Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips

Abstract

partial abstract: The $q$-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}}c_{Z,G,j}(λ_{Z,G,j})^{L_x}$, where $v$ is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet ($v=-1$) is the chromatic polynomial $P(G,q)$. Using coloring and transfer matrix methods, we give general formulas for $C_{X,G}=\sum_{j=1}^{N_{X,G,λ}}c_{X,G,j}$ for $X=Z,P$ on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient $c_{Z,G,j}$ of degree $d$ in $q$: $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, where $U_n(x)$ is the Chebyshev polynomial of the second kind, we determine the number of $λ_{Z,G,j}$'s with coefficient $c^{(d)}$ in $Z(G,q,v)$ for these cyclic strips of width $L_y$ to be $n_Z(L_y,d)=(2d+1)(L_y+d+1)^{-1} {2L_y \choose L_y-d}$ for $0 \le d \le L_y$ and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues $λ_{Z,G,j}$ is calculated to be $N_{Z,L_y,λ}={2L_y \choose L_y}$. We point out that $N_{Z,L_y,λ}=2N_{DA,tri,L_y}$ and $N_{P,L_y,λ}=2N_{DA,sq,L_y}$, where $N_{DA,Λ,n}$ denotes the number of directed lattice animals on the lattice $Λ$.