Potts Model Partition Functions for Self-Dual Families of Strip Graphs

Abstract

We consider the $q$-state Potts model on families of self-dual strip graphs $G_D$ of the square lattice of width $L_y$ and arbitrarily great length $L_x$, with periodic longitudinal boundary conditions. The general partition function $Z$ and the T=0 antiferromagnetic special case $P$ (chromatic polynomial) have the respective forms ${\sum }_{j=1}^{N_{F,L_y,\lambda }}{c}_{F,L_y,j}({\lambda }_{F,L_y,j}{)}^{L_x}$, with $F=Z,P$. For arbitrary $L_y$, we determine (i) the general coefficient $c_{F,L_y,j}$ in terms of Chebyshev polynomials, (ii) the number $n_F(L_y,d)$ of terms with each type of coefficient, and (iii) the total number of terms $N_{F,L_y,\lambda }$. We point out interesting connections between the $n_Z(L_y,d)$ and Temperley-Lieb algebras, and between the $N_{F,L_y,\lambda }$ and enumerations of directed lattice animals. Exact calculations of $P$ are presented for $2\le L_y\le 4$. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), $W(q)$. Generalizing $q$ from ${\mathbb{Z}}_{+}$ to $\mathbb{C}$, we determine the continuous locus $\mathcal{B}$ in the complex $q$ plane where $W(q)$ is singular. We find the interesting result that for all $L_y$ values considered, the maximal point at which $\mathcal{B}$ crosses the real $q$ axis, denoted $q_c$ is the same, and is equal to the value for the infinite square lattice, $q_c=3$. This is the first family of strip graphs of which we are aware that exhibits this type of universality of $q_c$.

36 pages, latex, three postscript figures

References in corpus (3)

• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Density of states, Potts zeros, and Fisher zeros of the Q-state Potts model for continuous Q
• Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice

Cited by in corpus (22)

• Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q\to \infty$ Limit
• $q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs
• Reliability Polynomials and their Asymptotic Limits for Families of Graphs
• Zeros of the Potts Model Partition Function on Sierpinski Graphs
• General Structural Results for Potts Model Partition Functions on Lattice Strips
• Transfer Matrices for the Partition Function of the Potts Model on Cyclic and Mobius Lattice Strips
• Transfer Matrices for the Partition Function of the Potts Model on Toroidal Lattice Strips
• Some Exact Results on the Potts Model Partition Function in a Magnetic Field
• Weighted Graph Colorings
• Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips
• Exact Results for Average Cluster Numbers in Bond Percolation on Lattice Strips
• Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings
• Transfer Matrices for the Zero-Temperature Potts Antiferromagnet on Cyclic and Mobius Lattice Strips
• Exact Chromatic Polynomials for Toroidal Chains of Complete Graphs
• Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs
• The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities
• Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices
• Partition Function Zeros of a Restricted Potts Model on Self-Dual Strips of the Square Lattice
• Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions
• Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs
• Chromatic Zeros on the Limit $G_{\infty }^{(p,\ell )}$ of the Family $G_m^{(p,\ell )}$ of Hierarchical Graphs
• Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips