Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips

Details Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips

2005-06-13

arxiv.org/abs/cond-mat/0506274v1

PHYSICA A 364, 231-262 (2006)

Physics

Condensed Matter

Statistical Mechanics

52 pages, latex, 10 figures

Scientific paper

10.1016/j.physa.2005.08.076

We present a method for calculating transfer matrices for the $q$-state Potts model partition functions $Z(G,q,v)$, for arbitrary $q$ and temperature variable $v$, on strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices of width $L_y$ vertices and of arbitrarily great length $L_x$ vertices, subject to toroidal and Klein bottle boundary conditions. For the toroidal case we express the partition function as $Z(\Lambda, L_y \times L_x,q,v) = \sum_{d=0}^{L_y} \sum_j b_j^{(d)} (\lambda_{Z,\Lambda,L_y,d,j})^m$, where $\Lambda$ denotes lattice type, $b_j^{(d)}$ are specified polynomials of degree $d$ in $q$, $\lambda_{Z,\Lambda,L_y,d,j}$ are eigenvalues of the transfer matrix $T_{Z,\Lambda,L_y,d}$ in the degree-$d$ subspace, and $m=L_x$ ($L_x/2$) for $\Lambda=sq, tri (hc)$, respectively. An analogous formula is given for Klein bottle strips. We exhibit a method for calculating $T_{Z,\Lambda,L_y,d}$ for arbitrary $L_y$. In particular, we find some very simple formulas for the determinant $det(T_{Z,\Lambda,L_y,d})$, and trace $Tr(T_{Z,\Lambda,L_y})$. Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included.

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