Griffiths-type theorems for short-range spin glass models

Abstract

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap $R^{1,2}$ implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter $\lambda$. In ${\mathbb{Z}}_2$ invariant models such as the standard Edwards-Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards-Anderson order parameter. This generalization of Griffiths' theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model.

31 pages, 3 figures, There is a 25-minute video that explains the main results of the present work: https://youtu.be/BF3hJiY1xvI

References in corpus (4)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• The three dimensional Ising spin glass in an external magnetic field: the role of the silent majority
• Absence of replica symmetry breaking in the random field Ising model
• Scaling Analysis of Domain-Wall Free-Energy in the Edwards-Anderson Ising Spin Glass in a Magnetic Field