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arXiv:1309.6987

Full replica symmetry breaking in spin-glass-like systems without reflection symmetry

T. I. Schelkacheva, Elena E. Tareyeva, N. M. Chtchelkatchev

Published Sep 26, 2013• 0 citations
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Abstract

We investigate the expansion near T_c of the free energy corresponding to the generalized Sherrington--Kirkpatrick model with arbitrary diagonal operators $\hat{U}$ standing instead of Ising spins. We focus on the case when U is an operator without reflection symmetry. The principal prescription for obtaining a full replica symmetry breaking solution is derived in general case in the neighbourhood of the transition temperature. An illustrating example is considered demonstrating the implicitly build solution.

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We investigate the expansion near T_c of the free energy corresponding to the generalized Sherrington--Kirkpatrick model with arbitrary diagonal operators U^\hat{U}U^ standing instead of Ising spins. We focus on the case when U is an operator without reflection symmetry. The principal prescription for obtaining a full replica symmetry breaking solution is derived in general case in the neighbourhood of the transition temperature. An illustrating example is considered demonstrating the implicitly build solution.
We solve the fermionic version of the Ising spin glass for arbitrary filling µ and temperature T taking into account replica symmetry breaking. Using a simple exact mapping from µ to the anisotropy parameter D , we also obtain the solution of the S = 1 Sherrington-Kirkpatrick model. An analytic expression for T = 0 gives an improved critical value for the first-order phase transition. We revisit the question of stability against replica-diagonal fluctuations and find that the appearance of complex eigenvalues of the Almeida-Thouless matrix is not an artefact of the replica-symmetric approximation.
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.
An extensive study of the mean-field static solution of the random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable replica-symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a full-replica-symmetry-breaking spin-glass phase. The nature of the transition can be either of the second order (such as in the Sherrington-Kirkpatrick model) or, at temperatures below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter and accompanied by a latent heat. In this last case, coexistence of phases takes place. The thermodynamics is worked out in the full-replica-symmetry-breaking scheme, and the related Parisi equations are solved using a pseudospectral method down to zero temperature.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
We investigate stability of replica symmetry breaking solutions in generalized ppp-spin models. It is shown that the kind of the transition to the one-step replica symmetry breaking state depends not only on the presence or absence of the reflection symmetry of the generalized "spin"-operators U^\hat{U}U^ but on the number of interacting operators and their individual characteristics.