三维精确可解统计模型──Baxter－Bazhanov模型的可积性条件

从手征Ｐｏｔｔｓ模型推导出三维精确可解Ｂａｘｔｅｒ－Ｂａｚｈａｎｏｖ模型的“可逆性”及“星一方”关系,从而说明其可积性条件──四面体方程是手征Ｐｏｔｔｓ模型星──三角关系的一个结论．若把玻尔兹曼权参变数表示为Ｚａｍｏｌｏｄｃｈｉｋｏｖ角变量形式,其附加条件自然成立．值得指出的是,由本文处理方法可以得出三维可解统计模型的星－三角关系,它包含了Ｂａｚｈａｎｏｖ和Ｂａｘｔｅｒ的结论．     

The vertex formulation of the Bazhanov-Baxter model

In this paper we formulate an integrable model on the simple cubic lattice. TheN-valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex-type tetrahedron equation. In the thermodynamic limit our model is equivalent to the Bazhanov-Baxter model. In the case whenN=2 we reproduce Korepanov's and Hietarinta's solutions of the tetrahedron equation as special cases.     

Integrable boundary Boltzmann weights and surface free energy inversion relations of the chiral Potts model

The integrability of the chiral Potts model with boundaries is considered in this paper. The boundary star-triangle relation determining the boundary Boltzmann weights for the chiral Potts model is presented. By solving the boundary star-triangle relation the boundary Boltzmann weights are obtained. The fusion procedure is then applied to derive the functional relations of the transfer matrices of the model with boundaries. From these functional relations the inversion relations of the surface free energies are extracted when the system size is big enough. Surprisingly, the inversion relation of the local surface free energy is as simple as those of other non-chiral models, but it has still to be solved.     

A class of elliptic solutions to the star-triangle relation

The star-triangle relation for the N-state IRF model is investigated. A class of elliptic solutions is found. The corresponding lattice model splits into mutually non-interacting diagonal lattices with the interactions between nearest-neighbour spins.     

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.     

Three-dimensional vertex model in statistical mechanics from Baxter-Bazhanov model

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube, and the Wu-Kadanoff-Wegner duality between the cube- and vertex-type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which corresponds to the three-dimensional Baxter-Bazhanov model. The vertex-type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. We write down the symmetry relations of the weight functions under the actions of the symmetry groupG of the cube. The six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected, with the six spaces in which the vertextype tetrahedron equation is defined.     

Analytical properties of the anisotropic cubic Ising model

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.     

Lattice Boltzmann Study of Mixed Convection in aCubic Cavity

The problem of the mixed convection in a cubic cavity is studied with lattice Boltzmann method. A multiple-relaxation-time lattice Boltzmann model for incompressible flow in the cubic cavity and another thermal lattice Boltzmann model for solving energy/temperature equation are proposed. The present models are first validated through a comparison with some available results, and then, we present a detailed parameter study on the mixed
convection in the cubic cavity. The numerical results show that the flow and temperature patterns change greatly with variations of the Reynolds and Richardson numbers.     

Remarks on the star-triangle relation in the Baxter-Bazhanov model

We show that the restricted star-triangle relation introduced by Bazhanov and Baxter can be obtained either from the star-triangle relation of the chiral Potts model or from the star-square relation proposed by Kashaev, Mangazeev, and Stroganov, and give a response to a guess of Bazhanov and Baxter.     

On discrete three-dimensional equations associated with the local Yang-Baxter relation

The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this Letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.Some related operator solutions to the tetrahedron equation are also constructed.St. Petersburg Branch of the Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, RussiaURA 14-36 du CNRS, associée à l'ENS de Lyon, au LAPP d'Annecy et à l Universitè de Savoie, France     

Duality relations and equivalences for models with O(N) and cubic symmetry

Formal expression for high-temperature series are derived for models with O(N) and cubic symmetry, with a special form of nearest neighbor interactions on the honeycomb lattice. By deriving low-temperature series for a class of generalized solid-on-solid and cubic models, a duality relation is established. Equivalences between cubic and SOS type models are also found. In the large-N limit, the series reduce to those of the hard hexagon model.     

Surface Critical Behavior of Two-Dimensional Dilute Ising Models

Ising models with nearest neighbor ferromagnetic random couplings on a square lattice with a (1, 1) surface are studied, using Monte Carlo techniques and a star-triangle transformation method. In particular, the critical exponent of the surface magnetization is found to be close to that of the perfect model, Β₁ = 1/2. The crossover from surface to bulk critical properties is discussed.     

A goldstone mode in the Kawasaki-Ising model

The hydrodynamic regime of superfluids is dominated by a Goldstone mode corresponding to a spontaneously brokenU(1) symmetry. In this article we map the Kawasaki-Ising model for a classical lattice gas into a quantum model for a superfluid and establish a connection between the normal density fluctuations of the first and the Goldstone mode of the second. The fact that the quantum model we obtain describes a superfluid derives from an inequality by Penrose and Onsager which gives a lower bound to the Bose-Einstein condensate density. Mathematically, the Goldstone mode can be described by means of a quantum extension of the local algebra of the Ising model. The classification of its irreducible representations requires an additionalU(1) phase factor and the correspondingU(1) gauge symmetry is spontaneously broken for all finite values of the temperature and of the density.     

