Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips

Abstract

We determine the general structure of the partition function of the q-state Potts model in an external magnetic field, Z(G, q, v, w) for arbitrary q, temperature variable v, and magnetic field variable w, on cyclic, Möbius, and free strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices with width Ly and arbitrarily great length Lx. For the cyclic case we prove that the partition function has the form Z(Λ, Ly ×Lx, q, v, w) = P Ly, where Λ denotes the lattice type,c (d) are specified polynomials of degree d in q, T Z,Λ,Ly ,d is the corresponding transfer matrix, and m = Lx (Lx/2) for Λ = sq, tri (hc), respectively. An analogous formula is given for Möbius strips, while only T Z,Λ,Ly ,d=0 appears for free strips. We exhibit a method for calculating T Z,Λ,Ly ,d for arbitrary Ly and give illustrative examples. Explicit results for arbitrary Ly are presented for T Z,Λ,Ly ,d with d = Ly and d = Ly − 1. We find very simple formulas for the determinant det(T Z,Λ,Ly,d ). We also give results for self-dual cyclic strips of the square lattice.

Exact Partition Functions for the q-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs

“…As is evident in the exact solution for the circuit graph in [9] and in our new solutions, certain λ Z,Λ,Ly,d,j terms occur with multiplicities greater than 1, which are polynomials in s. The numbers n Zh (L y , d, s) were derived for s = 1 (as Theorem 2.1 with Table 1) in [3], and results for arbitrary s ∈ I s were given in [9]. We have…”

“…In Refs. [3] and [4,8] we studied this case further and presented a number of properties of the resultant partition function for various families of graphs. As discussed in the introduction, a further generalization is to take the external magnetic field to favor or disfavor a set of several spin values in an interval I s rather than just a single value.…”

Some Exact Results on Bond Percolation

“…For the family of wheel graphs we thus need the coefficientsκ (1) = q − 1,κ (2) = (q − 1)(q − 3),c (0) = 1, andc (1) = q − 2. We have (from Table 5 of [22]) n Zh (1, G D , 1) = 3, n Zh (1, G D , 2) = 1 and (from Table 1 of [22]) n Zh (1, 0) = 2, n Zh (1, 1) = 1. The three λ Z,G D ,1,1,j , j = 1, 2, 3, are the roots of the following cubic equation: The twoλ Z,G D ,1,0,j are the roots of a quadratic equation,…”

“…(7.17) of) Ref. [22]. It is Z(G D , L y ×L x , q, v, w) = In the general notation of [22], the wheel graph is W h N = G D , L x × L y with L x = N − 1 and L y = 1.…”

“…[22]. It is Z(G D , L y ×L x , q, v, w) = In the general notation of [22], the wheel graph is W h N = G D , L x × L y with L x = N − 1 and L y = 1. For the family of wheel graphs we thus need the coefficientsκ (1) = q − 1,κ (2) = (q − 1)(q − 3),c (0) = 1, andc (1) = q − 2.…”