Analyticity and uniqueness of the invariant equilibrium states for general spin 1/2 classical lattice systems

Asano-Ruelle-Slawny method is generalized to discuss analyticity and uniqueness of the correlation functions in terms of the group structure associated with any lattice systems. The use of Poisson formula for abelian groups gives a simple method to obtain explicit domains where the above properties are verified.     

On the uniqueness of the equilibrium state for Ising spin systems

We show that for an Ising spin system of arbitrary spin with a ferromagnetic pair interaction and a periodic external magnetic field there is a unique equilibrium state if and only if the magnetization is continuous with respect to a uniform change in the external field. Hence, if the critical temperatureT _c is defined as the temperature where the spontaneous magnetization (which is a non-increasing function of the temperature) becomes positive, then the equilibrium state is unique forT>T _c and is non-unique forT<T _c (when the external field is zero). This implies that the correlation functions have a cluster property forT>T _c .We also show that for an anti-ferromagnet consisting of two sublattices there is a unique equilibrium state if and only if the staggered magnetization is continuous with respect to a change in the staggered field.Supported in part by U.S.A.F.O.S.R. under contract F 44620-71-C-0013, P001.     

Existence and uniqueness for random one-dimensional lattice systems

Existence and uniqueness are shown for the fixed point problem pertinent to hopping transport in one-dimension with random transfer rates. Continuity properties of the solution are exhibited. The connection with Dyson's treatment of the linear harmonic chain with random masses is established.     

Existence and uniqueness of Gibbs states for a statistical mechanical polyacetylene model

One-dimensional polyacetylene is studied as a model of statistical mechanics. In a semiclassical approximation the system is equivalent to a quantumXY model interacting with unbounded classical spins in one-dimensional lattice spaceZ. By establishing uniform estimates, an infinite-volume-limit Hilbert space, a strongly continuous time evolution group of unitary operators, and an invariant vector are constructed. Moreover, it is proven that any infinite-limit state satisfies Gibbs conditions. Finally, a modification of Araki's relative entropy method is used to establish the uniqueness of Gibbs states.     

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.     

Canonical and grand canonical Gibbs states for continuum systems

It is shown that for a large class of interactions any canonical Gibbs state satisfying a natural temperedness condition is a mixture of Gibbs states with appropriate activities, and vice versa. Some general results on Gibbs states and canonical Gibbs states are established. In particular, a differential characterization of Gibbs states is given.     

Calculation method for the continuum states of atomic systems

In the present work, we develop a calculational method of solving the scattering equations for spherically symmetric potentials by expanding the solutions on Coulomb functions. We utilize a multistep integration scheme together with the standard partial wave analysis in a region where the potential term dominates. The method applies to any physical problem expressed as [? ² + V(r) + k ²]ψ(r) = 0, while the extension of the method to more general scattering problems is briefly discussed. At present, we demonstrate a two-step Coulomb-fitted integration scheme by calculating the short-range scattering phase shifts for various potentials V (r).     

Thermodynamic equilibrium in isolated spin systems

A computational study of the redistribution of magnetization in systems of 2–15 coupled spins is presented, exhibiting the principles that govern the application of thermodynamics and statistical mechanics to isolated systems and emphasizing the meaning of equilibrium. In general, the equilibrium distribution is not uniform, i.e., the systems are not ergodic, because a nonuniform initial distribution sets the values of a large number of auxiliary nonphysical constants of the motion.     

Ground states of quantum spin systems

We prove that ground states of quantum spin systems are characterized by a principle of minimum local energy and that translationally invariant ground states are characterized by the principle of minimum energy per unit volume.     

A uniqueness theorem for anticommutation relations and commutation relations of quantum spin systems

Under quite natural assumptions we prove that a classical spin lattice has essentially two natural quantum extensions: The quantum spin lattice and the Fermi system. Moreover we derive a transformation from a commuting pair of Fermi systems to a purely anticommuting Fermi system.     

Stability and equilibrium states of infinite classical systems

We prove that any stationary state describing an infinite classical system which is stable under local perturbations (and possesses some strong time clustering properties) must satisfy the classical KMS condition. (This in turn implies, quite generally, that it is a Gibbs state.) Similar results have been proven previously for quantum systems by Haag et al. and for finite classical systems by Lebowitz et al.Supported by N.S.F. Grant MPS 71-03375 A03. Part of this work carried out at the Courant Institute where it was supported by N.S.F. Grant GP-37069X.Supported in part by AFOSR Grant #73-2430 and N.S.F. Grant MP S75-20638.Supported by N.S.F. Grant # GP33136X-2. Part of this work was carried out at the Institute for Advanced Study.     

