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Journal of Physics A Mathematical and General

Parisi solutions for the anisotropic spin glass problem

D Elderfield, D Sherrington

Published Aug 1, 1982• 35 citations
Publisher

Abstract

A Heisenberg spin glass with local uniaxial anisotropy is studied at mean field level in terms of a Parisi symmetry breaking scheme. It is shown that replica symmetry breaking leads to important modifications to the replica symmetric expressions for the susceptibilities chi T , chi L which signal the transitions in this system.
DOI
10.1088/0305-4470/15/8/011

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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.
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.
Using generalized Parisi's replica symmetry breaking scheme, the quantum spin glasstheory proposed by our previous work is extended to the first-step replica symmetry breaking. A set of new self-consistent equations of spin self-interact ions and spin glass order pa rameters are obtained. The corresponding local susceptibility is reduced. It is found that thereplica symmetry solution for vector spin glass model at low temperature must be brokenthrough calculating the free-energy of the replica symmetry breaking.
The fermionic Ising spin glass models in a transverse field are investigated in a Grassmann path integral formalism. The Parisi's scheme of one-step replica symmetry breaking (RSB) is used within the static ansatz. This formalism has already been applied in a theory in which mmm (Parisi's block-size parameter) is taken as a constant [Santos and Theumann Phys. Rev. B 72, 014447 (2005)]. Now, it is extended to consider mmm as a variational parameter. In this case, the results show that RSB is present when $T\ensuremath{\rightarrow}0$, in which the system is driven by quantum fluctuations.
Using generalized Parisi's replica symmetry breaking scheme, the quantum spin glass theory proposed by our previous work is extended to the first-step replica symmetry breaking. A set of new self-consistent equations of spin self-interact ions and spin glass order parameters are obtained. The corresponding local susceptibility is reduced. It is found that the replica symmetry solution for vector spin glass model at low temperature must be broken through calculating the free-energy of the replica symmetry breaking.