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Journal of Physics C Solid State Physics

Stability of a crystal-field split spin glass

Peter Mottishaw, D Sherrington

Published Sep 20, 1985• 46 citations
Publisher

Abstract

A generalisation of the Sherrington-Kirkpatrick Ising spin glass model to spins S z =0,+or-1, . . . +or-S with exchange J ij S i z S j z , where the J ij are randomly distributed, and crystal field D(S i z ) 2 , is examined with regard to the character of its transitions and the stability of its phases, in order to clarify and correct earlier work. Both continuous and discontinuous transitions are found. Conventional Parisi theory with renormalised parameters suffices near the continuous transition away from the tricritical point, but near the tricritical point and the first-order transitions, new effects are predicted.

DOI
10.1088/0022-3719/18/26/028

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A generalisation of the Sherrington-Kirkpatrick Ising spin glass model to spins S z =0,+or-1, . . . +or-S with exchange J ij S i z S j z , where the J ij are randomly distributed, and crystal field D(S i z ) 2 , is examined with regard to the character of its transitions and the stability of its phases, in order to clarify and correct earlier work. Both continuous and discontinuous transitions are found. Conventional Parisi theory with renormalised parameters suffices near the continuous transition away from the tricritical point, but near the tricritical point and the first-order transitions, new effects are predicted.
We study a quantum generalization of the infinite-range Sherrington-Kirkpatrick spin-glass model with biaxial crystal-field effects described by two uniaxial anisotropy parameters D x and D y . For spin dimensionality S = 1 we report an analytical and numerical analysis in the ( T , D x , D y ) parameter space (with T being the temperature). For D ≡ D x = D y the model effectively becomes classical and identical with the crystal-field-split spin-glass Ising model (introduced by Ghatak and Sherrington) showing a discontinuous phase transition to the spin-glass phase on a portion of the T - D line.
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.
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.
The Sherrington-Kirkpatrick and the nearest-neighbour X-Y spin-glass models are studied.The effect of a small uniform symmetry breaking field on the infinite-ranged Sherrington-Kirkpatrick Ising spin-glass model is discussed.It is shown that the off-diagonal contributions to the fluctuation-dissipation formula vanish for a symmetric distribution of interactions and the limit of zero field.This contradicts the result of Young and Kirkpatrick based on numerical evidence for small samples.The dynamics of the Sherrington-Kirkpatrick spin-glass are investigated.Below the critical temperature this model exhibits non-ergodic behaviour.Consequently, in order to obtain timedependent correlation functions it is necessary to average over the initial states with a Boltzmann weight.This necessitates the use of "replicas".It is shown that in Parisi's scheme the "dynamical" Edwards-Anderson order parameter is given by q(x =1), whereas the statistical mechanics order parameter is given by r 1 q(x)dx.The result for the statistical mechanics order parameter j 0 implies that there is no violation of the fluctuation-dissipation theorem once an average over samples is performed.It is also shown that the system in general keeps its memory of the initial state for all time.However, this memory is "lost" if one makes a replica symmetric ansatz for the order parameters.This illus trates the connection between replica symmetry breaking and nonergodicity.The nearest neighbour X-Y spin-glass in both two-and threedimensions is investigated by Monte Carlo simulation.In two--11ln[ln Xcn + 1-42] versus In T SG } X2 (Tq ) for T (3D) 4.11 4.12a 4.12b 4.12c 4.13a ' 4.13b > Plots of the data for T (3D) 4.13c 4.14a 4.14b 4.15a 4.15b Plots of £ ^ for 3D SG A plot of the data for £ assuming that LCD = 3 SG A plot of the data for £ assuming that LCD = 3 SG and p__ = 1.2 SG A5.1a A5.
We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
I study in detail the quantum Sherrington-Kirkpatrick (SK) model, i.e., the infinite-range Ising spin glass in a transverse field, by solving numerically the effective one-dimensional model that the quantum SK model can be mapped to in the thermodynamic limit. I find that the replica symmetric solution is unstable down to zero temperature, in contrast to some previous claims, and so there is not only a line of transitions in the (longitudinal) field-temperature plane (the de Almeida-Thouless, AT, line) where replica symmetry is broken, but also a quantum de Almeida-Thouless (QuAT) line in the transverse field-longitudinal field plane at T=0. If the QuAT line also occurs in models with short-range interactions its presence might affect the performance of quantum annealers when solving spin glass-type problems with a bias (i.e., magnetic field).