Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

We consider quantum unbounded spin systems (lattice boson systems) in -dimensional lattice space Z. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly -dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.     

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 boundsf _n on the free energy per lattice sitef and thatf _n 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.     

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.     

On a cluster expansion for lattice spin systems: A finite-size condition for the convergence

A study is made of the statistical mechanics of classical lattice spin systems with finite-range interactions in two dimensions. By means of a decimation procedure, a finite-size condition is given for the convergence of a cluster expansion that is believed to be useful for treating the range of temperature between the critical oneT _c and the estimated thresholdT ₀ of convergence of the usual high-temperature expansion.     

Cluster expansion ford-dimensional lattice systems and finite-volume factorization properties

We consider classical lattice systems with finite-range interactions ind dimensions. By means of a block-decimation procedure, we transform our original system into a polymer system whose activity is small provided a suitable factorization property of finite-volume partition functions holds. In this way we extend a result of Olivieri.     

The equivalence of ensembles and the Gibbs phase rule for classical lattice systems

We study the relation between the microcanonical, canonical, and grand canonical ensembles in the thermodynamic limit when the system becomes infinite. They are equivalent if there is only one phase in the system. In general it is shown that there is a unique limit of the microcanonical state being a mixture of pure phases if the microcanonical restrictions determine the volume fractions of the phases uniquely, and then the Gibbs phase rule is valid. In this context we show how to define the set of order parameters associated with the state of the system in a natural way.     

Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states

Starting from classical lattice systems ind2 dimensions with a regular zerotemperature phase diagram, involving a finite number of periodic ground states, we prove that adding a small quantum perturbation and/or increasing the temperature produce only smooth deformations of their phase diagrams. The quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds. For fermions, the interactions must be given by monomials of even degree in creation and annihilation operators. Our methods can be applied to some anyonic systems as well. Our analysis is based on an extension of Pirogov-Sinai theory to contour expansions ind+1 dimensions obtained by iteration of the Duhamel formula.     

Debye screening for two-dimensional Coulomb systems at high temperatures

The grand canonical ensemble of a two-dimensional Coulomb system with±1 charges is proved to have screening phenomena in its high-temperature region. The Coulomb potential in a finite region is assumed to be (–)^–1, where is the Laplacian with zero boundary conditions on. The hard-core condition is not assumed. The model is set up by separating (–)^–1 into a shortrange part and a long-range part depending on a parameter. The self-energies are subtracted only for the short-range part and therefore a choice of is a choice of subtraction of self-energies. The method of proof is in general the same as that of Brydges-Federbush Debye screening, except that here a modification for the short-range part of the potentials is needed.     

Convergent expansions for asymptotically degenerate inverse correlation lengths of classical lattice spin systems

We obtain convergent expansions for the inverse correlation length associated with various spin-spin correlation functions for some weakly coupled multicomponent classical lattice spin systems. In terms of the lattice quantum field theory associated with the models the expansions provide a convergent perturbation theory for particle masses which are asymptotically degenerate in the limit of zero coupling.Reseach partially supported by CNPq, Brazil.     

Phase transitions and reflection positivity for a class of quantum lattice systems

We prove a form of reflection positivity in planes containing sites for a class of quantum lattice systems. As an application, a proof is given of a phase transition for the Fisher-stabilized Ising antiferromagnet in an external magnetic field with parallel and transverse components, both by the method of infrared bounds and by a suitable version of the Peierls argument. We also discuss the spherical model in an appendix.     

Uniqueness and exponential decay of correlations for some two-dimensional spin lattice systems

We consider a two-dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice. The configuration space of the spin system is a direct product of mixing subshifts of finite type. The potential is defined on the set of all squares in Z² and decays exponentially with the linear size of the square. Via the polymer expansion technique we prove that for sufficiently high temperatures the limit Gibbs distribution is unique and has an exponential decay of correlations.     

The quasiclassical Langevin equation and its application to the decay of a metastable state and to quantum fluctuations

We report on investigations on the consequences of the quasiclassical Langevin equation. This Langevin equation is an equation of motion of the classical type where, however, the stochastic Langevin force is correlated according to the quantum form of the dissipation-fluctuation theorem such that ultimately its power spectrum increases linearly with frequency. Most extensively, we have studied the decay of a metastable state driven by a stochastic force. For a particular type of potential well (piecewise parabolic), we have derived explicit expressions for the decay rate for an arbitrary power spectrum of the stochastic force. We have found that the quasiclassical Langevin equation leads to decay rates which are physically meaningful only within a very restricted range. We have also studied the influence of quantum fluctuations on a predominantly deterministic motion and we have found that there the predictions of the quasiclassical Langevin equations are correct.     

