Colorings of odd or even chirality on hexagonal lattices

TL;DR

This paper introduces two classes of colorings with odd or even chirality on hexagonal lattices, revealing their invariant properties, nonergodicity issues, and how modified algorithms can equilibrate their entropies.

Contribution

It defines parity-based coloring classes on hexagonal lattices, analyzes their dynamics, and proposes an algorithm to enable parity changes, showing entropy equivalence.

Findings

Parity is an invariant in loop dynamics.

Standard algorithms are nonergodic due to parity conservation.

Modified algorithms can equilibrate even and odd classes.

Abstract

We define two classes of colorings that have odd or even chirality on hexagonal lattices. This parity is an invariant in the dynamics of all loops, and explains why standard Monte-Carlo algorithms are nonergodic. We argue that adding the motion of "stranded" loops allows for parity changes. By implementing this algorithm, we show that the even and odd classes have the same entropy. In general, they do not have the same number of states, except for the special geometry of long strips, where a Z$_2$ symmetry between even and odd states occurs in the thermodynamic limit.