Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

Abstract

We determine the general structure of the partition function of the $q$-state Potts model in an external magnetic field, $Z(G,q,v,w)$ for arbitrary $q$, temperature variable $v$, and magnetic field variable $w$, on cyclic, Möbius, and free strip graphs $G$ of the square (sq), triangular (tri), and honeycomb (hc) lattices with width $L_y$ and arbitrarily great length $L_x$. For the cyclic case we prove that the partition function has the form $Z(\Lambda ,L_y\times L_x,q,v,w)={\sum }_{d=0}^{L_y}{\tilde{c}}^{(d)}Tr[(T_{Z,\Lambda ,L_y,d}{)}^m]$, where $\Lambda$ denotes the lattice type, ${\tilde{c}}^{(d)}$ are specified polynomials of degree $d$ in $q$, $T_{Z,\Lambda ,L_y,d}$ is the corresponding transfer matrix, and $m=L_x$ ($L_x/2$) for $\Lambda =sq,tri(hc)$, respectively. An analogous formula is given for Möbius strips, while only $T_{Z,\Lambda ,L_y,d=0}$ appears for free strips. We exhibit a method for calculating $T_{Z,\Lambda ,L_y,d}$ for arbitrary $L_y$ and give illustrative examples. Explicit results for arbitrary $L_y$ are presented for $T_{Z,\Lambda ,L_y,d}$ with $d=L_y$ and $d=L_y-1$. We find very simple formulas for the determinant $det(T_{Z,\Lambda ,L_y,d})$. We also give results for self-dual cyclic strips of the square lattice.

Reference added to a relevant paper by F. Y. Wu

References in corpus (1)

• Some Exact Results on the Potts Model Partition Function in a Magnetic Field

Cited by in corpus (6)

• Classical phase transitions in a one-dimensional short-range spin model induced by entropy depletion or complex fields
• Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings
• Weighted-Set Graph Colorings
• Exact Partition Functions for the $q$-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs
• Parallel family trees for transfer matrices in the Potts model
• Some Exact Results on Bond Percolation