Character decomposition of Potts model partition functions, I: Cyclic geometry

Abstract

We study the Potts model (defined geometrically in the cluster picture) on finite two-dimensional lattices of size L×N , with boundary conditions that are free in the L-direction and periodic in the N -direction. The decomposition of the partition function in terms of the characters K 1+2l (with l = 0, 1, . . . , L) has previously been studied using various approaches (quantum groups, combinatorics, transfer matrices). We first show that the K 1+2l thus defined actually coincide, and can be written as traces of suitable transfer matrices in the cluster picture. We then proceed to similarly decompose constrained partition functions in which exactly j clusters are non-contractible with respect to the periodic lattice direction, and a partition function with fixed transverse boundary conditions.

Eigenvalue amplitudes of the Potts model on a torus

“…The construction and structure of the transfer matrix T can be taken over from the cyclic case [2]. In particular, we recall that T acts towards the right on states of connectivities between two time slices (left and right) and has a block-trigonal structure with respect to the number of bridges (connectivity components linking left and right) and a block-diagonal structure with respect to the residual connectivity among the non-bridged points on the left time slice.…”

“…In section 2, we define appropriate generalisations of the quantities we used in the cyclic case [2] and we expose all the mathematical background we will need. Then, in section 3,…”

Q-colourings of the triangular lattice: exact exponents and conformal field theory

“…Two distinct representation theoretical mechanisms are responsible for this phenomenon. First, the multiplicity (also known as amplitude, or quantum dimension) of certain eigenvalues of the corresponding TM vanishes at Q = B k , as can be seen from a combinatorial decomposition of the Markov trace [34,35]. Second, the representation theory of the quantum group U q (sl 2 ) for q a root of unity (with √ Q = q+q −1 ) guarantees that other eigenvalues are equal in norm at Q = B k [32,36], and as their combined multiplicity is zero, they vanish from the spectrum as well.…”

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

“…The construction and structure of the transfer matrix T can be taken over from the cyclic case [2]. In particular, we recall that T acts towards the right on states of connectivities between two time slices (left and right) and has a block-trigonal structure with respect to the number of bridges (connectivity components linking left and right) and a blockdiagonal structure with respect to the residual connectivity among the non-bridged points on the left time slice.…”

“…In a companion paper [2], we studied the case of cyclic boundary conditions (periodic in the N-direction and non-periodic in the L-direction). We decomposed Z into linear combinations of certain restricted partition functions (characters) K l (with l = 0, 1, .…”