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Journal of Physics C Solid State Physics

CPA approach to the random Ising model in a transverse field

Peter Prelovšek, R. Pirc

Published Mar 28, 1977• 16 citations
Publisher

Abstract

The dynamics of a two-component Ising model in a transverse field (IMTF) with randomness during both in the transverse field and the Ising coupling is treated within the single-site coherent potential approximation (CPA). The authors deal primarily with the polarized phase; they replace the fluctuating longitudinal component of the molecular field by two effective parameters, one for each species, which are determined self-consistently. The CPA spectra of the collective spin excitations are found analytically in the dilute case using a Hubbard model density for the Ising interaction, and are also calculated numerically for the general two-component IMTF. It is shown that the static susceptibility cannot diverge within the single site CPA. Characterizing the phase transition by the softening of the lowest mode in the spectrum. The stability conditions for the polarized phase at T=O are then determined.

DOI
10.1088/0022-3719/10/6/016

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The dynamics of a two-component Ising model in a transverse field (IMTF) with randomness during both in the transverse field and the Ising coupling is treated within the single-site coherent potential approximation (CPA). The authors deal primarily with the polarized phase; they replace the fluctuating longitudinal component of the molecular field by two effective parameters, one for each species, which are determined self-consistently. The CPA spectra of the collective spin excitations are found analytically in the dilute case using a Hubbard model density for the Ising interaction, and are also calculated numerically for the general two-component IMTF. It is shown that the static susceptibility cannot diverge within the single site CPA. Characterizing the phase transition by the softening of the lowest mode in the spectrum. The stability conditions for the polarized phase at T=O are then determined.
The coherent potential approximation has been applied to the bond diluted Ising model in a transverse field. Green function techniques decoupled in a generalised RPA have been used. The static and dynamic properties of this system have been studied at T=O and it is concluded that the form of the CPA equations inhibits the appearance of a second-order phase transition, if the divergence of the static susceptibility is the only criterion locating the transition. A discussion of this problem and its relation to other models is also included.
Recently, there has been interest in the problems of disorder in the Ising model in a transverse field, which may apply to DKDP-KDP mixed systems and diluted vanadates. The site and bond diluted transverse Ising models have already been studied using Green's function techniques along with the coherent potential approximation. However, the situation as regards to the existence of a second order phase transition and the divergence of the static susceptibility is not very clear. In the present paper, a spin ½ Ising model in a transverse field and random bond (site) dilution is studied by variational cluster and perturbation expansion method. The critical temperature for the existence of a second order phase transition is calculated as a function of concentration of bonds (spins) and the transverse field. Variation of the critical concentration, for the existence of phase transition, is studied as a function of the transverse field. A phase diagram in the temperature-concentration-transverse field space is constructed. The static susceptibility is calculated and is found to diverge at the critical temperature, as is expected for the existence of a second order phase transition.
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.
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.
The Ising Model in a transverse field is investigated employing a field theoretical diagrammatic technique. The diagrams are classified with respect to a high density parameter 1/z, where z is the number of spins interacting with a given spin. The energy spectrum of collective excitations has been obtained both in the disordered phase and in the ordered phase. Summation of the chain diagrams (of order 1) reproduces the previous results. We have calculated the contributions of all diagrams of order 1/z and showed that the RPA commonly used has failed to include consistently a class of diagrams which reflect the kinematics of spin operators. The damping of the excitation waves is also obtained.