Transfer Matrices for the Partition Function of the Potts Model on Cyclic and Mobius 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 cyclic and Möbius 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. For the cyclic case we express the partition function as $Z(\Lambda ,L_y\times L_x,q,v)={\sum }_{d=0}^{L_y}{c}^{(d)}Tr[(T_{Z,\Lambda ,L_y,d}{)}^m]$, where $\Lambda$ denotes lattice type, $c^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,\Lambda ,L_y,d}$ is the transfer matrix in the degree-$d$ subspace, and $m=L_x$ ($L_x/2$) for $\Lambda =sq,tri(hc)$, respectively. An analogous formula is given for Möbius strips. We exhibit a method for calculating $T_{Z,\Lambda ,L_y,d}$ for arbitrary $L_y$. Explicit results for arbitrary $L_y$ are given for $T_{Z,\Lambda ,L_y,d}$ with $d=L_y$ and $d=L_y-1$. In particular, we find very simple formulas 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. We also present formulas for self-dual cyclic strips of the square lattice.

43 pages, latex, 6 figures

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Cited by in corpus (11)

• Logarithmic Minimal Models
• Phase diagram of the chromatic polynomial on a torus
• Character decomposition of Potts model partition functions. I. Cyclic geometry
• Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
• Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips
• Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips
• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Zeros of the Potts Model Partition Function in the Large-$q$ Limit
• Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices
• Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions
• Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips