From explosive to infinite-order transitions on a hyperbolic network

TL;DR

This paper investigates phase transitions on hyperbolic networks, revealing a transition from explosive to infinite-order types through exact solutions of the Potts model, highlighting the role of parameter q.

Contribution

It provides exact expressions for the Potts model on hyperbolic networks, demonstrating the transition from discontinuous to infinite-order transitions as q varies.

Findings

Discontinuous transition is generic for q<2.

A continuous transition occurs at q=2 (Ising model).

For q>2, the transition becomes a BKT-type infinite-order transition.

Abstract

We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous ("explosive") transition in ordinary percolation. To this end, we solve the $q$-state Potts model in the analytic continuation for non-integer $q$ with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in $q$ shows that the discontinuous transition is generic in the regime $q<2$ that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, $q=2$. For $q>2$ the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless (BKT) type.