Eigenvalue Spectrum of the Superintegrable Chiral Potts Model

Abstract

We compute the eigenvalues of the 3-state superintegrable chiral Potts model and of the associated spin chain by use of a functional equation. We find that the system has four phases, two of which are massless and two of which are massive.

Bethe equation of the τ(2)model and eigenvalues of a finite-size transfer matrix of a chiral Potts model with alternating rapidities

“…(2.26) 1 The identity (2.24) is formula (2.44) in [17] where rapidities are in (2.8). The same equality holds also in the degenerate model with rapidities in (2.9).…”

“…Furthermore, the Q-operator investigation of the five-parameter τ (2) -family appearing in [19] has shown that the τ (2) -model is indeed the same theory as the chiral Potts model of alternating rapidities when the degenerate forms are included [41]. This suggests the results in [1,11,12,10,31] about the eigenvalue spectrum could be better understood through a general setting of the chiral Potts model with alternating rapidities, including the degenerate cases. Indeed, after sorting out the technical details, one finds the functional relations in [17] about the chiral Potts model with alternating rapidities also valid in the degenerate cases (for the details, see [41]).…”

“…By The condition (ii) could be known for the specialists as it is supported by the numerical studies for N = 3 in Table I of [1], but it seems not appeared in literature before to the best of the author's knowledge. This novel constraint is consistent with the total momentum e iP L = 1 condition: [34] (63)), and it also agrees with the conclusion, P a ≡ 0 or P b ≡ 0, in the algebraic Bethe ansatz discussion of τ (2) -model ([37] section 3.2).…”

“…The homogeneous superintegrable CPM with rapidities in W k ′ and m = k = n = 0 has been discussed extensively in literature, and the eigenvalues of chiral Potts transfer matrix are given in [1] [8,9,10,11,12]:…”

The perturbations φ2,1 and φ1,5 of the minimal models M(p, p′) and the trinomial analogue of Bailey's lemma

“…This indicate that a non perturbative definition of the vacuum must be given and is often a signal that level crossing of the vacuum has occurred [32]. Similarly for the φ 2,1 perturbation they find a problem with vacuum definition for 3p > 2p ′ .…”