Spontaneous symmetry breaking on graphs and lattices

TL;DR

This paper explores how spontaneous symmetry breaking manifests on discrete structures like graphs and lattices, revealing the influence of spectral properties and geometry on symmetry phenomena.

Contribution

It introduces a framework for understanding spontaneous symmetry breaking on graphs using spectral and resistance-based measures, extending concepts beyond traditional continuous spaces.

Findings

Spontaneous symmetry breaking is linked to spectral dimension and resistance metrics.

Certain graph geometries prevent symmetry breaking due to large fluctuations.

Discrete structures offer a clearer view of infrared phenomena in symmetry breaking.

Abstract

Spontaneous symmetry breaking is a cornerstone of modern physics, defining a wealth of phenomena in condensed-matter and high-energy physics, and beyond. It requires an infinite number of degrees of freedom, and even then, for continuous symmetries, it only works if the spatial dimension is not too low, following the classic results of Coleman, Hohenberg, Mermin and Wagner. While usually discussed in the context of quantum and statistical field theories, and in particular, effective field theories, there are advantages in addressing the same kind of phenomena on discrete geometric structures rather than conventional manifolds. When the space is discretized into a lattice, a lucid picture of conventional spontaneous symmetry breaking springs up, with the ultraviolet issues of continuum quantum field theory out-of-sight, and the key effect, which is infrared in nature, revealed through elementary harmonic oscillator networks. From there, it is natural to generalize lattices to other graphs/networks. In this setting, the presence of spontaneous symmetry breaking is controlled by fractional generalizations of resistance distance and the Kirchhoff index, and most broadly by the spectral dimension. Predictably, because of the richness of discrete geometric structures in comparison with continuous manifolds, a broader array of geometries emerge where spontaneous breaking of continuous symmetries is blocked by large fluctuations.