On the Degeneracy of Spin Ice Graphs, and its Estimate via the Bethe Permanent

TL;DR

This paper explores the degeneracy of spin ice graphs on arbitrary structures using graph theory, establishing that all but 1D cases are degenerate and employing the Bethe permanent to estimate frustration levels.

Contribution

It introduces a novel application of the Bethe permanent to estimate frustration in spin ice graphs and clarifies the conditions for degeneracy across different graph structures.

Findings

All spin ice graphs except 1D are degenerate.

Bethe permanent provides a lower bound for frustration.

Schrijver inequality often yields a tighter upper bound.

Abstract

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. Via the mapping of spin ices to the Ising model, we clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the 1D Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such technique can be used also to obtain an upper bound, we find that in all the examples we studied but one, another upper bound based on Schrijver inequality is tighter.