Algebraic Topology of Spin Glasses

TL;DR

This paper uses algebraic topology to analyze frustration in high-dimensional Ising spin glasses, explaining why spin glass phases exist in three or more dimensions but not in two, based on the stability of topological domain walls.

Contribution

It introduces a topological framework for understanding frustration and domain walls in spin glasses, linking global frustration to the existence of spin glass phases in higher dimensions.

Findings

Topologically nontrivial domain walls relate to global frustration.

Thermal fluctuations destroy order in 2D but not in 3D or higher.

Topological effects explain the dimensional dependence of spin glass phases.

Abstract

We study topology of frustration in $d$-dimensional Ising spin glasses with $d\ge 2$ with nearest-neighbor interactions. We prove the following: For any given spin configuration, the domain walls on the unfrustration network are all transverse to a frustrated loop on the unfrustration network, where a domain wall is defined to be a connected element of the collection of all the $(d-1)$-cells which are dual to the bonds having an unfavorable energy, and the unfrustration network is the collection of all the unfrustrated plaquettes. These domain walls are topologically nontrivial because they are all related to the global frustration of a loop on the unfrustration network. Taking account of the thermal stability for the domain walls, we can explain the numerical results that three or higher dimensional systems exhibit a spin glass phase, whereas two-dimensional ones do not. Namely, in two dimensions, the thermal fluctuations of the topologically nontrivial domain walls destroy the order of the frozen spins on the unfrustration network, whereas they do not in three or higher dimensions. This may be interpreted as a global, topological effect of the frustrations.