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Mean-Field Theory of Spin Glasses

Hidetoshi Nishimori

Published Jul 19, 2001• 6 citations
Publisher

Abstract

Abstract This chapter discusses the problem of spin glasses. If the interactions between spins are not uniform in space, the analysis of the previous chapter does not apply. In particular, when the interactions are ferromagnetic for some bonds and antiferromagnetic for others, the spin orientation cannot be uniform in space, unlike the ferromagnetic system. Under such a circumstance it sometimes happens that spins become randomly frozen — random in space but frozen in time. This is the intuitive picture of the spin glass phase. The chapter investigates the condition for the existence of the spin glass phase as an extension of the mean-field theory. In particular, the properties of the so-called replica-symmetric solution are explained in detail for the Sherrington–Kirkpatrick (SK) model.

DOI
10.1093/acprof:oso/9780198509417.003.0002

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•Jul 19, 2001•1 citation
Abstract This chapter discusses the problem of spin glasses. If the interactions between spins are not uniform in space, the analysis of the previous chapter does not apply. In particular, when the interactions are ferromagnetic for some bonds and antiferromagnetic for others, the spin orientation cannot be uniform in space, unlike the ferromagnetic system. Under such a circumstance it sometimes happens that spins become randomly frozen — random in space but frozen in time. This is the intuitive picture of the spin glass phase. The chapter investigates the condition for the existence of the spin glass phase as an extension of the mean-field theory. In particular, the properties of the so-called replica-symmetric solution are explained in detail for the Sherrington–Kirkpatrick (SK) model.
In this chapter we will put the concepts we have developed in Chapter 2 to work in constructing the mean field theory for an Ising spin glass. The term ‘mean field theory’ (henceforth frequently abbreviated ‘MFT’) can be interpreted in many ways. Here we will take it to mean the exact solution of a model in which the forces are of infinitely long range, so that each spin interacts equally strongly with every other one. For spin glasses, this is the Sherrington–Kirkpatrick (SK) model (1975), which we solve heuristically (though unfortunately incorrectly) in Section 3.1. The error is a somewhat subtle one, as is evident from the fact that we obtain exactly the same result in the more systematic calculations of Sections 3.2 (a direct summation of the leading terms in N −1 in high-temperature perturbation theory) and 3.3 (which uses the replica formalism). In both these approaches, we can also see how the theory itself reveals that it is wrong, since they lead to negative values of quantities which are necessarily positive. We then study the correct mean field theory, with replica symmetry breaking, in Section 3.4, including the remarkable nature of the broken ergodicity it implies and an examination of its stability. Further physical insight into the problem is obtained in Section 3.5 from the mean field equations first introduced by Thouless, Anderson and Palmer (TAP) (1977).
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.
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.
We consider a system of linear chains of Ising spins with short-range nearest-neighbor ferromagnetic interactions of strength K along the chains, and with infinite-range random interchain interactions of a Sherrington-Kirkpatrick type. Also included in the model is a local Gaussian random field with variance \ensuremath{\Delta}. In the replica symmetric case, the system is mapped exactly on the one-dimensional Ising model in a random field and a recursion method for calculating the spin-correlation functions is developed. For \ensuremath{\Delta}=0, an analytic relation for freezing temperature Tf{\mathit{T}}_{\mathit{f}}Tf​(K) is obtained. The replica-symmetric spin-glass phase is shown to be stable above the freezing temperature Tf{\mathit{T}}_{\mathit{f}}Tf​(K,\ensuremath{\Delta}), which is determined numerically.
Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.