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Physical Review Letters

Transfer-matrix inverison identities for exactly solvable lattice-spin models

Paul A. Pearce

Published Apr 13, 1987• 37 citations
Publisher

Abstract

A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.

DOI
10.1103/physrevlett.58.1502

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A new functional equation satisfied by the commuting row-to-row transfer matrices is derived for the eight-vertex model. Functional equations of the same form are satisfied by hard hexagons, magnetic hard squares, the self-dual Potts models, the Andrews-Baxter-Forrester models, and others. These new functional equations are called inversion identities because they generalize the inversion relation for local transfer matrices. It is conjectured that all solvable models satisfying Yang-Baxter equations possess such inversion identities and that, in general, these functional equations can be solved for the transfer-matrix eigenvalues.
Abstract This chapter defines transfer matrices, and the existence of a one-parameter family of commuting transfer matrices is defined as the condition of integrability. The local star-triangle (Yang–Baxter) equation is introduced for vertex, spin, and face models and used to demonstrate the commutation of the transfer matrices. The star-triangle equation is solved for the six-vertex, eight-vertex, SOS, RSOS, hard hexagon, and chiral Potts models. The commutation of the transfer matrix with the related quantum spin chain is derived.
Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras [Formula: see text], [Formula: see text], AL, DL and E6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.
A general analytic method for calculating finite-size corrections, central charges, and scaling dimensions of solvable lattice models is presented. The approach is to solve the special functional equations or inversion identities satisfied by the commuting row transfer matrices of these lattice models at criticality. For purposes of illustration, the method is applied to calculate the central charge c=4/5 and leading magnetic scaling dimension x=2/15 of hard hexagons. These numbers are rational due to special values of Rogers dilogarithms.
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
Baxter's method of solving the eight-vertex model in lattice statistical mechanics is examined from an elementary point of view. It is shown that the algebraic operations in the method can be carried out without recourse to elliptic functions. These include: construction of certain subspaces invariant uti.der the action of the transfer matrix; reduction of the transfer matrix eigenvalue problem to an equivalent ice-type problem and construction of certain matrices which commute with the transfer matrix and satisfy a functional matrix equation.