rule. They indicate spectrum of an unusual kind: dense point spectrum for which the adjoint operators have no point spectrum at all, only residual spectrum. This may serve as a lesson in what one might expect in more general situations.

STATIONARY PATTERNS IN CNN-LIKE ENSEMBLES WITH MODIFIED CELL OUTPUT FUNCTIONS

Stationary pattern formation in ensembles of coupled bistable elements is investigated both analytically and by means of numerical simulation. The models considered are similar to cellular nonlinear networks (CNNs) — a well-known class of collective dynamical systems intended mainly for image processing — but differ from them in the type of nonlinear functions contained in their equations of motion — cell output functions. The main subject of interest is the transformation of initial conditions, treated as a representation of a half-tone image, into a steady-state pattern. In the analytical part the location of attractors and their basins of attraction in the phase space are estimated for two types of CNN-like systems. In the simulation part the equations of two lattice systems — a CNN and a CNN-like system with local negative coupling — are integrated numerically with initial conditions taken in the form of a sample halftone image. The dependence of the patterns established in both systems upon parameters is studied and compared. The present paper proves, that CNN-like systems with modified cell output functions may be studied analytically, and results available for conventional CNNs may be adapted to this wider class of lattice systems. From the application point of view, it is shown, that the modification of cell output functions under certain conditions does not lead to a breakdown of the system functionality. Moreover, an example is presented, where such a modification allows to introduce new functionality into a system (namely, controlling contour lines width in an edge-detecting system).

Energy landscape properties studied using symbolic sequences

We investigate a classical lattice system with N particles. The potential energy V of the scalar displacements is chosen as a ϕ 4 on-site potential plus interactions. Its stationary points are solutions of a coupled set of nonlinear equations. Starting with Aubry’s anti-continuum limit it is easy to establish a one-to-one correspondence between the stationary points of V and symbolic sequences σ = ( σ 1 , … , σ N ) with σ n = + , 0 , − . We prove that this correspondence remains valid for interactions with a coupling constant ϵ below a critical value ϵ c and that it allows the use of a “thermodynamic” formalism to calculate statistical properties of the so-called “energy landscape” of V . This offers an explanation for why topological quantities of V may become singular, like in phase transitions. In particular, we find that the saddle index distribution is maximum at a saddle index n s max = 1 / 3 for all ϵ < ϵ c . Furthermore there exists an interval ( v ∗ , v max ) in which the saddle index n s as a function of the average energy v ̄ is analytical in v ̄ and it vanishes at v ∗ , above the ground state energy v gs , whereas the average saddle index n ̄ s as a function of the energy v is highly nontrivial. It can exhibit a singularity at a critical energy v c and it vanishes at v gs , only. Close to v gs , n ̄ s ( v ) exhibits power law behavior which even holds for noninteracting particles.

Long-range order in Gibbs lattice classical linear oscillator systems

The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generalized Peierls argument and two different contour bounds are our main tools.

Ground States in Relatively Bounded Quantum Perturbations of Classical Lattice Systems

We consider ground states in relatively bounded quantum perturbations of classical lattice models. We prove general results about such perturbations (existence of the spectral gap, exponential decay of truncated correlations, analyticity of the ground state), and also prove that in particular the AKLT model belongs to this class if viewed at large enough scale. This immediately implies a general perturbation theory about this model.

Entropy, Entanglement, and Area: Analytical Results for Harmonic Lattice Systems

We revisit the question of the relation between entanglement, entropy, and area for harmonic lattice Hamiltonians corresponding to discrete versions of real free Klein-Gordon fields. For the ground state of the d-dimensional cubic harmonic lattice we establish a strict relationship between the surface area of a distinguished hypercube and the degree of entanglement between the hypercube and the rest of the lattice analytically, without resorting to numerical means. We outline extensions of these results to longer ranged interactions, finite temperatures, and for classical correlations in classical harmonic lattice systems. These findings further suggest that the tools of quantum information science may help in establishing results in quantum field theory that were previously less accessible.

Violations of local equilibrium and linear response in classical lattice systems

We quantitatively and systematically analyze how local equilibrium, and linear response in transport are violated as systems move far from equilibrium. This is done by studying heat flow in classical lattice models with and without bulk transport behavior, in 1–3 dimensions, at various temperatures. Equations of motion for the system are integrated numerically to construct the non-equilibrium steady states. Linear response and local equilibrium assumptions are seen to break down in a similar manner. We quantify the breakdown through the analysis of both microscopic and macroscopic observables and examine its transformation properties under general redefinitions of the non-equilibrium temperature.

Sinai-Ruelle-Bowen Measures for Lattice Dynamical Systems

For weakly coupled expanding maps on the unit circle, Bricmont and Kupiainen showed that the Sinai-Ruelle-Bowen (SRB) measure exists as a Gibbs state. Via thermodynamic formalism, we prove that this SRB measure is indeed the unique equilibrium state for a Holder continuous potential function on the infinite dimensional phase space. For a more general class of lattice systems that are small perturbations of the uncoupled map lattice, we present the variational principle, the entropy formula, and the formula for the potential function for the SRB measures. For coupled map lattices with nearest neighbor interactions, we give an explicit formula of the potential function for the SRB measure and consequently, obtain the entropy in terms of coupling parameters.

Quantum criticality for few-body systems: path-integral approach.

We present the path-integral approach to treat quantum phase transitions and critical phenomena for few-body quantum systems. Allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. Imposing the constraint that any change in space time must preserve the scaling invariance of Brownian paths, we show that the mapped classical lattice system has a known scaling behavior when the particle is free, which breaks down when the strength of the interaction potential reaches a certain value. In principle, any quantity with known scaling behavior may be used to determine the transition point. We illustrate the method by numerically evaluating the correlation length and the radial mean distance for a system composed of a single particle in the presence of an attractive Pöschl-Teller potential in one and three dimensions. The method is general and has potential applicability for large systems.

On Algebraic Integrability of the Deformed Elliptic Calogero-Moser Problem

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the ϕ 4 type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms — the (Rashba)–Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.