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# Lower Bounds on the Ground State Entropy of the Potts Antiferromagnet on Slabs of the Simple Cubic Lattice 

Robert Shrock and Yan Xu C. N. Yang Institute for Theoretical Physics State University of New York Stony Brook, NY 11794 

We calculate rigorous lower bounds for the ground state degeneracy per site, 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 and that thus interpolate between the square (sq) and simple cubic (sc) lattices. We give a comparison with large- q series expansions for the sq and sc lattices and also present numerical comparisons. 

Nonzero ground state entropy (per lattice site), S$_{0}$ ̸ = 0, is an important subject in statistical mechanics, as an exception to the third law of thermodynamics and a phe- nomenon involving large disorder even at zero tempera- ture. Since S$_{0}$ = k$_{B}$ ln W , where W = lim$_{n}$ →∞ W 1 /n tot. and n denotes the number of lattice sites, S$_{0}$ ̸ = 0 is equiv- alent to W > 1, i.e., a total ground state degeneracy W$_{tot.}$ that grows exponentially rapidly as a function of n . One physical example is provided by H$_{2}$O ice, for which the residual entropy per site (at 1 atm. pressure) is measured to be S$_{0}$ = (0 . 41 ± 0 . 03) k$_{B}$ , or equivalently, W = 1 . 51 ± 0 . 05 [1]-[3]. A salient property of ice is that the ground state entropy occurs without frustration; i.e., each of the ground state configurations of the hydrogen atoms on the bonds between oxygen atoms minimizes the internal energy of the crystal [4]. 

A model that also exhibits ground state entropy with- out frustration and hence provides a useful framework in which to study the properties of this phenomenon is the q -state Potts antiferromagnet [5]-[7] on a given lattice Λ or, more generally, a graph G , for sufficiently large q . Consider a graph G = ( V,E ), defined by its vertex (site) and edge (bond) sets V and E . Denote the cardinalities of these sets as n ( G ) = | V | ≡ n and e ( G ) = | E | , and let { G } ≡ lim$_{n}$$_{(}$$_{G}$$_{)}$ →∞ G . An important connection with graph theory is the fact that the zero-temperature par- tition function of the q -state Potts antiferromagnet on the graph G satisfies Z ( G,q,T = 0) = P ( G,q ), where P ( G,q ) is the chromatic polynomial expressing the num- ber of ways of coloring the vertices of G with q colors such that no two adjacent vertices have the same color (called a proper q -coloring of G ) [8, 9]. Thus, 

In general, for certain special values of q , denoted q$_{s}$ , one has the following noncommutativity of limits [10] 

<img src='content_image/434597.jpg'>

and hence it is necessary to specify which order of limits 

PACS numbers: 

## I. INTRODUCTION 

<img src='content_image/434596.jpg'>

that one takes in defining W ( { G } ,q ). Here by W ( { G } ,q ) we mean the function obtained by setting q to the given value first and then taking n → ∞ . For the n → ∞ limit of a bipartite graph G$_{bip.}$ , an elementary lower bound is W ( { G$_{bip.}$ } ,q ) ≥ √$_{q}$ − 1, so that for q > 2, the Potts anti- ferromagnet has a nonzero ground state entropy on such a lattice. A better lower bound for the square lattice is W ( sq,q ) ≥ ( q $^{2}$− 3 q +3) / ( q − 1) [11]. In previous work [12]- [15] one of us and Tsai derived lower and upper bounds on W for a variety of different two-dimensional lattices. It was found that these lower bounds are quite close to the actual values as determined with reasonably good accu- racy from large- q series expansions and/or Monte Carlo measurements. 

In the present paper we generalize these lower bounds on two-dimensional lattice graphs by deriving lower bounds on W ( { G } ,q ) for sections of a three-dimensional lattice, namely the simple cubic lattice, which are of infi- nite extent in two directions (taken to lie along the x and y axes) and finite in the third direction, z . By compar- ison with large- q expansions and numerical evaluations, we show how the lower bounds for the W functions for these slabs interpolate between the values for the (re- spective thermodynamic limits of the) square and simple cubic lattices. These bounds are of interest partly be- cause one does not know the exact functions W ( sq,q ) or W ( sc,q ) for general q . 

Let us consider a section (slab) of the simple cubic lattice of dimensions L$_{x}$ × L$_{y}$ × L$_{z}$ vertices, which we de- note sc [( L$_{x}$ )$_{BCx}$ × ( L$_{y}$ )$_{BCy}$ × ( L$_{z}$ )$_{BCz}$], where the bound- ary conditions (BC) in each direction are indicated by the subscripts. The chromatic polynomial of this lattice will be denoted P ( sc [( L$_{x}$ )$_{BCx}$ × ( L$_{y}$ )$_{BCy}$ × ( L$_{z}$ )$_{BCy}$] ,q ). We will calculate lower bounds for W ( sc [( L$_{x}$ )$_{BCx}$ 

× ( L$_{y}$ )$_{BCy}$ × ( L$_{z}$ )$_{BCz}$] ,q ) in the limit L$_{x}$ → ∞ and L$_{y}$ → ∞ with L$_{z}$ fixed. These are independent of the bound- ary conditions imposed in the directions in which the slab is of infinite extent, and hence, for brevity of nota- 

## II. CALCULATIONAL METHOD 

