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Journal of Physics C Solid State Physics

The diluted transverse Ising model

T. Kaneyoshi

Published Jun 10, 1986• 25 citations
Publisher

Abstract

Two effective-field treatments are presented for the diluted transverse Ising model. The methods include the effects of correlation. The critical temperature, the longitudinal and transverse magnetisations, the initial transverse susceptibility and the two-site spin correlation function are calculated for a square lattice. It was found that, in particular, the transverse magnetisation and the initial transverse susceptibility exhibit behaviour which is interesting in the diluted case.

DOI
10.1088/0022-3719/19/16/018

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Two effective-field treatments are presented for the diluted transverse Ising model. The methods include the effects of correlation. The critical temperature, the longitudinal and transverse magnetisations, the initial transverse susceptibility and the two-site spin correlation function are calculated for a square lattice. It was found that, in particular, the transverse magnetisation and the initial transverse susceptibility exhibit behaviour which is interesting in the diluted case.
Within the framework of an effective-field theory with correlation evaluated for the Ising model with transverse field, the authors discuss the following cases: a diluted ferromagnetic square lattice with a transverse field; the magnetic behaviour of a transverse Ising model with diluted surface in a simple cubic lattice; and the surface magnetism of random transverse field at the surface. The surface magnetism is examined as a function of Delta =(J s /J)-1, the concentrations p and q of magnetic atoms at the surface, and Omega s and Omega b , the surface and bulk transverse field respectively. The phase diagrams for surface ordering are obtained and some characteristic behaviours of magnetism at the diluted surface are presented. The present framework is quite superior to the standard MFA, and whenever comparison is possible a satisfactory qualitative (and to a certain extent quantitative) agreement with results available in the literature is found.
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 effect of random bond (site) dilution on the phase transition of a three-dimensional transverse Ising model is studied by a cluster variation, along with perturbation expansion method. Complete phase diagrams in the temperature-concentration-transverse-field space are constructed for both the cases. The value of critical transverse field as a function of concentration, at zero temperature, exhibits a discontinuity at the critical value of concentration for the existence of phase transition, as conjectured by Harris. As is expected for the existence of a second-order phase transition, the static susceptibility diverges on the critical surface.
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 .