Stability of the Parisi Solution for the Sherrington-Kirkpatrick model near T=0

Abstract

To test the stability of the Parisi solution near T=0, we study the spectrum of the Hessian of the Sherrington-Kirkpatrick model near T=0, whose eigenvalues are the masses of the bare propagators in the expansion around the mean-field solution. In the limit $T\ll 1$ two regions can be identified. In the first region, for $x$ close to 0, where $x$ is the Parisi replica symmetry breaking scheme parameter, the spectrum of the Hessian is not trivial and maintains the structure of the full replica symmetry breaking state found at higher temperatures. In the second region $T\ll x\le 1$ as $T\to 0$, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full replica symmetry breaking state collapse. In this region only two eigenvalues are found: a null one and a positive one, ensuring stability for $T\ll 1$. In the limit $T\to 0$ the width of the first region shrinks to zero and only the positive and null eigenvalues survive. As byproduct we enlighten the close analogy between the static Parisi replica symmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it equals the static limit of the dynamic susceptibility computed via the modified fluctuation dissipation theorem.

24 pages, 6 figures, J. Phys. A (in press)

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Cited by in corpus (5)

• Field theory for zero temperature soft anharmonic spin glasses in a field
• Continuous replica-symmetry breaking in mean-field spin-glass models: Perturbation expansion without the replica trick
• Orientational glass: full replica symmetry breaking in generalized spin glass-like models without reflection symmetry
• The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution
• Nature of spin glass order in physical dimensions