The partially (an)isotropic Inozemtsev spin chain

Abstract: Traditionally, (quantum) integrable spin chains are studied under the assumption of short-range interactions between the spins, leading to the nearest-neighbour Heisenberg chains. The most famous integrable model with long-range spin interactions is the Haldane--Shastry chain, whose integrable structure was uncovered at IPhT about thirty years ago. The Inozemtsev spin chain, which famously made a guest appearance in AdS/CFT integrability, interpolates between the Heisenberg XXX and Haldane--Shastry chains while being exactly solvable throughout. Although widely believe to be integrable, the algebraic structure underlying the Inozemtsev chain is not yet known.

One way to learn about something is to try and deform it. For spin chains, it is natural to try and break the SU(2) spin symmetry ('isotropy') down to spin-z symmetry ('partial (an)isotropy') in a way that preserves the key features. In my talk I will present a new long-range spin chain: the partially (an)isotropic Inozemtsev chain. It is integrable in that it has a hierarchy of commuting hamiltonians. In the long-range limit it becomes the (known) partially (an)isotropic generalisation of Haldane--Shastry, while in the short-range limit it gives a variant of Heisenberg XXZ with nontrivial boundary conditions. Underlying the model is a new quantum many-body system with spins that generalises the elliptic Ruijsenaars model.

Based on work in progress with Rob Klabbers (Humboldt U Berlin)

mathematical physics

Audience: researchers in the topic

Séminaire de physique mathématique IPhT