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Journal of Physics A Mathematical and General

SU(1,1)-invariant solution of the quantum Yang-Baxter equation

Vladimir Chernyak, Alexander E. Kozhekin, E.I. Ogievetsky

Published Mar 21, 1993• 1 citation
Publisher

Abstract

A new approach to the solution of the quantum Yang-Baxter equation is presented and a complete SU(1,1)-invariant factorized unitary scattering matrix is constructed.
DOI
10.1088/0305-4470/26/6/014

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We comment on a recent paper by Kennedy (J. Phys. A: Math. Gen. 25 (1992) 2809) in which a systematic search for integrable spin-S su(2)-invariant quantum chains for S<or=6 revealed four spin-S families of integrable chains along with an additional integrable chain at S=3. We identify these su(2)-invariant chains with known G-invariant R-matrices, where G is a simple Lie algebra, and give arguments that Kennedy's results may well constitute the complete classification of integrable spin-S su(2)-invariant chains.
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 .
A general functional definition of the infinite dimensional quantum RRR-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional RRR-matrix from the general definition is demonstrated in a particular case when the group SU(2)SU(2)SU(2) takes place.