Structural properties of Potts model partition functions and chromatic polynomials for lattice strips

Abstract

The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width L y and arbitrary length L x has the form Z(G,q,v)=∑ j=1 N Z,G,λ c Z,G,j(λ Z,G,j) L x , where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet ( v=−1) is the chromatic polynomial P( G, q). Using coloring and transfer matrix methods, we give general formulas for C X,G=∑ j=1 N X,G,λ c X,G,j for X= Z, P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient c Z, G, j of degree d in q: c (d)=U 2d( q /2) , where U n ( x) is the Chebyshev polynomial of the second kind, we determine the number of λ Z, G, j 's with coefficient c ( d) in Z( G, q, v) for these cyclic strips of width L y to be n Z(L y,d)=(2d+1)(L y+d+1) −1 2L y L y−d for 0⩽ d⩽ L y and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λ Z, G, j is calculated to be N Z,L y,λ = 2L y L y . Results are also presented for the analogous numbers n P ( L y , d) and N P, L y , λ for P( G, q). We find that n P ( L y ,0)= n P ( L y −1,1)= M L y −1 (Motzkin number), n Z ( L y ,0)= C L y (the Catalan number), and give an exact expression for N P, L y , λ . Our results for N Z, L y , λ and N P, L y , λ apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations N Z, L y , λ =2 N DA, tri, L y and N P, L y , λ =2 N DA, sq, L y , where N DA, Λ, n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths N Z, L y , λ ∼ L y −1/24 L y and N P, L y , λ ∼ L y −1/23 L y as L y →∞. Some general geometric identities for Potts model partition functions are also presented.