Transfer Matrices: Exactly Solved Models

Paul A. Pearce

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.

André Xuereb

S B Kelland

The q-state Potts model on a square lattice is transformed into a staggered ice model. A variational principle for the largest eigenvalue of the transfer matrix of this model is used to develop a set of matrix equations. In the limit of these matrices becoming infinite, the equations determine the partition function and order parameter of the Potts model exactly. We have solved the equations numerically for finite matrices, obtaining estimates of these quantities and, as a result, the critical exponent β.

R. J. Baxter

Robert H. Swendsen

The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the complex phenomena exhibited at phase transitions in general. The one-dimensional model is solved exactly using transfer matrices. Models in higher dimensions are treated in the mean field approximation.