Yang-Baxter and the Boost: splitting the difference

Abstract

In this paper we continue our classification of regular solutions of the Yang-Baxter equation using the method based on the spin chain boost operator developed in [1]. We provide details on how to find all non-difference form solutions and apply our method to spin chains with local Hilbert space of dimensions two, three and four. We classify all 16\times16 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>16</mml:mn> <mml:mo>×</mml:mo> <mml:mn>16</mml:mn> </mml:mrow> </mml:math> solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔰</mml:mi> <mml:mi>𝔲</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>⊕</mml:mo> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔰</mml:mi> <mml:mi>𝔲</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> symmetry, which include the one-dimensional Hubbard model and the S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> </mml:math> -matrix of the {AdS}_5 \times {S}^5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>d</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>5</mml:mn> </mml:msup> </mml:mrow> </mml:math> superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.