I will introduce the landscape of quantum-integrable long-range spin chains and the associated (quantum-)algebraic structures, and describe recent advances and open problems in the field.
Since their origin as quantum-mechanical models for magnetism, spin systems have made their way into various areas of mathematics, including operator algebras, probability theory and representation theory. I will focus on spin chains that are quantum integrable: their spectrum admits an exact characterisation thanks to an underlying quantum-algebraic structure. Unlike for the traditionally studied nearest-neighbour Heisenberg spin chains, the quantum integrability of long-range spin chains exploits connections to a different kind of integrable models: 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) invariance as well as an explicit description of the spectrum via Jack (or Macdonald) polynomials.
The Inozemtsev spin chain interpolates between Heisenberg and Haldane–Shastry while admitting an exact description of its spectrum throughout, this time in terms of eigenfunctions of the elliptic 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.