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Physical review. B, Condensed matter

Classical equivalents of one-dimensional quantum-mechanical systems

Mustansir Barma, B. Sriram Shastry

Published Oct 1, 1978• 115 citations

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.

DOI: 10.1103/physrevb.18.3351

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arXiv:cond-mat/0610689•Oct 25, 2006•0 citations

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.

Exact expressions are given for the free energy of the classical Heisenberg model in one lattice dimension and two or three spin dimensions with general anisotropy; the two-dimensional result, already known, is in terms of Mathieu functions and the three-dimensional result in terms of Lame wavefunctions. The method presented solves the eigenvalue problem for an appropriate transfer matrix by relating it to solutions of the Helmholtz equation in suitable curvilinear coordinate systems. High and low temperature limits are discussed for the three-dimensional spin case and the connection given with already known results for partial anisotropy.

The quantum mechanical analogue of the classical Perk - Schultz model is considered which comprises the Uimin - Sutherland model and the integrable t - J chain. The quantum transfer matrix of these systems is established and the eigenvalue equations are obtained by an algebraic Bethe ansatz. Only the largest eigenvalue is needed for the calculation of the free energy of the quantum chain at finite temperature. The Bethe ansatz equations for the leading eigenvalue are transformed into a set of integral equations for some appropriately defined auxiliary functions. Furthermore, the eigenvalue of the quantum transfer matrix is expressed in terms of these functions. The integral formulation allows for taking the limit of infinite Trotter - Suzuki number analytically. The low-temperature limit of the free energy is obtained analytically and for intermediate temperatures numerical results are presented.

The density-matrix renormalization group (DMRG) applied to transfer matrices allows it to calculate static as well as dynamical properties of one-dimensional quantum systems at finite temperature in the thermodynamic limit. To this end the quantum system is mapped onto a two-dimensional classical system by a Trotter-Suzuki decomposition. Here we discuss two different mappings: The standard mapping onto a two-dimensional lattice with checkerboard structure as well as an alternative mapping introduced by two of us. For the classical system an appropriate quantum transfer matrix is defined which is then treated using a DMRG scheme. As applications, the calculation of thermodynamic properties for a spin-1/2 Heisenberg chain in a staggered magnetic field and the calculation of boundary contributions for open spin chains are discussed. Finally, we show how to obtain real time dynamics from a classical system with complex Boltzmann weights and present results for the autocorrelation function of the XXZ-chain.