Classical equivalents of one-dimensional quantum-mechanical systems

Abstract

Trotter's formula is used to construct two-dimensional classical systems equivalent to some one-dimensional quantum-mechanical systems of interest. The finite-temperature properties of the completely anisotropic Heisenberg chain are expressed in terms of an eight-vertex model in which the vertex weights depend on the size of the lattice. Knowledge of only the largest eigenvalue of the transfer matrix of the eight-vertex model is not sufficient to find the free energy of the chain except in the limit of zero temperature, when Baxter's result for the ground-state energy is recovered. We also examine two models with two species of variables each, and point out that by constructing the equivalent classical problem the trace over one set of variables can be performed.