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．     

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.     

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.     

Free energy of the chiral Potts model in the scaling region

We explicitly calculate the free energy of the general solvableN-state chiral Potts model in the scaling region, forT<T _c . We do this from both of the two available results for the free energy, and verify that they are mutually consistent. Ift=T _c –T, then we find that - _c /t has a Taylor expansion in powers oft ^2/N (together with higher-order non-scaling terms of ordert, ort logt).     

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.     

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.     

Lattice gas generalization of the hard hexagon model. I. Star-triangle relation and local densities

In the solvable hard hexagon model there is at most one particle in every pair of adjacent sites, and the solution automatically leads to various mathematical identities, in particular to the Rogers-Ramanujan relations. These relations have been generalized by Gordon. Here we construct a solvable model with at most two particles per pair of adjacent sites, and find the solution involves the next of Gordon's relations. We conjecture the corresponding solution for a model with at mostn particles per pair of adjacent sites: this involves all Gordon's relations, as well as others that we will discuss in a subsequent paper.     

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.     

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.     

The inversion relation method for some two-dimensional exactly solved models in lattice statistics

The partition-functions-per-site of several two-dimensional models (notably the eight-vertex, self-dual Potts and hard-hexagon models) can be easily obtained by using an inversion relation for local transfer matrices, together with symmetry and analyticity properties. This technique is discussed, the analyticity properties compared, and some equivalences (and nonequivalences) pointed out. In particular, the critical hard-hexagon model is found to have the same as the self-dualq-state Potts model, withq=(3 + 5)/2 = 2.618 .... The Temperley-Lieb equivalence between the Potts and six-vertex models is found to fail in certain nonphysical antiferromagnetic cases.     

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.     

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.     

The Riemann Surface of the Chiral Potts Model Free Energy Function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice inversion relation method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T _pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T _pq . In terms of the t _p ,t _q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N–1)-dimensional lattice (for N>2). The function T _pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.     

Knot invariants associated with a particularN→∞ continuous limit of the Baxter-Bazhanov model

First we briefly recall the definition of the three-dimensional Baxter-Bazhanov lattice model. The spins of this model are elements ofZ _N and theR-matrix is associated to the algebraU _q sl(n) ifq is a primitiveNth root of unity. Then we construct a particularN limit of the model, in which it is meaningful to interpret the spins as elements ofR and which gives the free Gaussian boson model. Finally, we study special limits of the rapidity variables in which we obtain braid group representations and we show that forn odd the associated knot invariants are given by the inverse of products of Alexander polynomials, evaluated at certain roots of unity.     

Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.     

Explicit solutions for a nonlinear model on the honeycomb and triangular lattices

We study a simple nonlinear model de?ned on the honeycomb and triangular lattices. We propose a bilin-earization scheme for the ?eld equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the Hirota bilinear difference equation and the Ablowitz-Ladik system. This result is used to derive the two sets of explicit solutions: the N-soliton solutions and ones constructed of the Toeplitz determinants.     

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

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.     

Triangular lattice Potts models

In this paper we study the 3-state Potts model on the triangular lattice which has two- and three-site interactions. Using a Peierls argument we obtain a rigorous bound on the transition temperature, thereby disproving a conjecture on the location of its critical point. Low-temperature series are generated and analyzed for three particular choices of the coupling constants; a phase diagram is then drawn on the basis of these considerations. Our analysis indicates that the antiferromagnetic transition and the transition along the coexistence line are of first order, implying the existence of a multicritical point in the ferromagnetic region. Relation of the triangularq-state Potts model with other lattice-statistical problems is also discussed. In particular, an Ashkin-Teller model and the hard-hexagon lattice gas solved by Baxter emerge as special cases in appropriate limits.Supported in part by NSF grant No. DMR 78-18808.     

Integrable boundary conditions for the B_2 model

New integrable B_2 model with off-diagonal boundary reflections is proposed. The general solutions of the reflection matrix for the B_2 model are obtained by using the fusion technique. Wefind that the reflection matrix has 7 free boundary parameters, which are used to describe the degree of freedom of boundary couplings, without breaking the integrability of the system. The new quantization conditions will induce the novel structure of the energy spectrum and the boundary states. The corresponding boundary effects can be studied based on the results in this paper. Meanwhile, the reflection matrix of high rank models associated with Bnalgebra can also be obtained by using the method suggested in this paper.