Phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction

TL;DR

This paper analytically derives the phase diagram of a one-dimensional three-state Potts model with mean-field interaction, revealing complex transition structures including first-order lines, triple points, and a critical point, without second-order transitions.

Contribution

It provides an analytical solution for the phase diagram of the model, identifying unique transition features and the asymptotic behavior of transition temperatures at large negative coupling.

Findings

Complex phase diagram with first-order lines, triple points, and a critical point.

Absence of second-order phase transition lines due to non-symmetry-breaking order parameter.

Analytical determination of one first-order transition line and asymptotic independence of transition temperature for large negative coupling.

Abstract

We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-$1$ Blume-Emery-Griffiths model and solving it by using an Hubbard-Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling $K$ and the temperature $T$. We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase-transition lines. We also prove that, when the nearest-neighbour coupling $K$ is large and negative, the first-order transition temperature becomes asymptotically independent of $K$.