Phase transition of clock models on hyperbolic lattice studied by corner transfer matrix renormalization group method

TL;DR

This study investigates phase transitions in N-state clock models on hyperbolic lattices using the corner transfer matrix renormalization group method, revealing different transition types depending on N and scaling behaviors.

Contribution

It introduces a detailed analysis of clock models on hyperbolic lattices, demonstrating the nature of phase transitions and specific heat behaviors for various N values.

Findings

N=3 exhibits a first order phase transition.

N>=4 shows a mean-field-like second order transition.

For N>=5, specific heat peaks scale as N^{-2} at low temperatures.

Abstract

Two-dimensional ferromagnetic N-state clock models are studied on a hyperbolic lattice represented by tessellation of pentagons. The lattice lies on the hyperbolic plane with a constant negative scalar curvature. We observe the spontaneous magnetization, the internal energy, and the specific heat at the center of sufficiently large systems, where the fixed boundary conditions are imposed, for the cases N>=3 up to N=30. The model with N=3, which is equivalent to the 3-state Potts model on the hyperbolic lattice, exhibits the first order phase transition. A mean-field like phase transition of the second order is observed for the cases N>=4. When N>=5 we observe the Schottky type specific heat below the transition temperature, where its peak hight at low temperatures scales as N^{-2}. From these facts we conclude that the phase transition of classical XY-model deep inside the hyperbolic lattices is not of the Berezinskii-Kosterlitz-Thouless type.