Universal asymptotic correlation functions for point group $\boldsymbol{C_{6v}}$ and an observation for triangular lattice $\boldsymbol{Q}$-state Potts model

TL;DR

This paper derives universal asymptotic correlation functions for systems with $C_{6v}$ symmetry, supported by Monte Carlo simulations of the triangular lattice $Q$-state Potts model, and explores their implications for equilibrium crystal shapes.

Contribution

It introduces a minimal universal form for correlation functions with $C_{6v}$ symmetry and demonstrates its broad applicability through numerical simulations and geometric analysis.

Findings

Universal correlation form with two parameters for $C_{6v}$ symmetry

Numerical evidence supports the form's applicability to lattice models

Derived universal shape of equilibrium crystals with genus 1 algebraic curve

Abstract

We investigate universal forms for asymptotic correlation functions of off-critical systems that possess $C_{6v}$ symmetry following the argument for $C_{4v}$ symmetry in Phys.~Rev.~E{\bf 102},~032141. Unlike the $C_{4v}$ case, a minimal form exists that contains only two free parameters: the normalization constant and modulus. Using this form as a building block, we can construct next asymptotic forms to the minimal one. We perform large-scale Monte Carlo simulations of the triangular lattice $Q$-state Potts model above the transition temperature and successfully obtain numerical evidence to support a wide applicability of the minimal form to lattice models, including unsolvable ones. From the calculated minimal form, we derive the universal shape of equilibrium crystals in the honeycomb lattice Potts model described by an algebraic curve of genus 1. Although the curve differs from those obtained in the $C_{4v}$ case, the latters also have genus 1. We indicate that the birational equivalence concept can play an important role in comparing asymptotic forms for different point group symmetries, for example, $C_{6v}$ and $C_{4v}$.