The Matrix Product Ansatz for integrable U(1)^N models in Lunin-Maldacena backgrounds

TL;DR

This paper develops a Matrix Product Ansatz to exactly solve a broad class of integrable U(1)^N spin chains, including models relevant to string theory and supersymmetric gauge theories, revealing new integrable models.

Contribution

It introduces a novel MPA framework for N-state U(1)^N models, connecting algebraic structures to integrability and uncovering previously unknown integrable models.

Findings

Exact solutions for N-state spin chains with U(1)^N symmetry.

Identification of a new class of integrable models.

Analysis of Yang-Baxter equations in specific sectors.

Abstract

We obtain through a Matrix Product Ansatz (MPA) the exact solution of the most general $N$-state spin chain with $U(1)^N$ symmetry and nearest neighbour interaction. In the case N=6 this model contain as a special case the integrable SO(6) spin chain related to the one loop mixing matrix for anomalous dimensions in ${\cal N} = 4$ SYM, dual to type $IIB$ string theory in the generalised Lunin-Maldacena backgrounds. This MPA is construct by a map between scalar fields and abstract operators that satisfy an appropriate associative algebra. We analyses the Yang-Baxter equation in the N=3 sector and the consistence of the algebraic relations among the matrices defining the MPA and find a new class of exactly integrable model unknown up to now.