Transfer matrices for the partition function of the Potts model on lattice strips with toroidal and Klein-bottle boundary conditions

Abstract

We present a method for calculating transfer matrices for the q-state Potts model partition functions Z(G, q, v), for arbitrary q and temperature variable v, on strip graphs G of the square (sq), triangular (tri), and honeycomb (hc) lattices of width L y vertices and of arbitrarily great length L x vertices, subject to toroidal and Klein bottle boundary conditions. For the toroidal case we express the partition function Corresponding results are given for the equivalent Tutte polynomials for these lattice strips and illustrative examples are included.

Tutte polynomials of bracelets

“…In order to illustrate Theorem 4 we shall work out the details for the cases B = K 2 , L = {11, 22}, and B = K 2 , L = {11, 12, 22}, where the bracelets are the cubic and quartic plane ladders CPL n and QPL n respectively. The results have previously been obtained by other methods [6,15], and more recently Chang and Shrock [7,8] have used methods that differ from the ones used here only in notation and presentation.…”

“…Chang and Shrock [6][7][8] have obtained significant results about the expansion of ZP when the graphs G n are lattice graphs with various boundary conditions. In our terms, these graphs are bracelets in which B is a path or cycle.…”

“…If B has two vertices, the square lattice graph is the graph we call the cubic plane ladder CPL n , and the triangular lattice graph is the quartic plane ladder QPL n . In the papers [7,8] the method used by Chang and Shrock was to obtain the entries of the relevant transfer matrices by explicit calculations, using diagrammatic representations of the various possibilities for the states. By this means they calculated the Tutte polynomials of many kinds of lattice graphs, including CPL n and QPL n (see (4.17), (5.23), (8.1), (8.3) in [7]).…”

Character decomposition of Potts model partition functions, II: Toroidal geometry

“…1 An example occurs for the square lattice of width L = 4, where an eigenvalue at level 1 coincides with an eigenvalue at level 2 [3], without any apparent reason.…”

“…Note that the four first amplitudes in this list have been obtained by Chang and Schrock [3] using a different method.…”