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Journal of Physics C Solid State Physics

The dilute Ising model in a transverse field

Roger James Elliott, I D Saville

Published Dec 7, 1974• 22 citations
Publisher

Abstract

The dilute Ising model in a transverse field which is related to phase transitions in a number of dilute systems has been studied by series expansion techniques to obtain the susceptibility for the face-centred cubic lattice. The surface of critical points in the temperature-concentration-transverse field space is obtained together with some indication of the critical exponent gamma .

DOI
10.1088/0022-3719/7/23/016

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Journal of Physics A Mathematical and General•Apr 1, 1984•24 citations
The dilute Ising model in a transverse field which is related to phase transitions in a number of dilute systems has been studied by series expansion techniques to obtain the susceptibility for the face-centred cubic lattice. The surface of critical points in the temperature-concentration-transverse field space is obtained together with some indication of the critical exponent gamma .
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.
The Ising model (exchange constant J) with a transverse field Gamma can be regarded as a simple model of hydrogen bonded ferroelectrics and of certain magnetic systems. The transition temperature T C decreases as Gamma increases and falls to zero at a critical field Gamma C . This transition is studied in the quadratic and simple cubic lattices with nearest neighbour interactions by a power series expansion of the susceptibility at high T up to fifth order in (J/kT) to all orders in ( Gamma /kT). The coefficients are calculated by diagram counting and time ordering perturbation methods. The critical Gamma against T curve is derived and the critical exponent gamma of the susceptibility. This appears to vary smoothly with Gamma but it is argued that this is an artifact due to the use of a short series and that the critical exponents are probably independent of Gamma as long as T C remains finite but change discontinuity at T C =0.
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.
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 .
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.