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.

arXiv comment24 pages, 6 figures, J. Phys. A (in press)

References in corpus (15)

• Analysis of the infinity-replica symmetry breaking solution of the Sherrington-Kirkpatrick model
• The spherical 2+p spin glass model: an analytically solvable model with a glass-to-glass transition
• Amorphous-amorphous transition and the two-step replica symmetry breaking phase
• Replica Fourier Transforms on Ultrametric Trees, and Block-Diagonalizing Multi-Replica Matrices
• Interface energies in Ising spin glasses
• Low temperature solution of the Sherrington-Kirkpatrick model
• Double Criticality of the Sherrington-Kirkpatrick Model at T=0
• Block diagonalizing ultrametric matrices
• Scaling and Renormalization Group in Replica Symmetry Breaking space: Evidence for a simple analytical solution of the SK model at zero temperature
• Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model
• On Equilibrium Dynamics of Spin-Glass Systems
• Sherrington-Kirkpatrick model near $T=T_c$: expanding around the Replica Symmetric Solution
• New technique for replica symmetry breaking with application to the SK-model at and near T=0
• Low temperature spin glass fluctuations: expanding around a spherical approximation
• Low Temperature Mass Spectrum in the Ising Spin Glass

Cited by in corpus (8)

• TASI lectures on complex structures
• 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
• The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution
• Orientational glass: full replica symmetry breaking in generalized spin glass-like models without reflection symmetry
• Nature of spin glass order in physical dimensions
• The Sherrington-Kirkpatrick model near T_c and near T=0
• Derivation of the spin-glass order parameter from stochastic thermodynamics