Structural Properties of Potts Model Partition Functions and Chromatic Polynomials for Lattice Strips

Abstract

partial abstract: The $q$-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $Z(G,q,v)={\sum }_{j=1}^{N_{Z,G,\lambda }}{c}_{Z,G,j}({\lambda }_{Z,G,j}{)}^{L_x}$, where $v$ is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet ($v=-1$) is the chromatic polynomial $P(G,q)$. Using coloring and transfer matrix methods, we give general formulas for $C_{X,G}={\sum }_{j=1}^{N_{X,G,\lambda }}{c}_{X,G,j}$ for $X=Z,P$ on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient $c_{Z,G,j}$ of degree $d$ in $q$: $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, where $U_n(x)$ is the Chebyshev polynomial of the second kind, we determine the number of ${\lambda }_{Z,G,j}$'s with coefficient $c^{(d)}$ in $Z(G,q,v)$ for these cyclic strips of width $L_y$ to be $n_Z(L_y,d)=(2d+1)(L_y+d+1{)}^{-1}(\genfrac{}{}{0ex}{}{2L_y}{L_y-d})$ for $0\le d\le L_y$ and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues ${\lambda }_{Z,G,j}$ is calculated to be $N_{Z,L_y,\lambda }=\left(\genfrac{}{}{0ex}{}{2L_y}{L_y}\right)$. We point out that $N_{Z,L_y,\lambda }=2N_{DA,tri,L_y}$ and $N_{P,L_y,\lambda }=2N_{DA,sq,L_y}$, where $N_{DA,\Lambda ,n}$ denotes the number of directed lattice animals on the lattice $\Lambda$.

53 pages, latex

References in corpus (9)

• Spanning Trees on Graphs and Lattices in d Dimensions
• Exact Potts Model Partition Functions on Ladder Graphs
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• T=0 Partition Functions for Potts Antiferromagnets on Square Lattice Strips with (Twisted) Periodic Boundary Conditions
• T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology
• Ground-State Degeneracy of Potts Antiferromagnets on Two-Dimensional Lattices: Approach Using Infinite Cyclic Strip Graphs
• Ground State Entropy of the Potts Antiferromagnet on Strips of the Square Lattice
• Ground State Entropy of Potts Antiferromagnets on Cyclic Polygon Chain Graphs
• Ground State Entropy of the Potts Antiferromagnet with Next-Nearest-Neighbor Spin-Spin Couplings on Strips of the Square Lattice

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• Distribution of Transverse Distances in Directed Animals
• The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities
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• Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions
• Ground State Entropy of the Potts Antiferromagnet on Homeomorphic Expansions of Kagome Lattice Strips
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