The complete list of non-symmetric eight-vertex constant solutions to the Yang-Baxter equation is obtained and new solutions dependent on more than one variable are presented. The corresponding spin Hamiltonians are derived by Sutherland's method (1970).
We solved the Yang-Baxter equation for the R-matrices of three-state vertex models with ice condition, and obtained a complete list of solvable nineteen-vertex models and associated quantum spin Hamiltonians of spin one.
All regular 4×4 solutions of the Yang-Baxter equation
In this article we present all nonsingular upper triangular solutions to the constant quantum Yang-Baxter equation R j1j2 k1k2 R k1j3 l1k3 R k2k3 l2l3 = R j2j3 k2k3 R j1k3 k1l3 R k1k2 l1l2 in the three-state case. The upper triangular ansatz implies 729 equations for 45 variables. Fortunately, many of the equations turned out to be simple, allowing us to start breaking the problem into smaller ones. In the end we had a total of 552 solutions, but many of them were either inherited from two-state solutions or subcases of others. The final list contains 35 nontrivial solutions, most of them new.
A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Gröbner-basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well-known C, P, and T reflections, and rotations and scalings R̃=κ(Q⊗Q)R(Q⊗Q)−1.
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.