arXiv:cond-mat/0606425 · Physical Review B

Equilibrium dynamics of spin-glass systems

A. Crisanti, Luca Leuzzi

Published Apr 5, 2007• 15 citations

Abstract

We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical $2+p$ spin-glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, H\"orner, and Sommers [Z. Phys. B: Condens. Matter 92, 257 (1993)] dynamical solution of the spherical $p$-spin spin-glass model, reproducing a stable static limit that coincides, in the case of a one step replica symmetry breaking ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off equilibrium. We formally extend our analysis to any number of replica symmetry breakings $R$. In the limit $R\ensuremath{\rightarrow}\ensuremath{\infty}$, both formulations lead to the Parisi antiparabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The formulation does not contain the additional order parameter $\ensuremath{\Delta}$ of the Sompolinsky theory.

DOI: 10.1103/physrevb.75.144301

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arXiv:cond-mat/0606425 · Physical Review B

Equilibrium dynamics of spin-glass systems

A. Crisanti, Luca Leuzzi

Published Apr 5, 2007• 15 citations

arXiv HTML PDF

Abstract

We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical

2+p

spin-glass model we test the asymptotic static limit of the Sompolinsky solution showing that it fails to yield a thermodynamically stable solution. We then present an alternative formulation, based on the Crisanti, H"orner, and Sommers [Z. Phys. B: Condens. Matter 92, 257 (1993)] dynamical solution of the spherical

p

-spin spin-glass model, reproducing a stable static limit that coincides, in the case of a one step replica symmetry breaking ansatz, with the solution at the dynamic free energy threshold at which the relaxing system gets stuck off equilibrium. We formally extend our analysis to any number of replica symmetry breakings

R

. In the limit

R\ensuremath{\rightarrow}\ensuremath{\infty}

, both formulations lead to the Parisi antiparabolic differential equation. This is the special case, though, where no dynamic blocking threshold occurs. The formulation does not contain the additional order parameter

\ensuremath{\Delta}

of the Sompolinsky theory.

DOI: 10.1103/physrevb.75.144301

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The study of the equilibrium and the out-of-equilibrium dynamics of spin glasses is crucial to understand some of the characterizing properties of disordered systems, including slow relaxation times and aging. The current chapter focuses on recent developments on this relevant topic. In the first part of the chapter, Andrea Crisanti discusses the relation between the equilibrium dynamics of mean-field spin glasses and RSB, briefly reviewing the pioneering ideas of Sommers and Sompolinsky and the subsequent theory by Crisanti, Horner and Sommers. Jorge Kurchan and Silvio Franz focus on the slow out-of-equilibrium glassy dynamics, in both mean-field models and finite-dimensional models, connecting the RSB scheme of Parisi with the concept of multithermalization. Finally, Andrea Maiorano overviews recent results on the presence (and absence) of weak ergodicity breaking in mean-field spin glasses, a pivotal assumption in many theoretical studies focusing on aging. © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.The study of the equilibrium and the out-of-equilibrium dynamics of spin glasses is crucial to understand some of the characterizing properties of disordered systems, including slow relaxation times and aging. The current chapter focuses on recent developments on this relevant topic. In the first part of the chapter, Andrea Crisanti discusses the relation between the equilibrium dynamics of mean-field spin glasses and RSB, briefly reviewing the pioneering ideas of Sommers and Sompolinsky and the subsequent theory by Crisanti, Horner and Sommers. Jorge Kurchan and Silvio Franz focus on the slow out-of-equilibrium glassy dynamics, in both mean-field models and finite-dimensional models, connecting the RSB scheme of Parisi with the concept of multithermalization. Finally, Andrea Maiorano overviews recent results on the presence (and absence) of weak ergodicity breaking in mean-field spin glasses, a pivotal assumption in many theoretical studies focusing on aging. © 2023 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.

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We present a critical analysis of the Sompolinsky theory of equilibrium dynamics. By using the spherical

2+p

p

R

. In the limit

R\ensuremath{\rightarrow}\ensuremath{\infty}

\ensuremath{\Delta}

of the Sompolinsky theory.

{\mathit{P}}_{\mathit{d}}

(q) which plays a similar role for the dynamics to that played by the usual function for the statics.

n<~4,

a full fixed-point instability with a consequent runaway towards a strong-coupling regime occurs in the parameter space while, for

d<4

and

n>4,

compatibly with the modified Harris criterion, an additional marginally stable fixed point exists which might control a new anisotropic random critical behavior. In any case, the traditional replica-symmetric Dorogovtsev-like solution for classical and quantum systems is unstable against the replica symmetry-breaking modes.