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Journal of Liaoning Normal University

The Theoretical Calculation of the Properties of Transverse Ising Model

Tian Li-zhen

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

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.

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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 paper we investigate the effect of a random transverse field, distributed according to a trimodal distribution, on the phase diagram and magnetic properties of a two-dimensional lattice (square with z=4),  ferromagnetic Ising system consisting of magnetic atoms with spin-1. This study is done using the effectivefield theory (EFT) with correlations method. The equations are derived using a probability distribution method based on the use of exact Van der Waerden identities. We present our numerical results, such as the phase diagrams, the thermal variations of the transverse magnetization, the internal energy, the magnetic specific heat as a function of different values of p, the concentration of the random transverse field. As a result, the critical values of transverse field , temperature and concentration were obtained for the square lattice.
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.
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.
A transverse Ising spin system, in the presence of time-dependent longitudinal field, is studied by the effective-field theory (EFT). The effective-field equations of motion of the average magnetization are given for the simple cubic lattice (Z = 6) and the honeycomb lattice (Z = 3). The Liapunov exponent λ is calculated for discussing the stability of the magnetization and it is used to determine the phase boundary. The dynamic phase transition diagrams in h 0 /ZJ — Γ/ZJ plane and in h 0 /ZJ — T/ZJ plane have been drawn, and there is no dynamical tricritical point on the dynamic phase transition boundary. The effect of the thermal fluctuations upon the dynamic phase boundary has been discussed.