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.

arXiv commentFinal version. 40 pages, 3 tables. Keywords: Yang-Baxter Equation, Lattice Integrable Models, Eight-Vertex Model, Bethe Ansatz, Differential and Algebraic Geometry

References in corpus (13)

• The Bethe-Ansatz for N=4 Super Yang-Mills
• Review of AdS/CFT Integrability: An Overview
• Braiding Operators are Universal Quantum Gates
• Gauge Theory and Integrability, II
• On $A_{n-1}^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)},A_{2n}^{(2)},A_{2n-1}^{(2)}$ and $D_{n+1}^{(2)}$ Reflection $K$-Matrices
• On the solutions of the Yang-Baxter equations with general inhomogeneous eight-vertex $R$-matrix: Relations with Zamolodchikov's tetrahedral algebra
• Yang-Baxter equation and reflection equations in integrable models
• Reflection matrices for the $U_q[sl(m|n{)}^{(1)}]$ vertex model
• Yang-Baxter equation for the asymmetric eight-vertex model
• Reflection matrices for the $U_q[sl(r|2m{)}^{(2)}]$ vertex model
• Reflection matrices with $U_q[os{p}^{(2)}(2|2m)]$ symmetry
• On the multiparametric {\cal U}_q[D_{n+1}^{(2)}] vertex model
• Reflection $K$-matrices for a nineteen vertex model with $U_q[\mathrm{osp}{\left(2|2\right)}^{\left(2\right)}]$ symmetry

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