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 $L_x\times L_y=2\mathrm{}\times 3$ and $2\times 4$ and boundary conditions (a) $({\mathrm{FBC}}_x,{\mathrm{FBC}}_y,{\mathrm{FBC}}_z)$ and (b) $({\mathrm{PBC}}_x,{\mathrm{FBC}}_y,{\mathrm{FBC}}_z),$ 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 $Z_{+}$ to $C,$ 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 $K_{m,m}.$

• Received 12 February 2001

DOI:https://doi.org/10.1103/PhysRevE.64.011111