Exact T=0 Partition Functions for Potts Antiferromagnets on Sections of the Simple Cubic Lattice

Abstract

We present exact solutions for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial $P$) on tube sections of the simple cubic lattice of fixed transverse size $L_x\times L_y$ and arbitrarily great length $L_z$, for sizes $L_x\times L_y=2\times 3$ and $2\times 4$ and boundary conditions (a) $(FB{C}_x,FB{C}_y,FB{C}_z)$ and (b) $(PB{C}_x,FB{C}_y,FB{C}_z)$, where $FBC$ ($PBC$) denote free (periodic) boundary conditions. In the limit of infinite-length, $L_z\to \infty$, we calculate the resultant ground state degeneracy per site $W$ (= exponent of the ground-state entropy). Generalizing $q$ from ${\mathbb{Z}}_{+}$ to $\mathbb{C}$, we determine the analytic structure of $W$ and the related singular locus $\mathcal{B}$ which is the continuous accumulation set of zeros of the chromatic polynomial. For the $L_z\to \infty$ limit of a given family of lattice sections, $W$ is analytic for real $q$ down to a value $q_c$. We determine the values of $q_c$ for the lattice sections considered and address the question of the value of $q_c$ for a $d$-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph $K_{m,m}$.

28 pages, latex, six postscript figures, two Mathematica files

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