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Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips | | Shu-Chiuan Chang
; Robert Shrock
; | | Date: |
13 Jun 2005 | | Subject: | Statistical Mechanics | cond-mat.stat-mech | | Abstract: | 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 $Z(Lambda, L_y imes L_x,q,v) = sum_{d=0}^{L_y} sum_j b_j^{(d)} (lambda_{Z,Lambda,L_y,d,j})^m$, 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. | | Source: | arXiv, cond-mat/0506274 | | Services: | Forum | Review | PDF | Favorites | |
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