Europe/Lisbon —

Anton Kapustin

Anton Kapustin, Caltech

M. Berry showed how to attach a line bundle and a connection on it to a family of quantum Hamiltonians with a non-degenerate ground state, under the assumption that the Hilbert space is finite-dimensional. The first Chern class of this line bundle is a topological invariant of the family. It is far from obvious if this construction can be generalized to quantum many-body Hamiltonians. Indeed, naive generalizations fail because ground states of different Hamiltonians typically correspond to inequivalent representations of the algebra of observables. Nevertheless, it is possible to construct such invariants by making use of a certain differential graded Lie algebra (DGLA) attached to a quantum lattice system. For example, it turns out that to any family of gapped Hamiltonians on a 1d lattice one can attach a “quantized” degree-3 cohomology class on the parameter space. In this talk I will outline a construction of this DGLA as well as the construction of higher Berry classes. The talk is based on a work in progress with Nikita Sopenko.