A new Q-matrix in the Eight-Vertex Model

We construct a Q-matrix for the eight-vertex model at roots of unity for crossing parameter η = 2mK/L with odd L, a case for which the existing constructions do not work. The new Q-matrix Q̂ depends on the spectral parameter v and also on a free parameter t. For t = 0 Q̂ has the standard properties. For t 6= 0, however, it does not commute with the operator S and not with itself for different values of the spectral parameter. We show that the six-vertex limit of Q̂(v, t = iK′/2) exists. An essential tool in Baxter’s solution of the eight-vertex model [1, 2, 3, 4] is the Q-matrix which satisfies the TQ equation T (v)Q(v) = [ρh(v − η)]Q(v + 2η) + [ρh(v + η)]Q(v − 2η) (1) and commutes with T . Here T (v) is the transfer matrix of the eight-vertex model (A.1). Combined with periodicity properties of Q(v) in the complex vplane equ. (1) leads to the derivation of Bethe’s equations and the solution of the model. For generic values of the crossing parameter η the transfer matrix T has a non degenerate spectrum. For rational values of η/K however this is not the case. This leads to the existence of different Q-matrices which all satisfy equ. (1). In ref.[1] Baxter constructs a Q-matrix valid for 2Lη = 2m1K + im2K ′ (2) with integer m1,m2, L. In ref.[2] Baxter derived a Q-matrix valid for generic values of η. As these Q-matrices are different we distinguish them by writing Q72 and Q73 respectively for the constructions in ref. [1] and ref. [2]. It turned out, however, that Q72 has interesting properties beyond its role in equ. (1) because of its restriction to rational values of η/K. In ref. [5] it is conjectured that Q72(v) satisfies the following functional relation: For N even and η = m1K/L where either L even or L and m1 odd eQ72(v − iK )