Gibbs states of a one dimensional quantum lattice

A one dimensional infinite quantum spin lattice with a finite range interaction is studied. The Gibbs state in the infinite volume limit is shown to exist as a primary state of a UHF algebra. The expectation value of any local observables in the state as well as the mean free energy depend analytically on the potential, showing no phase transition. The Gibbs state is an extremal KMS state.On leave from Research Institute for Mathematical Sciences, Kyoto University Kyoto, Japan.     

A characterization of Gibbs states of lattice boson systems

We consider lattice boson systems interacting via potentials which are superstable and regular. By using the Wiener integral formalism and the concept of conditional reduced density matrices we are able to give a characterization of Gibbs (equilibrium) states. It turns out that the space of Gibbs states is nonempty, convex, and also weak-compact if the interactions are of finite range. We give a brief discussion on the uniqueness of Gibbs states and the existence of phase transitions in our formalism.     

On the Gibbs states for one-dimensional lattice Boson systems with a long-range interaction

We consider an infinite chain of interacting quantum (anharmonic) oscillators. The pair potential for the oscillators at lattice distanced is proportional to {d ²[log(d+1)]F(d)}^–1 where _rZ [rF(r)]^–1 < . We prove that for any value of the inverse temperature> 0 there exists a limiting Gibbs state which is translationally invariant and ergodic. Furthermore, it is analytic in a natural sense. This shows the absence of phase transitions in the systems under consideration for any value of the thermodynamic parameters.     

On the equivalence of KMS and Gibbs conditions for states of quantum lattice systems

The equivalence of a Gibbsian equilibrium condition and the KMS condition is proven for one-dimensional quantum lattice systems with a finite range interaction at arbitrary temperature, and for quantum lattice systems of arbitrary dimension, with a finite body interaction, at high temperature.     

Low temperature expansions for the Gibbs states of weakly interacting quantum Ising lattice systems

Low temperature expansions for the Gibbs states of weakly interacting transverse Ising-like models are developed, by conditioning the states on a sub-algebra of observables. The conditioned states have effective classical Hamiltonians which are estimated by the solution to a simple implicit equation. Provided the interaction is sufficiently weak but fixed independent of temperature, and the temperature is sufficiently low, exponential clustering of the correlation functions holds.     

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.As a by-product, a new proof is given of the fact that for a large class of potentials, a canonical Gibbs measure is a mixture of grand canonical measures. Thus the equivalence of ensembles is obtained in the sense of identical correlation functions.     

On surface states of one dimensional semiinfinite systems

Surface state energies are computed for a one dimensional semi-infinite system, which accommodates simultaneously (1) surface distortion (2) adsorbed impurity (3) finite distance of separation between vacuum potential discontinuity and surface atom and (4) negative (attractive) bulk potential. The locations of surface state energies in the band gaps are indicated; further, nature of variations of surface state energies with relevant parameters, covering a wide range, is studied. The results are discussed critically; besides, they are compared with similar ones obtained on the basis of two other models, one due to Aerts and the other due to Neuberger and Rutherford Fischer.The authors are thankful to Prof. G. B. Mitra, Head of the Department of Physics, for his endless interest and encouragement. They are further grateful to the esteemed referee who reviewed the paper and suggested so kindly several modifications.     

Translation symmetry breaking in four dimensional lattice gauge theories

We consider lattice gauge theories with finite abelian groupG in the weak coupling regime. It is shown that there is only one translation invariant equilibrium state for the infinite system. In four dimensions we construct a nontranslation invariant equilibrium state, describing an infinite system with localized magnetic flux tube, starting and ending at infinity.     

Discrete symmetry analysis of lattice systems

Discrete dynamical systems and mesoscopic lattice models are considered from the point of view of their symmetry groups. Some peculiarities in behavior of discrete systems induced by symmetries are pointed out. We reveal also the group origin of moving soliton-like structures similar to “spaceships” in cellular automata.     

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 Physics 25(1):57–72 (1981). The techniques of this paper are based on those employed by Gross.     

On uniqueness of KMS states of one-dimensional quantum lattice systems

We present a proof of the theorem on the uniqueness of KMS states of one-dimensional quantum lattice systems, which is based on some equicontinuity.     

Equilibrium states for a one dimensional lattice gas

The thermodynamic equilibrium state can be defined directly for an infinite system via an equilibrium condition or via the variational principle. Both definitions are used to calculate the equilibrium state for a one dimensional lattice gas with finite range interactions.     

Gibbs states and equilibrium states for finitely presented dynamical systems

We studyfinitely presented dynamical systems (which generalize Axiom A systems) and show that the notions of equilibrium states and Gibbs states (for Hölder continuous functions) are equivalent. Our results extend those of Ruelle, Haydn, and others on Axiom A dynamical systems and statistical mechanics.     

Simulation of two dimensional systems with two dimensional electrostatics

Some of the problems associated with Monte Carlo and molecular dynamics simulations of three dimensional ionic and dipolar systems are discussed, with emphasis on the use of periodic boundary conditions (PBC). It is shown that analogous problems may arise in two dimensional systems provided that the interactions are two dimensional electrostatic interactions, that is, interactions derived from the two dimensional Laplace equation. The PBC hamiltonian is evaluated by considering the appropriate two dimensional lattice sums, and a computable form for the effective pair interactions in PBC developed. The idea of an external dielectric constant is introduced and its effects included in the PBC hamiltonian. Formulae for evaluating the dielectric constant from a simulation with any external dielectric constant are given. Perturbation formulae showing the effects on the structure and mean square dipole moment of a dipolar system which are caused by a change in external dielectric constant are derived. A formula for the shift in mean square dipole moment of an ionic system is also developed. The problem of interpreting results from such simulations is discussed briefly.     

Metastable states of quantum lattice systems

We extend the characterisation of metastability, as given in [1] for classical systems, to show that quantal lattice systems with suitable long range forces can support metastable states.     

Quantum fluctuations in quantum lattice systems with continuous symmetry

We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems atT=0. Our analysis is based on Pitaevskii and Stringari's idea that the uncertainty relation can be employed to show quantum fluctuations. For one-dimensional systems, it is shown that the ground state is invariant under a continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than|r| ^–d+1 whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice systems having not-too-long-range interactions.     

Spontaneous translation symmetry breakdown for quantum lattice systems

We study the set of equilibrium states for quantum lattice states in the presence of a translation symmetry of the model. We derive a characterization of the spontaneous breaking of this symmetry, i.e., the decomposition of an invariant equilibrium state into a mixture of noninvariant equilibrium states, in terms of the separability in mean energy of these states for a class of perturbed dynamics.     

On the absence of spontaneous breakdown of continuous symmetry for equilibrium states in two dimensions

Using the Bogoliubov inequality, we extend previously known results concerning the absence of continuous symmetry breakdown for equilibrium states of certain quantum and classical lattice, and continuum systems in two space dimensions.Partially supported by the N.S.F. under grant MCS 7801433.Partially supported by the N.S.F. under grant MCS 7906633.