Recent advances for integrable long-range spin chains

Speaker: Dr Jules Lamers
Affiliation: University of Melbourne

Abstract

I will introduce the landscape of quantum-integrable long-range spin chains, give an overview of the associated (quantum-)algebraic structures, and describe recent advances and open problems in the field.

Unlike for the traditionally studied nearest-neighbour Heisenberg spin chains, the exact solvability / quantum integrability of long-range spin chains exploits connections to quantum-many body systems of Calogero–Sutherland (or Ruijsenaars–Macdonald) type. For the so-called Haldane–Shastry spin chains this can be understood in terms of affine Hecke algebras, which yield Yangian (or quantum-loop) symmetries as well as eigenvectors featuring Jack (or Macdonald) polynomials. The Inozemtsev spin chain interpolates between Heisenberg and Haldane–Shastry. Its spectrum admits an exact characterisation in terms of eigenfunctions of the elliptic quantum Calogero–Sutherland model; here the underlying quantum-algebraic structure is not understood yet.

My talk is based on joint work with R. Klabbers (Nordita), with V. Pasquier and D. Serban (IPhT CEA/Saclay), and work in progress.