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．     

Star-triangle relation for a three-dimensional model

The solvablesl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising-type model on the body-centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to the spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.On leave of absence from the Institute for High Energy Physics, Protvino, Moscow Region, 142284, Russia.     

Interfacial tension of the chiral Potts model

We calculate the interfacial tension of theN-state chiral Potts model by solving the functional relations for the transfer matrices of the model with skewed boundary conditions. Our result is valid for the general physical model (with positive Boltzmann weights) and at all subcritical temperatures. The interfacial tension has been calculated previously for the superintegrable chiral Potts model with skewed boundary conditions. UsingZ-invariance, Baxter has argued that the interfacial tension of this model should be the same as the interfacial tension of the general physical model. We show that this is indeed the case.     

Equivalence of the Two Results for the Free Energy of the Chiral Potts Model

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries, and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behavior can be obtained The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.     

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.     

Star-triangle relations in the exactly solvable statistical models

A regular method for analysis of lattice spin models with a nearest neighbour interaction is proposed. Star-triangle relations in the form of functional equations are used. Parametric families of transfer matrices commuting due to star-triangle relations are constructed. The eigenvalues of transfer matrices as functions of the spectral parameter are shown to obey two functional equations. The solution of these equations for the maximal eigenvalue yields the partition function of the model. The method is applied for evaluation of the partition function of the critical Potts models, the Ising model, the Ashkin-Teller model equivalent to the eight-vertex model.     

Corner transfer matrices of the chiral Potts model

We present some symmetry and factorization relations satisfied by the corner transfer matrices (CTMs) of the chiral Potts model. We show how the single-spin expectation values can be expressed in terms of the CTMs, and in terms of the related boost operator. Low-temperature calculations lead naturally to the variables that uniformize the Boltzmann weights of the model.     

Free energy of the solvable chiral Potts model

Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent is 1–2/N.     

Corner transfer matrices of the chiral Potts model. II. The triangular lattice

We consider a two-dimensional edge-interaction model satisfying the star-triangle relations. For the triangular lattice, the corner transfer matrices are functions of three rapidities: we show that they possess various factorization properties and satisfy certain equations. We indicate how these equations can be solved for the Ising model. We then consider the three-state chiral Potts model and obtain low-temperature solutions to the equations. The conjectured formula for the order parameter (the spontaneous magnetization) is verified to one more order in a series expansion.     

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.     

The Ising Model with Boundaries: Integrability and Functional Relations

In this Letter, we analyse the boundary conditions of the planar Ising model and determine the boundary Boltzmann weights in terms of bulk Boltzmann weights. The commutativity of the transfer matrices and their functional relations are shown.     

Chiral Potts model with skewed boundary conditions

We obtain the transfer matrix functional relations for the chiral Potts model with skewed boundary conditions and find that they are the same as for periodic boundary conditions, but with modified selection rules. As a start toward calculating the interfacial tension in general, we here evaluate it in a low-temperature limit, performing a Bethe-ansatz-type calculation. Finally, we specialize the relations to the superintegrable case, verifying the ansatz proposed by Albertiniet al.     

Cyclic monodromy matrices forsl(n) trigonometricR-matrices

The algebra of monodromy matrices forsl(n) trigonometricR-matrix is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product ofL-operators. Cocommutativity of representations is discussed and intertwiners for factorizable representations are written through the Boltzmann weights of thesl(n) chiral Potts model.     

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.     

A Monte Carlo study of the asymmetric clock or chiral Potts model in two dimensions

The phase diagram of the two-dimensional, three-state chiral Potts or asymmetric clock model is studied using Monte Carlo techniques. The phase boundaries are compared to those obtained using the finite-size renormalization group and the free fermion approximation. The incommensurate phase is described in detail and crossover effects near the Lifshitz point are discussed.     

Transfer Matrix Functional Relations for the Generalized τ2 (t q ) Model

The N-state chiral Potts model in lattice statistical mechanics can be obtained as a “descendant” of the six-vertex model, via an intermediate “Q” or “τ₂ (t _q )” model. Here we generalize this to obtain a column-inhomogeneous τ₂ (t _q ) model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do not need the usual chiral Potts relations between the Nth powers of the rapidity parameters a _p , b _p , c _p , d _p of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.     

Baxter's Solution for the Free Energy of the Chiral Potts Model

In honor of Baxter's sixtieth birthday, we would like to review some of his work on the free energy of the chiral Potts model. In spite of the enormous complexity and difficulty of the problem, Baxter, using functional relations was able to calculate not only the free energy, but also the interfacial tension. We here show that the integral for the free energy simplifies in the superintegrable case and is identical to his earlier results using entirely different approaches. His calculations are extended to include other regions. We also attempt to clarify some of his reasoning as several steps may be mysterious at first glance.     

Commuting transfer matrices in the chiral Potts models: Solutions of star-triangle equations with genus>1

We present solutions of the star-triangle equations which are uniformized by curves of genus greater than one. One particular solution which is relevant to the commensurate-incommensurate transition in the chiral three-state Potts model has genus 10 if T≠T_c and genus 1 if T=T_c. Solutions for N>3 are also presented.     

Functional Relations for the Order Parameters of the Chiral Potts Model

Following the method of Jimbo, Miwa, and others, we obtain functional relations for the order parameters of the chiral Potts model. We have not yet solved these relations. Here we discuss their properties and show how one should beware of spurious solutions.     

Integrable models through representations of the hecke algebra

We present new solutions to the Yang-Baxter equation through representations of the Hecke algebra. The generators of the Hecke algebra are considered as Boltzmann weights for face models. The heights live in a graph. These models were conjectured to be integrable by Di Francesco and Zuber. We prove integrability for some of the suggested models by building explicitly the Boltzmann weights. Some relations that the Boltzmann weights satisfy and the consequence on partition functions with various boundary conditions are also discussed.