Abstract
We classify all regular solutions of the Yang-Baxter equation of eight-vertex type. Regular solutions correspond to spin chains with nearest-neighbor interactions. We find a total of four independent solutions. Two are related to the usual six- and eight-vertex models that have R matrices of difference form. We find two new solutions of the Yang-Baxter equation, which are manifestly of nondifference form. These new solutions contain the S-matrices of the AdS_{2} and AdS_{3} integrable models as a special case. This can be used as a starting point to study and classify integrable deformations of these holographic integrable systems.
Highlights
Introduction.—The Yang-Baxter equation is an important equation that appears in many different areas of physics [1,2,3,4]
The main idea behind this method is to use the Hamiltonian rather than the corresponding R matrix as a starting point. We applied this method to solutions of the Yang-Baxter equation that were of difference form Rðu; vÞ 1⁄4 Rðu − vÞ
We demonstrate our method by classifying all solutions of the Yang-Baxter equation of eight-vertex type
Summary
With R0 denoting the derivative with respect to the second variable, expanding the Yang-Baxter equation around u2 1⁄4 u3 ≡ v yields. The other conserved charges of the integrable model are given by the higher derivatives of the transfer matrix. The interaction range of Qr is r and from (1) it follows that 1⁄2Qr; Qs 1⁄4 0: ð5Þ This tower of conserved charges is the defining property of an integrable system. Boost operator: Instead of taking derivatives of the transfer matrix, there is an alternative way to compute the higher conserved charges Qr1⁄43;4;:. 1⁄2R13R12; H23ðvÞ 1⁄4 R13R012 − R013R12; ð10Þ with Rij 1⁄4 Rijðu; vÞ These equations are special cases of the Sutherland equation [22] and they form a set of coupled first order differential equations. We can verify whether the solutions of the Sutherland equations satisfy the Yang-Baxter equation and formally prove integrability. Local basis transformation: If Rðu; vÞ is a solution of the Yang-Baxter equation, we can generate a different regular solution by defining
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