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arXiv:cond-mat/0211707 · Physical Review Letters

Free Energy Fluctuations in Ising Spin Glasses

Timo Aspelmeier, M A Moore

Published Apr 30, 2003• 24 citations
arXivHTMLPDF

Abstract

The sample-to-sample fluctuations of the free energy in finite-dimensional Ising spin glasses are calculated, using the replica method, from higher order terms in the replica number n. It is shown that the Parisi symmetry breaking scheme does not give the correct answers for these higher order terms. A modified symmetry breaking scheme with the same stability is shown to resolve the problem.

DOI
10.1103/physrevlett.90.177201

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The sample-to-sample fluctuations of the free energy in finite-dimensional Ising spin glasses are calculated, using the replica method, from higher order terms in the replica number n. It is shown that the Parisi symmetry breaking scheme does not give the correct answers for these higher order terms. A modified symmetry breaking scheme with the same stability is shown to resolve the problem.
Parisi's replica-symmetry-breaking scheme is applied to the quantum Sherrington-Kirkpatrick Ising spin glass in a transverse magnetic field. The order-parameter function q(x) is calculated near Tf{\mathit{T}}_{\mathit{f}}Tf​, while for all temperatures the first stage of replica-symmetry breaking is calculated. Internal energy, entropy, and magnetic susceptibility are calculated in order to investigate the quality of both the replica-symmetric and the replica-symmetry-broken theories.
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 one-dimensional Ising spin-glass model with power-law long-range interactions is a useful proxy model for studying spin glasses in higher space dimensions and for finding the dimension at which the spin-glass state changes from having broken replica symmetry to that of droplet behavior. To this end we have calculated the exponent that describes the difference in free energy between periodic and antiperiodic boundary conditions. Numerical work is done to support some of the assumptions made in the calculations and to determine the behavior of the interface free-energy exponent of the power law of the interactions. Our numerical results for the interface free-energy exponent are badly affected by finite-size problems.
The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.
The replica-symmetry-breaking (RSB) solution of the infinite-range Ising spin glass in the presence of a transverse field is obtained. The quenched free energy and the phase boundary of the glass transition temperature versus the transverse field are calculated at first-step RSB without using the static approximation. We demonstrate that replica symmetry (RS) has to be broken in the spin-glass phase by comparing the free energies of the RSB and RS solutions. No evidence is found to support an intermediate spin-glass phase with replica symmetry.