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Physical review. B, Condensed matter

Monte Carlo method for the Ising model in a transverse field

Rick J. Creswick, Horácio A. Farach, James M. Knight, Charles P. Poole

Published Sep 1, 1988• 15 citations
Publisher

Abstract

The partition function for the Ising model in a transverse field can be expressed as a sum over the usual Ising states by evaluation of the diagonal matrix elements between Ising states with the use of the cumulant expansion. The cumulant sum is evaluated to all orders in \ensuremath{\beta} and second order in \ensuremath{\Gamma}, the transverse field. The effective Hamiltonian is then employed in Monte Carlo simulation to evaluate the average energy and magnetization in the longitudinal and transverse directions. Comparison with exact results on a one-dimensional chain shows that the method works quite well. Monte Carlo results for the average energy, specific heat, and the transverse and longitudinal components of the magnetization for the transverse Ising model on the simple cubic lattice are presented.

DOI
10.1103/physrevb.38.4712

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The partition function for the Ising model in a transverse field can be expressed as a sum over the usual Ising states by evaluation of the diagonal matrix elements between Ising states with the use of the cumulant expansion. The cumulant sum is evaluated to all orders in \ensuremath{\beta} and second order in \ensuremath{\Gamma}, the transverse field. The effective Hamiltonian is then employed in Monte Carlo simulation to evaluate the average energy and magnetization in the longitudinal and transverse directions. Comparison with exact results on a one-dimensional chain shows that the method works quite well. Monte Carlo results for the average energy, specific heat, and the transverse and longitudinal components of the magnetization for the transverse Ising model on the simple cubic lattice are presented.
The Monte Carlo simulation approach is applied to the classical (S= infinity ) Ising model in a transverse field. The results are presented for the case of a 3D simple cubic lattice. Thermodynamic spin averages, correlation functions and the internal energy are considered above as well as below the transition temperature, with an emphasis on the critical regime. The comparison with predictions of various theoretical approaches is performed. In the region where the transverse field does not exceed its critical value, the longitudinal quantities are shown to exhibit an Ising-type behaviour. On the other hand, the transverse functions reveal some new qualitative features, which so far have not been well represented theoretically.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.
The magnetic properties of the transverse Ising model with coordination number 6 (simple cubic lattice) and 4 (square lattice) are studied by using Bethe-Peierls approximation. For different transverse fields, the quantities of the two systems, such as the free energy, transverse and longitudinal magnetizations, internal energy and the magnetic specific heat, and so on, are calculated. It is found that the transverse field influences the quantities of the system very much and has the inhibiting effect on the spontaneous magnetization.
In this article new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground- state wave functions are especially discussed.
Series expansions are presented for the magnetisation, susceptibility, magnetic field derivatives of the susceptibility and 'specific heat' of the zero-temperature transverse Ising model. Coefficients in these series have been calculated to tenth order in lambda 2 (where lambda is the transverse field) for the linear chain, to eighth order for the square lattice and to seventh order for the triangular lattice. These series yield estimates of the low-temperature critical exponents alpha ', beta ', gamma ' and Delta ' of the two- and three-dimensional Ising models. They provide good evidence for the symmetry of exponents above and below the critical point, e.g. gamma '= gamma .