content_image/434827.jpg content_image/434826.jpg content_image/434814.jpg content_image/434818.jpg content_image/434819.jpg content_image/434815.jpg content_image/434816.jpg content_image/434817.jpg
the fact that the effective coordination number increases monotonically as a function of L$_{z}$ . 

The use of periodic boundary conditions in the z direc- tion minimizes finite-size effects, so that for a given L$_{z}$ , W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q ) would be expected to be closer to W ( sc, q ) than W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ) [30]. Again, to the extent that the lower bounds are close to the actual W functions for these respective slabs, one would expect W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q )$_{ℓ}$ to be closer than W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q )$_{ℓ}$ to W ( sc, q ). Our results agree with this expectation. In contrast to W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{F}$ , q ), W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q ) is not, in general, a non-increasing function of L$_{z}$ , as was discussed in general in [30] (see Fig. 1 therein). Thus, values of W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q ), and hence, a for- tiori , W ( S$_{(}$$_{L}$$_{z}$$_{)}$$_{P}$ , q )$_{ℓ}$, may actually lie slightly below those for W ( sc, q ), as is evident for the W ( S$_{3}$$_{P}$ , q )$_{ℓ}$ entries in Table I. 

In this paper we have calculated rigorous lower bounds for the ground state degeneracy per site W , equivalent to the ground state entropy S$_{0}$ = k$_{B}$ ln W , of the q -state Potts antiferromagnet on slabs of the simple cubic lattice that are infinite in two directions and finite in the third. Via comparison with large- q expansions and numerical evaluations, we have shown how the results interpolate between the square (sq) and simple cubic (sc) lattices. 

Acknowedgments: This research was supported in part by the NSF grant PHY-06-53342. RS expresses his grat- itude to coauthors on previous related works, in partic- ular, S.-H. Tsai and J. Salas, as well as N. Biggs, S.-C. Chang, and M. Roˇ cek. 

We note the following results on E d lattices and lat- tice sections: W (Λ$_{bip.}$ , 2) = 1 for any bipartite lattice; W ( sq, 3) = (4 / 3) 3 / 2 [37]; and W ( { L } , q ) = W ( { C } , q ) = q − 1, where L$_{n}$ and C$_{n}$ denote the n -vertex line and cir- cuit graphs. For the infinite-length square-lattice strip of width 2, W ( sq [2$_{F}$ × ∞ ] , q ) = W ( sq [2$_{P}$ × ∞ ] , q ) = √ q 2 − 3 q + 3, where, as in the text, the subscripts F and P denote free and periodic boundary conditions in the direction in which the strip is finite. For the infinite- length strip of the square lattice with (transverse) width 3 and free transverse boundary conditions, sq [3$_{F}$ × ∞ ] [31–33] 

where 

<img src='content_image/434827.jpg'>

## VIII. CONCLUSIONS 

## IX. APPENDIX 

<img src='content_image/434826.jpg'>

with 

<img src='content_image/434814.jpg'>

For the infinite-length strip of the square lattice with width 4 and free transverse boundary conditions, sq [4$_{F}$ × ∞ ] [31, 35] 

where λ$_{4}$$_{F}$$_{,max}$ is the largest root of the cubic equation 

with 

<img src='content_image/434818.jpg'>

and 

<img src='content_image/434819.jpg'>

<img src='content_image/434815.jpg'>

<img src='content_image/434816.jpg'>

<img src='content_image/434817.jpg'>

