On the Kert\'esz line: Thermodynamic versus Geometric Criticality

TL;DR

This paper investigates the relationship between thermodynamic and geometric criticality in the Potts model with external fields, deriving the Kertész line that marks the transition between percolating and non-percolating phases.

Contribution

It extends the understanding of critical behavior to finite external fields, analytically and numerically calculating the Kertész line for the Potts model.

Findings

Kertész line separates percolating and non-percolating regimes.

Geometric analysis aligns with thermodynamic phase transition structure.

Analytical and numerical methods confirm the Kertész line's location.

Abstract

The critical behaviour of the Ising model in the absence of an external magnetic field can be specified either through spontaneous symmetry breaking (thermal criticality) or through cluster percolation (geometric criticality). We extend this to finite external fields for the case of the Potts' model, showing that a geometric analysis leads to the same first order/second order structure as found in thermodynamic studies. We calculate the Kert\'esz line, separating percolating and non-percolating regimes, both analytically and numerically for the Potts model in presence of an external magnetic field.