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Journal of Physics C Solid State Physics

Numerical simulation of the classical Ising model in a transverse field statical properties

Peter Prelovšek, Igor Sega

Published May 28, 1978• 17 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/11/10/021

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Journal of Liaoning Normal University•Jan 1, 2005•0 citations
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 molecular-dynamics technique is applied to the study of the classical ($S=\ensuremath{\infty}$) Ising model in a transverse field on a simple-cubic lattice. The results for the longitudinal- and transverse-relaxation shape function are presented for a set of temperatures in the disordered phase at one value of the transverse field. The longitudinal spectra at $T=\ensuremath{\infty}$ and in the critical regime are analyzed via the Mori's continued-fraction expansion and compared with various approximate results within this approach. The relation of the observed central mode to a domain-wall motion is also discussed. However, due to heavy-damping phenomena no well-defined propagating modes are observed in the critical region. Finally, transverse correlation functions are tested against their diffusive character.
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.
Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.
Both weak-coupling and strong-coupling series expansions are calculated for the Ising model in a transverse field at zero temperature in three dimensions. Series are obtained for the ground-state energy, the magnetization, the susceptibility, and the energy gaps, on the simple cubic, the body-centred cubic and the face-centred cubic lattices. The analysis of the critical behaviour is consistent with the behaviour predicted by renormalization group theory for the four-dimensional simple Ising model. There is a remarkable degree of universality between all three lattices.
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.