Functional Relations for Transfer Matrices of the Chiral Potts Model

Abstract

It has recently been shown that the solvable N-state chiral Potts model is related to a vertex model with N-state spins on vertical edges, two-state spins on horizontal edges. Here we generalize this to a “j-state by N-state” model and establish three sets of functional relations between the various transfer matrices. The significance of the “super-integrable” case of the chiral Potts model is discussed, and results reported for its finite-size corrections at criticality.

Off-diagonal Bethe Ansatz solution of the τ 2-model

“…where φ k is a generic complex number, 2 the functions a(u) and d(u) are given by (3.9) and (3.10), the function F k (u) is given by 9) and the function Q(u) is a trigonometric polynomial of degree (p − 1)N…”

“…Many papers have appeared in literature for such connections and many efforts have been made to obtain the solutions of chiral Potts model by solving the τ 2 -model with a recursive functional relation [9][10][11][12]. However, it was found that only in the super-integrable sub-sector [2] the algebraic Bethe Ansatz method can be applied on this model to obtain Baxter's T − Q [13] solutions and Bethe Ansatz equations, while for the generic τ 2 -model, though its integrability [1] was proven, there is no simple Q-operator solution in terms of Baxter's T − Q relation.…”

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).…”

Bethe ansatz solutions of the τ 2-model with arbitrary boundary fields

“…The finite-size inhomogeneous τ 2 -model also known as the Baxter-Bazhanov-Stroganov model (BBS model) [1][2][3][4] is a N -state spin lattice model, which is intimately related to some other integrable models under certain parameter constraints such as the chiral Potts model [5][6][7][8][9][10] and the relativistic quantum Toda chain model [11]. Lots of papers have appeared to explain such connections and many efforts have been made to calculate the eigenvalues of the chiral Potts model by solving the τ 2 -model with a recursive functional relation [4,[12][13][14].…”