On the Solutions of the Yang-Baxter Equations with General Inhomogeneous Eight-Vertex R-Matrix: Relations with Zamolodchikov’s Tetrahedral Algebra

Abstract

We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent R ij (u)-matrices of the six-and eight-vertex models, where the only constraint is the particle number conservation by mod(2). A complete classification of the solutions is performed. We have obtained also two spectral parameter dependent particular solutions R ij (u, v) of YBE. The application of the non-homogeneous solutions to construction of Zamolodchikov's tetrahedral algebra is discussed. 1

Exceptional solutions to the eight-vertex model and integrability of anisotropic extensions of massive fermionic models

“…Thus, we have a 1 = a 2 and b 1 = b 2 . In such a case, there are in general twelve independent Yang-Baxter equations following from (4.6) (for a detailed exposition see [58]):…”

“…The parameter η is defined via the relation d 2 = ηd 1 , and is required to be a constant for consistency of the Yang-Baxter equations. A necessary condition for a general solution of the inhomogeneous YBE to exist is [58]:…”

“…Besides this real solution, we note, for completeness sake, that it is possible to find some quite non-trivial solutions involving complex coupling constants, the physical meaning of which is currently not clear to us. Nevertheless, since the general analysis conducted in [58] relied on non-vanishing and non-coinciding S-matrix elements, it is still possible to find some exceptional solutions to the YBE with g 4 = 0 and g 5 = 0 by violating any of these hypotheses. One such solution corresponds to: 37) or, in terms of the renormalized coupling constants:…”

“…There are, however, some exceptional cases for which the R-matrix factorization may also occur, corresponding to the situations in which Baxter's condition is not valid due to the degeneracy of the resulting Yang-Baxter equations. Such exceptional cases have been classified and discussed in details in [58] (see also [59,60]). For Belavin's model, we find one such exceptional solution for some particular relation between the coupling constants.…”

Classifying integrable spin-1/2 chains with nearest neighbour interactions

“…Nevertheless, certain solutions of the Yang-Baxter equation with Z 2 symmetry were classified in [7]. Moreover, solutions of {6,7,8}-vertex type were classified in [8]. Recently all solutions of difference form for {4,5,6,7,8}-vertex type models have been obtained in [9].…”