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Progress of Theoretical Physics

Second Order Treatment of a Pseudo-Spin System. II: Application to the Transverse-Field Ising Model

Carla Buzano, Mario Rasetti, M. Vadacchino

Published Sep 1, 1982• 3 citations
Publisher

Abstract

The transverse-field Ising model is solved within the scheme of the generalized mean-field method earlier proposed by the authors. The uncompatibility between the self-consistency condition and the thermal equilibrium is corrected through the introduction of a correlated mean-field, realized by a virtual-field parameter. Numerical results are discussed in detail, from which the specific heat and the transverse-field/critical-temperature phase boundary diagram are derived.

DOI
10.1143/ptp.68.716

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The transverse-field Ising model is solved within the scheme of the generalized mean-field method earlier proposed by the authors. The uncompatibility between the self-consistency condition and the thermal equilibrium is corrected through the introduction of a correlated mean-field, realized by a virtual-field parameter. Numerical results are discussed in detail, from which the specific heat and the transverse-field/critical-temperature phase boundary diagram are derived.
Abstract We evaluate a mean field method to describe the properties of the ground state of the Ising chain in a transverse magnetic field. Specifically, a method of the Bethe–Peierls type is used by solving spin blocks with a self-consistency condition at the borders. The computations include the critical point for the phase transition, exponent of magnetisation and energy density. All results are obtained using basic quantum mechanics at an undergraduate level. The advantages and the limitations of the approach are emphasised.
A mean-field approximation is used to investigate the quantum transverse Ising spin-glass model in the Edwards-Anderson version. We introduce two order parameters, diagonal and off diagonal in replica indices, and discuss their physical significance. The effective free-energy function is obtained, and the critical value of the transverse field $\ensuremath{\Gamma}$ at zero temperature is calculated. We find the behavior of the order parameters near the zero-temperature critical point. The phase diagram in the $\ensuremath{\Gamma}$,TTT plane is also given.
The classical transverse field Ising spin-glass model with short-range interactions is investigated beyond the mean-field approximation for a real d -dimensional lattice. We use an appropriate non-trivial modification of the Bethe-Peierls method recently formulated for the Ising spin-glass. The zero-temperature critical value of the transverse field and the linear susceptibility in the paramagnetic phase are obtained analytically as functions of dimensionality d . The phase diagram is also calculated numerically for different values of d . In the limit , known mean-field results are consistently reproduced.