Ground State Entropy of Potts Antiferromagnets: Bounds, Series, and Monte Carlo Measurements

TL;DR

This paper provides improved bounds, series expansions, and Monte Carlo measurements for the ground state entropy of Potts antiferromagnets on various lattices, revealing high accuracy and new insights into their behavior.

Contribution

It introduces more precise bounds, extensive series calculations, and Monte Carlo data for the ground state entropy of Potts antiferromagnets on different lattices, including honeycomb and Archimedean.

Findings

Bounds agree with series to many terms

Monte Carlo results closely match bounds and series

Next-nearest-neighbor couplings affect entropy estimates

Abstract

We report several results concerning $W(\Lambda,q)=\exp(S_0/k_B)$, the exponent of the ground state entropy of the Potts antiferromagnet on a lattice $\Lambda$. First, we improve our previous rigorous lower bound on $W(hc,q)$ for the honeycomb (hc) lattice and find that it is extremely accurate; it agrees to the first eleven terms with the large-$q$ series for $W(hc,q)$. Second, we investigate the heteropolygonal Archimedean $4 \cdot 8^2$ lattice, derive a rigorous lower bound, on $W(4 \cdot 8^2,q)$, and calculate the large-$q$ series for this function to $O(y^{12})$ where $y=1/(q-1)$. Remarkably, these agree exactly to all thirteen terms calculated. We also report Monte Carlo measurements, and find that these are very close to our lower bound and series. Third, we study the effect of non-nearest-neighbor couplings, focusing on the square lattice with next-nearest-neighbor bonds.