Improved lower bounds on the ground-state entropy of the antiferromagnetic Potts model

We present generalized methods for calculating lower bounds on the ground-state entropy per site, $S_0$, or equivalently, the ground-state degeneracy per site, $W=e^{S_0/k_B}$, of the antiferromagnetic Potts model. We use these methods to derive improved lower bounds on $W$ for several lattices.

Figure 1

Plot of the ratios $R_{\text{sq},q;(2,1)/(1,1)}$ (lower curve) and $R_{\text{sq},q;(3,1)/(1,1)}$ (upper curve) as functions of $q$ for $3\le q\le 6$.

Figure 2

Plot of the ratios $R_{\text{sq},q;(1,k)/(1,1)}$ for $k=2$ to 5 as functions of $q$ for $3\le q\le 6$. From bottom to top, the curves refer to $k=2$, 3, 4, and 5, respectively.

Figure 3

Plot of the ratios $R_{\text{tri},q;(2,1)/(1,1)}$ (lower curve) and $R_{\text{tri},q;(3,1)/(1,1)}$ (upper curve) as functions of $q$ for $4\le q\le 6$.

Figure 4

Plot of the ratios $R_{\text{tri},q;(1,k)/(1,1)}$ for $k=2$ to 5 as functions of $q$ for $4\le q\le 6$. From bottom to top, the curves refer to $k=2$, 3, 4, and 5, respectively.

Figure 5

Section of the $(3·6·3·6)$ (kagomé) lattice.