Classical Lattice Systems Research Articles

The residual entropy for a class of one-dimensional classical lattice models

The zero-temperature limit of the characteristic equation of the transfer matrix is derived for one-dimensional classical lattice systems with a finite one-point configuration space and finite-range interactions. In this way, the author obtains an explicit polynomial equation for the residual entropy, which only involves the counting of some specific periodic ground-state configurations. The formula is applied to some well known models and compared with similar calculations in the literature.

Nonperiodic order in classical lattice systems

Lattice gas models without periodic ground states are discussed. It is shown that for most interactions, that is, generically, ground states have long range order but not the perfect order of true periodicity.

The ground state and its entropy for a class of one-dimensional classical lattice systems

We calculate explicitly the zero-temperature limit of the finite temperature equilibrium state and its entropy for a class of one-dimensional Ising models with an interaction of range n. For that purpose we extend the usual transfer matrix method to ground states. The interaction consists of a ferromagnetic nearest neighbour term and a competing antiferromagnetic interaction with the nth neighbour.

Monte Carlo simulations for two-dimensional quantum systems

The Trotter-Suzuki transformation has been used to obtain the classical representation ford-dimensional lattice systems with boson and fermion degrees of freedom. A Monte Carlo algorithm for the equivalent (d+1)-dimensional classical system is presented. Numerical results are shown for the Heisenberg-spin-glass, the XY model and the spinless fermion lattice gas in two dimensions.

Tiling problems and undecidability in the cluster variation method

In cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small, finite subsystem (cluster), approximations being made for the influence of the surrounding system. In this context we study, for translation-invariant classical lattice systems, the conditions under which a state for a cluster admits an extension to a global translation-invariant state. This extension problem is related to undecidable tiling problems. The implication is that restrictions of global translation-invariant states cannot be characterized purely locally in general. This means that there is an unavoidable element of uncertainty in the application of a cluster approximation.

Non-translation Invariant Product States of Classical Lattice Systems

The notion of symmetric states is extended in such a way as to include the description of non-translation invariant mean field systems. A complete characterization of such states is given by means of non-translation invariant product states.

An order-by-order construction of low-temperature phase diagrams for classical lattice systems

An inductive algorithm is presented for the construction of phase diagrams by means of the low-temperature expansion technique. First the phase diagram is studied in the set of formal series. In each step, properties of this phase diagram are related to extremal elements of some family of convex sets. Approximations of the phase diagram in orderN are obtained by truncating all formal series at theNth term.

Some examples concerning the global Markov property

We present examples of interactions of classical lattice systems whose extremal Gibbs states fail to have the global Markov property. One of the examples is translation invariant.

Detailed balance and critical slowing down for classical lattice systems

For classical lattice systems we prove rigorously a relation between the fluctuations and the relaxation times for a dynamics given by a detailed balance semigroup. In particular we give a rigorous proof of the conventional theory of critical slowing down.

On some variational approximations in two-dimensional classical lattice systems

A new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster-variation methods, among them the well-known Bethe-Peierls and Kramers-Wannier approximations. The limiting behavior of a hierarchical sequence of cluster-variation approximations, the so-calledC hierarchy, is discussed. It is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsfn on the free energy per lattice sitef and thatfn → f asn → ∞. Our results are based on extension theorems for states given on subsets of the lattice, which might be of some independent interest, and on an application of transfer matrix concepts to the variational characterization of translation-invariant equilibrium states.

Discrete lattice systems and the equivalence of microcanonical, canonical and grand canonical Gibbs states

It is proven that a microcanonical Gibbs measure on a classical discrete lattice system is a mixture of canonical Gibbs measures, provided the potential is “approximately periodic,” has finite range and possesses a commensurability property. No periodicity is imposed on the measure. When the potential is not approximately periodic or does not have the commensurability property, the inclusion does not hold.

Stochastic simulation of classical lattice systems.

Stochastic simulation of classical lattice systems.

An infinite chain of inequalities for correlation functions of classical lattice systems

In quantum-mechanical systems the moments of the symmetrized time-dependent autocorrelation function satisfy an infinite set of chained inequalities. By analogy the same inequalities are derived for correlation functions of classical lattice systems.

Detailed balance and equilibrium

For classical lattice systems, an infinite set of jump-processes satisfying the condition of detailed balance is found. It is proved that any state invariant for these processes is an equilibrium state, providing a new characterization of DLR-states by means of the notion of detailed balance. This extends previous results, proved in one and two dimensions.

On solvable models in classical lattice systems

We consider one dimensional classical lattice systems and an increasing sequence ℒ n (n=1,2, ...) of subsets of the state space; ℒ n takes into account correlations betweenn successive lattice points. If the interaction range of the potential is finite, we prove that the equilibrium states defined by the variational principle are elements of {ℒ n } n<∞. Finally we give a new proof of the fact that all faithful states of ℒ n are DLR-states for some potential.

(P3, T8) Energy-entropy inequalities for classical lattice systems

Classical dynamical entropy is an important tool to analyse the efficiency of information transmission in communication processes. Quantum dynamical entropy was first studied by Connes, Størmer and Emch. Since then, there have been many attempts to formulate or compute the dynamical entropy for some models. Here we review four formulations due to 1.(a) Connes, Narnhofer and Thirring,2.(b) Ohya,3.(c) Accardi, Ohya and Watanabe,4.(d) Alicki and Fannes. We consider mutual relations between these formulations and we show some concrete computations for a model.

High-temperature differentiability of lattice Gibbs states by Dobrushin uniqueness techniques

We establish conditions for the differentiability, to any order, of the Gibbs states of classical lattice systems with arbitrary compact single-spin space and with interactions in the Dobrushin uniqueness region. The derivatives are expressed as series expansions and are shown to be continuous on the uniqueness region. We also provide a procedure for estimating the size of the derivatives. These results verify a conjecture of L. Gross and extend his results in “Absence of second-order phase transitions in the Dobrushin uniqueness region,”Journal of Statistical Physics25(1):57–72 (1981). The techniques of this paper are based on those employed by Gross.

Energy-entropy inequalities for classical lattice systems

A new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities. Their physical meaning is found to express thermodynamic stability. We demonstrate the applicability of the inequalities in specific models.

The mermin-wagner phenomenon and cluster properties of one- and two-dimensional systems

We give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one- and two-dimensional quantum and classical lattice and continuum systems. For a class of models verifying infrared bounds our conditions are necessary and sufficient. Using the same techniques we obtain “a priori” bounds on clustering for systems with continuous symmetry, improving results of Jasnow and Fisher.

Equivalence of Gibbs ensembles for classical lattice systems

Equivalence of Gibbs ensembles for classical lattice systems