Integrable systems, separation of variables and the Yang-Baxter equation

TL;DR

This paper reviews recent progress in quantum integrability, focusing on separation of variables, the Yang-Baxter equation, and new computational techniques for high-rank spin chains with applications in AdS/CFT.

Contribution

It introduces a novel link between separation of variables and quantum algebra representation theory, and develops new tools for solving the Yang-Baxter equation and scalar products in integrable models.

Findings

Constructed separated variables for $ ext{gl}(n)$ spin chains.

Developed the Functional SoV (FSoV) technique based on Baxter TQ equations.

Classified solutions to the Yang-Baxter equation, including 4x4 solutions with fermion number preservation.

Abstract

This article, based on the author's PhD thesis, reviews recent advancements in the field of quantum integrability, in particular the separation of variables (SoV) program for high-rank integrable spin chains and the boost mechanism for solving the Yang-Baxter equation. We begin with a general overview of quantum integrable systems with special emphasis on their description in terms of quantum algebras. We then provide a detailed account of the Yangian of $\mathfrak{gl}(n)$ in particular the Bethe algebra, fusion, and T- and Q-systems. We then introduce the notion of separation of variables in integrable systems and build on Sklyanin's work in rank 1 models and extend to higher rank. By exploiting a novel link between SoV and quantum algebra representation theory we construct the separated variables for $\mathfrak{gl}(n)$ spin chains for arbitrary compact representations of the symmetry algebra and develop various new tools along the way. Next, we build on the previous part and develop a new technique for the computation of scalar products in the SoV framework which we call Functional SoV or FSoV. Unlike the work in the previous, operatorial, part this approach is based on the Baxter TQ equations. After developing this technique we supplement it with a new operator construction providing a unified view of functional and operatorial SoV. Then, we generalise the results of the previous part from compact spin chains to non-compact spin chains. The final part of this work is based on the development of tools for solving the Yang- Baxter equation. We develop a bottom-up approach for this based on the so-called Boost automorphism and uses the spin chain Hamiltonian as a starting point. Our approach allows us to classify numerous families of solutions in particular a complete classification of 4 x 4 solutions which preserve fermion number which have applications in the AdS/CFT correspondence.