Eigenvalue amplitudes of the Potts model on a torus

Abstract

We consider the Q-state Potts model in the random-cluster formulation, defined on finite two-dimensional lattices of size L x N with toroidal boundary conditions. Due to the non-locality of the clusters, the partition function Z(L,N) cannot be written simply as a trace of the transfer matrix T\_L. Using a combinatorial method, we establish the decomposition Z(L,N) = \sum\_{l,D\_k} b^{l,D\_k} K\_{l,D\_k}, where the characters K\_{l,D\_k} = \sum\_i (\lambda\_i)^N are simple traces. In this decomposition, the amplitudes b^{l,D\_k} of the eigenvalues \lambda\_i of T\_L are labelled by the number l=0,1,...,L of clusters which are non-contractible with respect to the transfer (N) direction, and a representation D\_k of the cyclic group C\_l. We obtain rigorously a general expression for b^{l,D\_k} in terms of the characters of C\_l, and, using number theoretic results, show that it coincides with an expression previously obtained in the continuum limit by Read and Saleur.