Ground State Entropy of Potts Antiferromagnets and the Approach to the 2D Thermodynamic Limit

TL;DR

This paper investigates how the ground state degeneracy of the Potts antiferromagnet on infinite strips approaches the 2D thermodynamic limit, showing rapid convergence and monotonic behavior for different boundary conditions.

Contribution

It provides quantitative analysis of the convergence rate of ground state entropy for Potts antiferromagnets on strips with various boundary conditions, including proofs of monotonicity.

Findings

W approaches the 2D limit rapidly with increasing strip width.

Convergence is within 5% for moderate q and width 4.

Monotonic convergence for free boundary conditions, non-monotonic for periodic.

Abstract

We study the ground state degeneracy per site (exponent of the ground state entropy) $W(\Lambda,(L_x=\infty) \times L_y,q)$ for the $q$-state Potts antiferromagnet on infinitely long strips with width $L_y$ of 2D lattices $\Lambda$ with free and periodic boundary conditions in the $y$ direction, denoted FBC$_y$ and PBC$_y$. We show that the approach of $W$ to its 2D thermodynamic limit as $L_y$ increases is quite rapid; for moderate values of $q$ and $L_y \simeq 4$, $W(\Lambda,(L_x=\infty) \times L_y,q)$ is within about 5 % and ${\cal O}(10^{-3})$ of the 2D value $W(\Lambda,(L_x=\infty) \times (L_y=\infty),q)$ for FBC$_y$ and PBC$_y$, respectively. The approach of $W$ to the 2D thermodynamic limit is proved to be monotonic (non-monotonic) for FBC$_y$ (PBC$_y$). It is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states.