Symmetry Breaking and Convex Set Phase Diagrams for the q-state Potts Model

TL;DR

This paper links symmetry breaking phase transitions in the classical q-state Potts model to the geometric structure of probability spaces, providing a new way to visualize and analyze phase diagrams through convex sets.

Contribution

It introduces a geometric framework using convex sets of expectation values to understand phase transitions and symmetry breaking in classical spin systems.

Findings

Symmetry breaking corresponds to ruled surfaces in convex sets.

Critical exponents and susceptibilities can be derived from the geometry.

The approach offers an intuitive method to construct phase diagrams.

Abstract

We demonstrate that the occurrence of symmetry breaking phase transitions together with the emergence of a local order parameter in classical statistical physics is a consequence of the geometrical structure of probability space. To this end we investigate convex sets generated by expectation values of certain observables with respect to all possible probability distributions of classical q-state spins on a two-dimensional lattice, for several values of q. The extreme points of these sets are then given by thermal Gibbs states of the classical q-state Potts model. As symmetry breaking phase transitions and the emergence of associated order parameters are signaled by the appearance ruled surfaces on these sets, this implies that symmetry breaking is ultimately a consequence of the geometrical structure of probability space. In particular we identify the different features arising for continuous and first order phase transitions and show how to obtain critical exponents and susceptibilities from the geometrical shape of the surface set. Such convex sets thus also constitute a novel and very intuitive way of constructing phase diagrams for many body systems, as all thermodynamically relevant quantities can be very naturally read off from these sets.