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Journal of Physics C Solid State Physics

Glass-like phase for continuous distributions of exchange interactions-solvable model for the spin glass

Seiichiro Katsura

Published Apr 14, 1977• 10 citations
Publisher

Abstract

The random-bond Ising model with continuous (rectangular) distribution of exchange interactions is treated in the Bethe approximation and is shown to have the glass-like phase. The phase diagram similar to that by Sherrington and Kirkpatrick (1975) is obtained. The random-bond Ising model on the infinite Bethe lattice is regarded as an exactly solvable model showing the spin glass.

DOI
10.1088/0022-3719/10/7/002

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The random-bond Ising model with continuous (rectangular) distribution of exchange interactions is treated in the Bethe approximation and is shown to have the glass-like phase. The phase diagram similar to that by Sherrington and Kirkpatrick (1975) is obtained. The random-bond Ising model on the infinite Bethe lattice is regarded as an exactly solvable model showing the spin glass.
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A generalisation of the Sherrington-Kirkpatrick Ising spin glass model to spins S z =0,+or-1, . . . +or-S with exchange J ij S i z S j z , where the J ij are randomly distributed, and crystal field D(S i z ) 2 , is examined with regard to the character of its transitions and the stability of its phases, in order to clarify and correct earlier work. Both continuous and discontinuous transitions are found. Conventional Parisi theory with renormalised parameters suffices near the continuous transition away from the tricritical point, but near the tricritical point and the first-order transitions, new effects are predicted.
We study the three-dimensional quantum Ising spin glass in a transverse magnetic field following the evolution of the bond probability distribution under renormalization-group transformations. The phase diagram (critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ vs transverse field \ensuremath{\Gamma}) we obtain shows a finite slope near T=0, in contrast with the infinite slope for the pure case. Our results compare very well with the experimental data recently obtained for the dipolar Ising spin glass LiHo0.167{\mathrm{LiHo}}_{0.167}LiHo0.167​Y0.833{\mathrm{Y}}_{0.833}Y0.833​F4{\mathrm{F}}_{4}F4​, in a transverse field. This indicates that this system is more appropriately described by a model with short-range interactions than by an equivalent Sherrington-Kirkpatrick model in a transverse field.
The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P ( q ) is defined and calculated for the present quantum case. The P ( q ) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.