The inverse Mermin-Wagner theorem for classical spin models on graphs

TL;DR

This paper proves that classical spin models on certain inhomogeneous graphs can exhibit spontaneous magnetization at finite temperature, reversing the traditional Mermin-Wagner theorem under specific graph conditions.

Contribution

It introduces the inverse Mermin-Wagner theorem for graphs, establishing conditions for spontaneous symmetry breaking on TOA graphs with inhomogeneous structures.

Findings

Spontaneous magnetization exists at finite temperature on TOA graphs.

The result applies to models with O(n) symmetry, including the Ising model.

Provides a general condition for symmetry breaking on inhomogeneous graphs.

Abstract

In this letter we present the inversion of the Mermin-Wagner theorem on graphs, by proving the existence of spontaneous magnetization at finite temperature for classical spin models on transient on the average (TOA) graphs, i.e. graphs where a random walker returns to its starting point with an average probability $\bar F < 1$. This result, which is here proven for models with O(n) symmetry, includes as a particular case $n=1$, providing a very general condition for spontaneous symmetry breaking on inhomogeneous structures even for the Ising model.