On the number of ground states of the Edwards-Anderson spin glass model

TL;DR

This paper investigates the number of ground states in the Edwards-Anderson spin glass model on infinite graphs, establishing that on the half-plane the number is either two or infinite, and develops tools for higher dimensions.

Contribution

It proves the first non-trivial result on the entire set of ground states in a dimension greater than one for the EA model, showing the number is either two or infinite.

Findings

Number of ground states on the half-plane is either two or infinite.

Develops new tools applicable to higher-dimensional lattices.

First such result for non-trivial dimensions in spin glass models.

Abstract

Ground states of the Edwards-Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on Z^d for any d. This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation or the number of ground states in the disordered ferromagnet. It was recently shown by Newman, Stein and the two authors that, on the half-plane Z \times N, there is a unique ground state (up to global flip) arising from the weak limit of finite-volume ground states for a particular choice of boundary conditions. In this paper, we study the entire set of ground states on the infinite graph, proving that the number of ground states on the half-plane must be two (related by a global flip) or infinity. This is the first result on the entire set of ground states in a non-trivial dimension. In the first part of the paper, we develop tools of interest to prove the analogous result on Z^d.