A generalization of Fermat's principle for classical and quantum systems

The analogy between dynamics and optics had a great influence on the development of the foundations of classical and quantum mechanics. We take this analogy one step further and investigate the validity of Fermat's principle in many-dimensional spaces describing dynamical systems (i.e., the quantum Hilbert space and the classical phase and configuration space). We propose that if the notion of a metric distance is well defined in that space and the velocity of the representative point of the system is an invariant of motion, then a generalized version of Fermat's principle will hold. We substantiate this conjecture for time-independent quantum systems and for a classical system consisting of coupled harmonic oscillators. An exception to this principle is the configuration space of a charged particle in a constant magnetic field; in this case the principle is valid in a frame rotating by half the Larmor frequency, not the stationary lab frame.     

The self-consistent diagram approximation for lattice systems

We apply the self-consistent diagram approximation to calculate equilibrium properties of lattice systems. The free energy of the system is represented by a diagram expansion in Mayer-like functions with averaging over states of a reference system. The latter is defined by one-particle mean potentials, which are calculated using the variational condition formulated. As an example, numerical computations for a two-dimensional lattice gas on a square lattice with attractive interaction between nearest neighbours were carried out. The critical temperature, the phase coexistence curve, the chemical potential and particle and vacancy distribution functions coincide within a few per cent with exact or with Monte Carlo data. Received 18 March 1999 and Received in final form 8 November 1999     

The third law of thermodynamics and the degeneracy of the ground state for lattice systems

The third law of thermodynamics, in the sense that the entropy per unit volume goes to zero as the temperature goes to zero, is investigated within the framework of statistical mechanics for quantum and classical lattice models. We present two main results: (i) For all models the question of whether the third law is satisfied can be decided completely in terms of ground-state degeneracies alone, provided these are computed for all possible boundary conditions. In principle, there is no need to investigate possible entropy contributions from low-lying excited states, (ii) The third law is shown to hold for ferromagnetic models by an analysis of the ground states.Dedicated to Pierre Résibois. Work supported in part by NSF grant PHY-7825390 A01.     

Uniform upper bounds (and thermodynamic limit) for the correlation functions of symmetric coulomb-type systems

Uniform upper bounds are proven for the correlation functions in the strictly charge-neutral canonical and grand canonical ensembles for charge-symmetric two-component systems. For the grand canonical ensemble the increase of the correlation functions along the thermodynamic-limit sequence is shown as well, implying the existence of the states. The particles have bounded pair interactions of positive type. Both classical and quantum systems with Boltzmann statistics are considered. Coulomb systems with regularized interactions are included as a special case.     

On the Lifschitz singularity and the tailing in the density of states for random lattice systems

We prove rigorously the existence of a Lifschitz singularity in the density of states at zero energy in some random lattice systems of noninteracting bosons and fermions in any numberv of dimensions. The basic tool is a simple modification of the method of Fukushima to yield the correct upper and lower bounds for allv. We also comment on the mathematical difference between the models treated and the system of phonons with mass disorder in the harmonic approximation, whose behavior is known to be of Debye form, not Lifschitz, at low temperatures.Supported by the Swiss National Science Foundation.On leave of absence from the Institute de Fisica, University of São Paulo, Brazil.     

二维单斜点阵光子晶体的第一布里渊区及带隙计算

二维单斜点阵光子晶体在光学聚焦器件及光子晶体波导中有重要的应用价值，详细讨论了二维单斜点阵光子晶体的第一布里渊区及带隙计算，并与常规方法计算得出的二维正三角形晶格光子晶体的带隙结构进行了比较.最后讨论了临界条件下二维单斜点阵光子晶体的带隙结构，证明了本方法的有效性.     

Metastable systems driven by colored noise: The stationary state

The influence of the Bardeen-Herring back-jump correlations on the Fermi-Dirac statistics of the one-dimensional nonhomogeneous fermionic lattice gas is studied by the Monte Carlo simulation technique and semianalytically. The resulting distribution is obtained, exhibiting increased population of the lower levels in comparison to the Fermi-Dirac statistics.