Study of spin models with polyhedral symmetry on square lattice

TL;DR

This study explores 2D spin models with polyhedral symmetry, revealing how discretization affects phase transitions and critical temperatures through Monte Carlo simulations.

Contribution

It introduces and analyzes new polyhedral symmetry spin models on a square lattice, extending understanding of anisotropy effects in 2D systems.

Findings

Critical temperatures decrease with increasing number of spin states.

Systematic variation of phase transition properties across different polyhedral models.

Monte Carlo simulations effectively estimate critical exponents for these models.

Abstract

Anisotropy is important for the existence of true long range order in two dimensional (2D) systems. This is firmly exemplified by the $q$-state clock models in which discreteness drives the quasi-long range order into a true long range order at low temperature for $q> 4$. Previously we studied 2D edge-cubic spin model, which is one of the discrete counterpart of the continuous Heisenberg model, and observed two finite temperature phase transitions, each corresponds to the breakdown of octahedral ($O_h$) symmetry and $C_{3h}$ symmetry, which finally freezes into ground state configuration. The present study investigates discret models with polyhedral symmetry, obtained by e equally partioning the $4\pi$ of the solid angle of a sphere. There are five types of models if spins are only allowed to point to the vertices of the polyhedral structures such as Tetrahedron, Octahedron, Hexahedron, Icosahedron and Dodecahedron. By using Monte Carlo simulation with cluster algorithm we calculate order parameters and estimate the critical temperatures exponents of each model. We found a systematic decrease in critical temperatures as the number of spin states increases (from the Tetrahedral to Dodecahedral spin model).