The Concept of Spin Ice Graphs and a Field Theory for their Topological Monopoles and Charges

TL;DR

This paper develops a graph-theoretical framework and a field theory for understanding topological monopoles and charges in spin ice systems, linking their physical properties to graph spectra and Laplacian operators.

Contribution

It introduces a novel graph-theoretical approach and a field-theoretical model for spin ices, connecting topological charges to graph Laplacians and spectral properties.

Findings

Entropic interactions are governed by the inverse graph Laplacian.

Screening functions are derived from graph spectra as Green operators.

Application to various graph structures demonstrates the model's versatility.

Abstract

Spin ices can now be fabricated in a variety of geometries which control their collective behavior and exotic properties. Therefore, their proper framework is graph theory. We relate spin ice notions such as ice rule, ice manifold, Coulomb phases, charges and monopoles, to graph-theoretical notions, such as balance, in/out-degrees, and Eulerianicity. We then propose a field-theoretical treatment in which topological charges and monopoles are the degrees of freedom while the binary spins are subsumed into entropic interaction among charges. We show that for a spin ice on a graph in a Gaussian approximation, the kernel of the entropic interaction is the inverse of the graph Laplacian, and we compute screening functions from the graph spectra as Green operators for the screened Poisson problem on a graph. We then apply the treatment on star graphs, tournaments, cycles, and regular spin ice in different dimensions.