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Journal of Physics C Solid State Physics

Replica symmetry and massless modes in the Ising spin glass

Alan J. Bray, M A Moore

Published Jan 14, 1979• 138 citations
Publisher

Abstract

The n to 0 replica method is used to study the Ising spin glass below its transition temperature. The order parameter correlation functions are shown to contain massless modes ('replicon' modes) within mean-field theory. These become unstable at first order in perturbation theory about the mean-field solution. An alternative formulation of perturbation theory, however, yields a family of new solutions which break, in general, the symmetry between the replicas. Stability requirements single out a unique member of this family which exhibits (i) massless moded, and (ii) a restoration of replica symmetry. Within the framework of the new solution, perturbation theory for the order parameter (Q alpha beta ) breaks down for dimensionalities d<or=4 due to an infrared divergence. Four dimensions is interpreted as the 'lower critical dimensionality' of the theory, below which there can be no order of the Edwards-Anderson kind (1975).

DOI
10.1088/0022-3719/12/1/020

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arXiv:cond-mat/0505522•Physical Review B•Jul 19, 2005•4 citations
The n to 0 replica method is used to study the Ising spin glass below its transition temperature. The order parameter correlation functions are shown to contain massless modes ('replicon' modes) within mean-field theory. These become unstable at first order in perturbation theory about the mean-field solution. An alternative formulation of perturbation theory, however, yields a family of new solutions which break, in general, the symmetry between the replicas. Stability requirements single out a unique member of this family which exhibits (i) massless moded, and (ii) a restoration of replica symmetry. Within the framework of the new solution, perturbation theory for the order parameter (Q alpha beta ) breaks down for dimensionalities d<or=4 due to an infrared divergence. Four dimensions is interpreted as the 'lower critical dimensionality' of the theory, below which there can be no order of the Edwards-Anderson kind (1975).
The replica-symmetric cluster expansion for the Ising spin glass is reviewed and explained. The expansion allows computation of various quantities and correlations directly in finite dimensions, without recourse to the mean-field theory or the saddle point. The expansion for the free energy is recovered with the use of external fields that couple to the replica-symmetric Edwards-Anderson order parameter. The expansion scheme is applied to the Edwards-Anderson susceptibility. The expansion diverges in the mean-field [infinite dimensionality] limit. The divergent expansion, when naively summed, yields the negative replicon mass of the mean-field theory. This so-called Almeida-Thouless instability of the replica-symmetric mean-field theory is argued to be absent in finite dimensions. The argument, similar to the one already reported for the entropy, depends on the low-temperature behavior of terms in the expansion at finite dimensions. \textcopyright{} 1996 The American Physical Society.
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
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.
The asymptotical behavior of physical quantities, like the order parameter, the replicon, and longitudinal masses, is studied around the zero-field spin-glass transition point when a small external magnetic field is applied. An effective field theory to model this asymptotics contains a small perturbation in its Lagrangian which breaks the zero-field symmetry. A first-order renormalization group supplemented by perturbational results provides the scaling functions. The perturbative zero of the scaling function for the replicon mass defines a generic Almeida-Thouless surface stemming from the zero-field fixed point.
A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed, in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension du=6, and, in particular, explicit corrections in εlnt and (εlnt)2 with t=(Tc−T)/Tc and ε=6−d. This allows us to verify that standard scaling is satisfied with β=1+ε/2+0(ε2). We have also investigated the transverse (replicon) correlation function, particularly at zero overlap. Near du, G00 (p) ~ t/P4 (for p~0) is the most obvious obstacle to a meaningful theory in d=3. If we make the twofold assumption that (i) scaling applies, and (ii) the replicon propagator is dominated by the spectrum of the (small) transverse masses, we obtain the softer behavior G00(p) ~ (1/p2–η). (t/p1/v)β and a prediction for a soft replicon spectrum in tγk2γ/β γ/β instead of tk2 at du. We have checked tγ to one loop and work is in progress to check k2γ/β to the same order. Taking the above divergence in p−(d+2−η)/2 as the leading divergence defines the lower critical dimension d ℓ by d ℓ=2−η (d ℓ). Known values of η (at d=6, from the ε expansion at Tc, or from numerical work at d=4, 3) are compatible with d ℓ ~ 2.5±.3.
We analyze the infinite range Ising spin glass in a transverse-field $\ensuremath{\Gamma}$ below the critical temperature by a one step replica symmetry breaking theory. The set of nnn replicas is divided in rrr blocks of mmm replicas each. We present results for different values of the block-size parameter mmm. The spin operators are represented by bilinear combinations of fermionic fields and we compare the results of two models: In the four-states (4-S) model the diagonal Siz{S}_{i}^{z}Siz​ operator has two unphysical vanishing eigenvalues, that are suppressed by a restraint in the two-states (2-S) model. In the static approximation we obtain qualitatively similar results for both models. They both exhibit a critical temperature ${T}_{c}(\ensuremath{\Gamma})$ that decreases when $\ensuremath{\Gamma}$ increases, until it reaches a quantum critical point at the same value of ${\ensuremath{\Gamma}}_{c}$ and they are both unstable under replica symmetry breaking in the whole spin glass phase. Below the critical temperature we present results for the order parameters and free energy.