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

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

2000-05-15

arxiv.org/abs/cond-mat/0005232v1

Physica A 296, 131-182 (2001)

Physics

Condensed Matter

Statistical Mechanics

53 pages, latex

Scientific paper

10.1016/S0378-4371(01)00150-9

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 $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,\lambda}}c_{Z,G,j}(\lambda_{Z,G,j})^{L_x}$, 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 $C_{X,G}=\sum_{j=1}^{N_{X,G,\lambda}}c_{X,G,j}$ for $X=Z,P$ on cyclic and M\"obius 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 $\lambda_{Z,G,j}$'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 M\"obius strips of these lattices, the total number of distinct eigenvalues $\lambda_{Z,G,j}$ is calculated to be $N_{Z,L_y,\lambda}={2L_y \choose L_y}$. We point out that $N_{Z,L_y,\lambda}=2N_{DA,tri,L_y}$ and $N_{P,L_y,\lambda}=2N_{DA,sq,L_y}$, where $N_{DA,\Lambda,n}$ denotes the number of directed lattice animals on the lattice $\Lambda$.

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