Chiral Potts model as a descendant of the six-vertex model

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

“…The chiral Potts transfer matrix T p,p ′ (q) with p, p ′ in (2.8) or (2.9) can be constructed from the τ (2) -matrix as a Baxter's Q-operator (see [17,19] or [41] section 3.1). For each τ (2) -eigenvalue, one associates the t N -function P (t) in (4.25).…”

“…Furthermore, the knowledge was culminated in a recent proof of the order parameter by Baxter [16]. These results all rely on the technique of functional relations [17] about the transfer matrix and fusion matrices of the associated τ (2) -model in the extended study of chiral Potts model as a descendent of the six-vertex model found in [19]. The study is along the line of T Q-relation method invented by Baxter in solving the eigenvalue problem of the eight-vertex model [5,6,7].…”

“…While in the functional-relation symmetry study of XXZ chains associated to the cyclic U q (sl 2 ) representations labeled by the parameter ζ ∈ C * with q N = ζ N = 1, these XXZ chains are equivalent to certain chiral Potts models with alternating superintegrable rapidities [40]. 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.…”

“…. will denote rapidities in a curve in (2.8) or (2.9), and write their coordinates by x p , y p , µ p , t p , λ p whenever it will be necessary to specify the element p. The L-operator of τ (2) p,p ′ (t) in (2.1) is defined by the affine coordinates of p, p ′ in (2.8) or (2.9) with the parameters 19) with the variables σ, σ † in (2.16). Note that α q , α q are related to z(t) by the equality 20) which is the connection between the determinant of…”

The Superintegrable Chiral Potts Quantum Chain and Generalized Chebyshev Polynomials

“…Then, for the Z 3 -case, using functional relations for the transfer matrix, Albertini, McCoy and Perk [8] obtained also the excited level sectors which involve solutions of Bethe-equations. Finally, Baxter [11], exploiting functional relations derived from the relation of the 2-dimensional integrable chiral Potts model to the six-vertex model [22], solved the general Z N -superintegrable case. 2 .…”