Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

Abstract

The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nonetheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.

Yang-Baxter and the Boost: splitting the difference

“…where c i are constants. The resulting Hamiltonian is that of the XYZ spin chain [13,38] under our identifications.…”

“…This approach is usually supplemented with differentiating 1 the YBE and reducing the cubic functional equations to a system of coupled partial differential equations. This approach has recently been used to provide a full classification of R-matrix of size 4 × 4 so-called 8-and-lower-vertex models [38] obeying the difference property R(u, v) = R(u − v) and to obtain certain 9 × 9 models [39] whose R-matrix satisfies the so-called ice rule but it quickly becomes unwieldy as the size of the R-matrix increases.…”

The Free-Fermion Eight-Vertex Model: Couplings, Bipartite Dimers and Z-Invariance

“…The Z -invariant weights of the 8V model in the non-checkerboard case have been parameterized by Baxter [4][5][6] and Zamolodchikov [70]. Other techniques have appeared since to classify these solutions [35,50,62,66]. By considering checkerboard Yang-Baxter equations, more solutions can appear, as noted for instance in [59], although no complete parametrization is known.…”

Polynomials with Symmetric Zeros

“…In mathematics, we can cite a very interesting connection between Alexander polynomials and the theory of Salem numbers: more precisely, the Alexander polynomial associated with the so-called Pretzel Knot P(−2, 3, 7) is nothing but the Lehmer polynomial L(z) introduced in Section 5.1; it is indeed the Alexander polynomial with the smallest Mahler measure [73]. In physics, knot theory is connected with quantum groups and it also can be used to one construct solutions of the Yang-Baxter equation [74] through a method called baxterization of braid groups.…”