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Journal of Physics A Mathematical and General

The two-dimensional bond-diluted transverse Ising model at zero temperature

Raimundo R. dos Santos

Published May 1, 1981• 21 citations
Publisher

Abstract

The critical behaviour of the bond-diluted two-dimensional quantum transverse Ising model is investigated by an approximate cluster decimation transformation. The dependence of the critical field on the concentration shows a discontinuity at the percolation concentration, associated with the existence of long-range order in the percolating cluster at non-zero transverse field, according to Harris' conjecture (1974).

DOI
10.1088/0305-4470/14/5/014

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The critical behaviour of the bond-diluted two-dimensional quantum transverse Ising model is investigated by an approximate cluster decimation transformation. The dependence of the critical field on the concentration shows a discontinuity at the percolation concentration, associated with the existence of long-range order in the percolating cluster at non-zero transverse field, according to Harris' conjecture (1974).
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.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.
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.