Dobrushin states for classical spin systems with complex interactions

We consider a classical spin system on the hypercubic lattice with a general interaction of the form $$ H = \frac{\beta } {4}\sum\limits_{\begin{array}{*{20}c} {x,y:} \\ {|x - y| = 1} \\ \end{array} } {|s_x - s_y | - h} \sum\limits_x {x{}_x + } \sum\limits_A {\lambda _A \prod\limits_{y \in A} {S_y } } $$ are the spin variables, Β is the inverse temperature,h is the magnetic field, andλ _A are translation-invariant coupling constants satisfyingλ _A = 0 if diamA > l. No symmetry relating the configurationss ={sinx} and-s=-s _x is assumed. In dimension d-3, we construct low-temperature States which break the translation invariance of the system by introducing so-called Dobrushin boundary conditions which force a horizontal interface into the system. In contrast to previous constructions, our methods work equally well for complex interactions, and should therefore be generalizable to quantum spin systems.     

Bound states in the continuum in quasiperiodic systems

We first propose the existence of bound states in the continuums (BICs) in quasiperiodic systems. Owing to long-range correlation, destructive interference may occur in quasiperiodic systems with higher generation order. Occurrences of BICs in Fibonacci quantum wells studied by localization analysis and gap map method are proposed.     

Effects of temperature on the relaxation to equilibrium and stationary nonequilibrium states of some Langevin systems

We investigate the effects of temperature on the properties of the time relaxation to equilibrium and nonequilibrium steady states of correlation functions of some Langevin harmonic systems. We consider commonly used dissipative and conservative Langevin dynamics, and show that the time relaxation rate depends on the temperature in the case of thermal reservoirs at different temperatures connected to the system, but it does not happen in the case of relaxation to equilibrium, i.e., if all the heat bath are at the same temperature. Our formalism maps the initial stochastic problem on a noncanonical quantum field theory, and the calculations of the relaxation rates are based on a perturbative analysis. We argue to show the reliability of the perturbative computation.     

On the evolution of higher dimensional Heisenberg continuum spin systems

We consider the evolution of a classical Heisenberg ferromagnetic spin chain in its continuum limit in higher spatial dimensions. It is shown that the evolution of a radially symmetric chain could be identified with the motion of a helical space curve as in the linear case. The resulting invariant equations for the curvature (radial energy density) and torsion (related to current density) are shown to be equivalent to a generalized nonlinear Schrödinger equation, similar to the one derived by Ruijgrok and Jurkiewicz recently. Equivalent linear equations as well as special static solutions of point singular type are obtained. Similarity solutions, a class of which belong to Riccati type, are discussed in detail. For general higher dimensions, a potentially useful formulation is presented: Under stereographic projection of the unit sphere of spin, the equation of motion takes a neater form even with the inclusion of anisotropic interactions. Classes of explicit solutions are reported in higher dimensions. Propagating spin waves, static spin waves of point singular nature and of finite energy in some cases are also discussed.     

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.     

On the equivalence between KMS-states and equilibrium states for classical systems

It is shown that for any KMS-state of a classical system of non-coincident particles, the distribution functions are absolutely continuous with respect to Lebesgue measure; the equivalence between KMS states and Canonical Gibbs States is then established.Supported in part by NSF Grant MCS 75-21684Supported in part by NSF Grant MPS 72-04534Supported in part by NSF Grant MPS 75-20638     

On the uniqueness of the equilibrium state for antiferromagnetic Ising spin system in the phase transition region

We show that at low temperature an Ising spin system with antiferromagnetic interaction in a small enough external magnetic field has only one translationally invariant state.     

Analyticity and uniqueness for spin 1/2 classical ferromagnetic lattice systems at low temperatures

Analyticity and uniqueness of correlation functions is investigated for a number of systems by application of Ruelle's theorem on zeros of Asano contracted polynomials to the partition function. To answer the question when the partition function of a system is the Asano contraction of those of subsystems the groups appearing in the low and high temperature expansions are employed.On leave of absence from Department of Physics and Astronomy, Tel Aviv University Ramat-Aviv, Israel.