Optimised Dirac operators on the lattice: construction,properties and applications

We review a number of topics related to block variable renormalisation group transformations of quantum fields on the lattice, and to the emerging perfect lattice actions. We first illustrate this procedure by considering scalar fields. Then we proceed to lattice fermions, where we discuss perfect actions for free fields, for the Gross‐Neveu model and for a supersymmetric spin model. We also consider the extension to perfect lattice perturbation theory, in particular regarding the axial anomaly and the quark gluon vertex function. Next we deal with properties and applications of truncated perfect fermions, and their chiral correction by means of the overlap formula. This yields a formulation of lattice fermions, which combines exact chiral symmetry with an optimisation of further essential properties. We summarise simulation results for these so‐called overlap‐hypercube fermions in the two‐flavour Schwinger model and in quenched QCD. In the latter framework we establish a link to Chiral Perturbation Theory, both, in the p‐regime and in the ϵ‐regime. In particular we present an evaluation of the leading Low Energy Constants of the chiral Lagrangian – the chiral condensate and the pion decay constant – from QCD simulations with extremely light quarks.     

A lattice Maxwell system with discrete space–time symmetry and local energy–momentum conservation

A lattice Maxwell system is developed with gauge-symmetry, symplectic structure and discrete space–time symmetry. Noether's theorem for Lie group symmetries is generalized to discrete group symmetries for the lattice Maxwell system. As a result, the lattice Maxwell system is shown to admit a discrete local energy–momentum conservation law corresponding to the discrete space–time symmetry. A lattice model that respects all local conservation laws and geometric structures is as good as and probably more preferable than standard models on continuous space–time. It can also be viewed as an effective algorithm for the governing differential equations on continuous space–time.     

Critical line of the triangular potts model with two- and three-site interactions: An alternate derivation

A double star-triangle relation which is a generalization of the usual single star-triangle relation is introduced. Together with a duality transformation it is possible to rederive the critical line of the Potts model with two- and three-site interactions on alternate triangles.     

Decorated star-triangle relations and exact integrability of the one-dimensional Hubbard model

The exact integrability of the one-dimensional Hubbard model is demonstrated with the help of a novel set of triangle relations, the decorated star-triangle relations. The covering two-dimensional statistical mechanical model obeys the star-triangle or Yang-Baxter relation. A conjecture is presented for the eigenvalues of the transfer matrix.     

Entropy of a one-dimensional mixed lattice gas

This paper deals with the grand canonical entropy of a lattice gas mixture. The entropy is a function of the multisite densities corresponding to the interaction pattern of the system in question. It is first evaluated for a nearest-neighborinteraction, one-dimensional simple lattice gas to show how the structure of bulk fluid is locally maintained. Generalization requires one set of interrelations among multisite densities presented in closed form for an arbitrary lattice, and one set between Boltzmann factors and multisite densities which is written down for simply connected lattices. Application is made to two-row lattices, which turn out to have local behavior from this viewpoint, as do all single-row or Bethe lattices with complete range-p interactions. Nonlocal examples are also given, and suggestions made for approximation sequences in general lattices.     

The nuclear symmetry energy from relativistic Brueckner-Hartree-Fock model

The microscopic mechanisms of the symmetry energy in nuclear matter are investigated in the framework of the relativistic Brueckner-Hartree-Fock (RBHF) model with a high-precision realistic nuclear potential, pvCDBonn A. The kinetic energy and potential contributions to symmetry energy are decomposed. They are explicitly expressed by the nucleon self-energies, which are obtained through projecting the G-matrices from the RBHF model into the terms of Lorentz covariants. The nuclear medium effects on the nucleon self-energy and nucleon-nucleon interaction in symmetry energy are discussed by comparing the results from the RBHF model and those from Hartree-Fock and relativistic Hartree-Fock models. It is found that the nucleon self-energy including the nuclear medium effect on the single-nucleon wave function provides a largely positive contribution to the symmetry energy, while the nuclear medium effect on the nucleon-nucleon interaction, i.e., the effective G-matrices provides a negative contribution. The tensor force plays an essential role in the symmetry energy around the density. The scalar and vector covariant amplitudes of nucleon-nucleon interaction dominate the potential component of the symmetry energy. Furthermore, the isoscalar and isovector terms in the optical potential are extracted from the RBHF model. The isoscalar part is consistent with the results from the analysis of global optical potential, while the isovector one has obvious differences at higher incident energy due to the relativistic effect.     

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX _n ⁽¹⁾ =A _n ⁽¹⁾ ,B _n ⁽¹⁾ ,C _n ⁽¹⁾ ,D _n ⁽¹⁾ . The local states taken on by the fluctuation variables are the dominant integral weights ofX _n ⁽¹⁾ of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX _n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.