Exact Potts Model Partition Functions for Strips of the Honeycomb Lattice

TL;DR

This paper provides exact calculations of the Potts model partition function for honeycomb lattice strips of various widths and boundary conditions, including formulas, zeros plots, and thermodynamic quantities.

Contribution

It introduces general formulas for the number of terms in the partition function and computes explicit results for specific strip widths and boundary conditions.

Findings

Exact partition functions for honeycomb strips up to width 6.

Plots of zeros in the q-plane and v-plane for various parameters.

Thermodynamic quantities like internal energy and specific heat for infinite strips.

Abstract

We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on $n$-vertex strip graphs $G$ of the honeycomb lattice for a variety of transverse widths equal to $L_y$ vertices and for arbitrarily great length, with free longitudinal boundary conditions and free and periodic transverse boundary conditions. These partition functions have the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}} c_{Z,G,j}(\lambda_{Z,G,j})^m$, where $m$ denotes the number of repeated subgraphs in the longitudinal direction. We give general formulas for $N_{Z,G,j}$ for arbitrary $L_y$. We also present plots of zeros of the partition function in the $q$ plane for various values of $v$ and in the $v$ plane for various values of $q$. Explicit results for partition functions are given in the text for $L_y=2,3$ (free) and $L_y=4$ (cylindrical), and plots of partition function zeros are given for $L_y$ up to 5 (free) and $L_y=6$ (cylindrical). Plots of the internal energy and specific heat per site for infinite-length strips are also presented.