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 $λ$. 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.

Figures (3)

• Figure 1: Black dots and white dots represent the sites in $\Lambda_7$ and its boundary $\partial\Lambda_7$, respectively. Solid lines and dashed lines represent bonds in $\mathcal{B}_7$ and $\partial\mathcal{B}_7$, respectively.
• Figure 2: Four translated copies of $\Lambda_3$ are embedded into $\Lambda_9$. This corresponds to the optimal choice \ref{['e:Kopt']}.
• Figure 3: Schematic pictures of the low-temperature equilibrium states in the three-replica systems. The rounded rectangles labeled as 1, 2, and 3 indicate replicas. The dotted lines with $+0$ and $-0$ indicate coupling between the replicas with replica coupling parameters $\lambda\downarrow+0$ and $\lambda\uparrow-0$, respectively. (a) The replicated ferromagnetic Ising model in two or higher dimensions at low temperatures. The equilibrium state is the equal mixture of two states in which most spins in each replica are aligned with each other. The spins in replicas 1 and 2 are pointing in the same direction, while the spins in replica 3 in the opposite direction. Here, the replica permutation symmetry is broken, but it simply reflects the $\mathbb{Z}_2$ symmetry breaking that takes place in the single (non-replicated) system. (b) The duplicated random energy model at low temperatures. The equilibrium state is a pure state in which the replicas 1 and 2 are in the ground state, and the replica 3 is in the mixture of all other spin configurations. Here, the breaking of the replica permutation symmetry is the only symmetry breaking exhibited by the system and provides an intrinsic characterization of the spin glass order.

Theorems & Definitions (7)

• Lemma 2.1: Infinite-volume limits of the free energy densities
• Theorem 3.1: Griffiths' theorem for $\mathbb{Z}_2$ invariant model
• Theorem 3.2: Griffiths-type theorem for the general EA model
• Theorem 3.3: Spontaneous breakdown of replica permutation symmetry
• Corollary 3.4: Broadening of the overlap implies literal replica symmetry breaking
• Lemma 4.1
• Theorem C.1: Griffiths-type theorem for a general first-order phase transition