The four-state solution of the Yang-Baxter equation

Heather A. Dye

Arash Pourkia

We present a new class of solutions to the parameter-dependent Yang–Baxter equation across all dimensions, which includes a significant subclass of unitary and entangling solutions. In any dimension d ≥ 2 , we construct a well-structured matrix R ( u ) that exhibits intriguing and useful symmetries. A key feature of R ( u ) is its composition from four monomial-based layers: R ( u ) = R a ( u ) + R b ( u ) + R x ( u ) + R y ( u ) , where a ( u ) , b ( u ) , x ( u ) and y ( u ) are four sets of monomial complex functions of the form, a u n a , b u n b , x u n x and y u n y , respectively. This well-designed structure plays a crucial role in demonstrating that R ( u ) is indeed a solution to the Yang–Baxter equation and in establishing the conditions for its unitarity and entangling properties. Additionally, it allows us to identify interesting non-trivial subfamilies with these properties. As is widely recognized, unitary and entangling solutions to the Yang–Baxter equation serve as universal quantum logic gates for qudit quantum computing . Moreover, the search for new solutions to the Yang–Baxter equation in higher dimensions is a common endeavor in both mathematics and physics .

Michio Jimbo

Dmitry Chicherin, Sergey E. Derkachov, Vyacheslav Pavlovich Spiridonov

Marius de Leeuw, Chiara Paletta, Anton Pribytok +2 more

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 16 × 16 solutions which exhibit \mathfrak{su}(2)\oplus \mathfrak{su}(2) 𝔰 𝔲 ( 2 ) ⊕ 𝔰 𝔲 ( 2 ) symmetry, which include the one-dimensional Hubbard model and the S S -matrix of the {AdS}_5 \times {S}^5 A d S 5 × S 5 superstring sigma model. In all cases we find interesting novel solutions of the Yang-Baxter equation.

T Kennedy

The author considers solutions of the Yang-Baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin Hamiltonian which is isotropic in spin space, i.e. SU(2)-invariant. Four such solutions are known for each value of the spin S. (For S=1/2 they degenerate into the same solution, and for S=1 they only give three different solutions). For S<or=6 he shows that these are the only solutions which are SU(2)-invariant, except for S=3 when there is a fifth solution.