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Journal of Physics C Solid State Physics

Replica theory of quantum spin glasses

Alan J. Bray, M A Moore

Published Aug 30, 1980• 248 citations
Publisher

Abstract

The Sherrington-Kirkpatrick model, generalised to quantum spins, is discussed within an exact replica formalism and reduced to a (time-dependent) single-site problem. The existence of a phase transition is established for all values of the spin S.
DOI
10.1088/0022-3719/13/24/005

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In the present paper we present a detailed analysis of quantum effects in the spin-glass transition as described by a quantum Heisenberg analogue of the Sherrington-Kirkpatrick model. The spin operators are represented in terms of two fermion fields and the problem is reduced to that of n fermion levels at one site in a random time-dependent field and with an interaction delocalized in time. It is shown that within a Hartree-Fock approximation in a replica-symmetric theory one obtains a mean-field description of the transition with satisfactory zero-temperature properties. The transition is described by two order parameters: the static magnetic susceptibility and the spin-glass order parameter. The saddle-point equations obtained in this work are analogous to those of Sommers for a theory of classical spins.
This thesis is concerned with the mean field theory of spin glasses within the replica method approach. We have studied the Sherrington-Kirkpatrick model [33,4δ] both with Ising and m-component spins. It is shown that the replica symmetric solution gives an instability in the condensed phase which requires a breaking of replica symmetry in the spin glass phase. The various broken replica symmetry solutions proposed thus far (Bray and Moore [52], Blandin et al [53], Parisi [54]) are reviewed and particular attention is given to the one proposed by Parisi. The order parameter susceptibility and the stability of the solution are studied within this breaking of replica symmetry scheme which give results in agreement with the predictions of TAP [47] and Bray and Moore [52]. The breakdown of linear response theory in the spin glass phase is also discussed.
We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.
The renormalization group treatment of spin glasses has been a theoretical challenge since shortly after the formulation of the Sherrington–Kirkpatrick model. This chapter brings three perspectives on this topic. Section 4.1, by T. Lubensky, reviews the early replica symmetric description of the problem, which predate and in some ways anticipate the discovery of the instability of that treatment by de Almeida and Thouless. The perturbative renormalization group approach to the ensuing transition is presented in Sec. 4.2, contributed jointly by T. Temesvari and I. Kondor. The findings of this approach motivate the consideration of non-perturbative renormalization group schemes, which are discussed in Sec. 4.3 written by M. C. Angelini. © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
In the replica formalism for an Ising spin glass the interaction energy between two sites depends on the number of replicas of the two sites that have parallel or antiparallel spins; therefore, the energy may take n+1 different values even when permutation symmetry is assumed in the space of n replicas. This insight is used to propose a new replica-symmetric solution to the renormalization-group equations for a one-dimensional spin-glass model with long-range interactions. In a simple approximation this solution gives a value of (2\ensuremath{\epsilon}${)}^{\mathrm{\ensuremath{-}}1/2}$ for the correlation-length exponent [\ensuremath{\epsilon}=1-\ensuremath{\sigma} and (1+\ensuremath{\sigma})/2 is the range parameter for the random interactions]. The new solution is argued to be equivalent to the replica-symmetric ansatz for the Sherrington-Kirkpatrick model.