Recent work has revealed a host of ‘dualities’ between strongly interacting models. As apparent from the canonical example of Kramers and Wannier, such dualities are much subtler than a one- to-one mapping of energy levels, but rather are non-invertible. I describe an algebraic structure in the XXZ spin chain and three other Hamiltonian that yields non-invertible maps between them and also guarantees all are integrable. Several of these models also possess useful non-invertible symmetries, with the spontaneous breaking of one resulting in an unusual ground-state degeneracy. The maps are found expicitly using topological defects coming from fusion categories and the lattice analogue of the orbifold construction.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Topological Quantum Computation.
