Phase transition of $q$-state clock models on heptagonal lattices

TL;DR

This paper investigates the phase transitions of $q$-state clock models on heptagonal lattices with negative curvature, revealing a persistent intermediate phase and boundary effects that influence the nature of the transitions.

Contribution

It demonstrates the existence of an intermediate phase for all $q \\ge 2$ on heptagonal lattices, contrasting with planar systems, and analyzes boundary effects on phase transitions.

Findings

Presence of an intermediate phase for all $q \\ge 2$

Boundary effects break mean-field behavior in the bulk

Two-stage transition explained by spin-wave gap and boundary contributions

Abstract

We study the $q$-state clock models on heptagonal lattices assigned on a negatively curved surface. We show that the system exhibits three classes of equilibrium phases; in between ordered and disordered phases, an intermediate phase characterized by a diverging susceptibility with no magnetic order is observed at every $q \ge 2$. The persistence of the third phase for all $q$ is in contrast with the disappearance of the counterpart phase in a planar system for small $q$, which indicates the significance of nonvanishing surface-volume ratio that is peculiar in the heptagonal lattice. Analytic arguments based on Ginzburg-Landau theory and generalized Cayley trees make clear that the two-stage transition in the present system is attributed to an energy gap of spin-wave excitations and strong boundary-spin contributions. We further demonstrate that boundary effects breaks the mean-field character in the bulk region, which establishes the consistency with results of clock models on boundary-free hyperbolic lattices.