Spectrum and completeness of the three-state superintegrable chiral Potts model

Abstract

We find the rules which count the energy levels of the 3 state superintegrable chiral Potts model and demonstrate that these rules are complete. We then derive the complete spectrum of excitations in the thermodynamic limit in the massive phase and demonstrate the existence of excitations which do not have a quasi-particle form. The physics of these excitations is compared with the BCS superconductivity spectrum and the counting rules are compared with the closely related S = 1 XXZ spin chain.

Quasi-Particles in the Chiral Potts Model

“…This was studied in [11] where it was found that as L → ∞ there are three allowed values for the imaginary parts of λ j…”

“…Moreover this result may be extended to order 1/L by use of the counting rules of [11], where we note that the rewriting of the excitations associated with the ± signs in (2.22) in terms of + and −2s excitations is possible, because any set of roots v l = v, ωv and ω 2 v satisfy (2.19) independently of the value of v. Thus the reduction in the number of states which results from choosing all the ± signs to be negative is exactly compensated for by elimination of the exclusion rule −v…”

“…to study both the superintegrable case (1.5) [11] and the scalar case (1.15) (which is the critical three-state Potts model) [16] [17]. In the first case, in the massive regime, the spectrum consists of two quasi-particles.…”

The symmetry of the Bazhanov–Stroganov model associated with the superintegrable chiral Potts model

“…The Drinfeld polynomial of a L(sl 2 )-degenerate eigenspace of the model is equivalent to the polynomial [4,5,[10][11][12][13][14] which characterizes a subspace with the Isinglike spectrum of the superintegrable chiral Potts model. …”

On State Counting and Characters

“…The more interesting questions that remain to be answered involve cases where the dependence of the energy functional on the roots of the Bethe equations is non-trivial, as in the case of the superintegrable chiral Potts model [36], where similar completeness studies have been made [37].…”