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Ferroelectrics

New effective field theory for the transverse Ising model

F. C. Sá Barreto, I. P. Fittipaldi, Boštjan Žekš

Published Oct 1, 1981• 196 citations
Publisher

Abstract

A new type effective field approach is used to derive the thermodynamical properties of the transverse Ising model. The method is based on an exact formal identity for the two-state transverse Ising model and utilizes an exponential operator technique. The lines of critical points in the Ω-T plane are derived and the critical transverse fields ΩC are obtained analytically, for lattices with coordination numbers z = 2, 4 and 6. This simple method yields results which represent a remarkable improvement on the usual mean field approximation. Our results are also compared with those from series expansion method.

DOI
10.1080/00150198108219575

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Monte Carlo method for the Ising model in a transverse field

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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.

Physical review. B, Condensed matter•Sep 1, 1988•15 citations
A new type effective field approach is used to derive the thermodynamical properties of the transverse Ising model. The method is based on an exact formal identity for the two-state transverse Ising model and utilizes an exponential operator technique. The lines of critical points in the Ω-T plane are derived and the critical transverse fields ΩC are obtained analytically, for lattices with coordination numbers z = 2, 4 and 6. This simple method yields results which represent a remarkable improvement on the usual mean field approximation. Our results are also compared with those from series expansion method.
Abstract A method in Ising models is introduced by using the effective field theory (EFT) with correlations. In the present method, for the spin‐1 Ising ferromagnetic systems with three coordination numbers q (q = 3, 4 and 6), we have derived a set of linear equations by taking as a basis the thermal averages of a central spin and a perimeter spin at the site i, which is defined within spin identities and differential operator technique. By solving numerically the set of linear equations derived individually for the Ising systems, we have evaluated the phase diagrams of both the spin‐1 Ising ferromagnetic systems with a single‐ion anisotropy and the spin‐1 transverse Ising ferromagnetic systems with a transverse field. The present results of critical transverse field for the spin‐1 transverse Ising systems improve those obtained by the pair approximation. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
On the basis of the effective Hamiltonian method the static properties of the transverse Ising model with substitutional disorder have been investigated. By a perturbation expansion of the density matrix the self-consistent equation for the effective field is obtained in an analytical form. The transition temperature and the susceptibility etc. are derived from the partition function and numerically calculated. It is shown that for the diluted systems this approximation gives much lower transition temperatures and higher critical concentrations than the simple ones previously derived. In the two-component system with a strong transverse field a finite critical concentration is predicted in contrast to the previous studies. The application of this approximation to the “pseudo” one-dimensional systems having weak interchain correlations is also discussed.
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.