Spin chain are quantum-mechanical models for magnetic materials. Special examples are (quantum) integrable: they have many conserved charges whose spectrum and properites can be determined exactly thanks to a rich underlying algebraic structure. While one traditionally assumes that only neighbouring spins interact, there are also integrable spin chains with long-range interactions. Their integrability hinges on connections to quantum many-body systems of Calogero--Sutherland type. This is by now rather well understood in the (truly long-range) trigonometric case, but much less so in the elliptic (intermediate-range) case.
In this talk I will introduce spin chains, give an overview of the key results for integrable long-range chains, and outline how integrability works for these models.
Based on recent and ongoing work with R Klabbers (Humboldt U Berlin) and D Serban (IPhT).
