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

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 $Z(\Lambda ,L_y\times L_x,q,v)={\sum }_{d=0}^{L_y}{\sum }_j{b}_j^{(d)}({\lambda }_{Z,\Lambda ,L_y,d,j}{)}^m$, 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.

52 pages, latex, 10 figures

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Cited by in corpus (8)

• Phase diagram of the chromatic polynomial on a torus
• Eigenvalue amplitudes of the Potts model on a torus
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Classical phase transitions in a one-dimensional short-range spin model induced by entropy depletion or complex fields
• What is the probability of connecting two points ?
• Character decomposition of Potts model partition functions. II. Toroidal geometry
• Zeros of the Potts Model Partition Function in the Large-$q$ Limit
• Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions