Spontaneous symmetry breaking on graphs and lattices

TL;DR

The paper investigates spontaneous symmetry breaking (SSB) of continuous global symmetries on discrete geometries, showing that discretization provides a UV-safe setting and that infrared behavior is controlled by the Laplacian spectrum. On lattices, ground-state fluctuations are captured by $W^2=\frac{1}{V}\sum_{Q=1}^{V-1}\frac{1}{\omega_Q}$, yielding Coleman-type results: in $1$D fluctuations diverge (no SSB) while in $d>1$ they remain finite (SSB occurs); massive fields localize and Schrödinger-type Goldstone bosons always localize, with higher-derivative theories allowing SSB in higher dimensions. On general graphs, SSB is governed by the spectral dimension $d_s$ via $\overline{W^2}=\int_0^{\infty}\frac{d\lambda}{\lambda^{p/2}}\rho(\lambda)$ and by fractional resistance distances $\Omega^{(\gamma)}_{IJ}$ and Kirchhoff indices $K_\gamma$, with fractal and heterogeneous networks providing tunable $d_s$ and SSB behavior. The results suggest a unified, ultraviolet-safe framework for SSB on diverse discrete geometries and motivate mathematical study of Laplacian spectra near zero to predict and design symmetry-breaking properties in complex networks.

Abstract

Spontaneous symmetry breaking is a cornerstone 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 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.