Unitary preparation of many body Chern insulators: Adiabatic bulk boundary correspondence

We approach the long-standing problem of preparing an out-of-equilibrium many-body Chern insulator (CI) and associated bulk-boundary correspondence unitarily. Herein, this is addressed by constructing a dynamical many-body Chern invariant exploiting the property of the bulk macroscopic electric polarisation (Resta polarisation) of the CI. This Chern invariant defined from observable correlations is also established to topologically classify many body Chern states in equilibrium. The non-equilibrium behavior of the invariant is probed by ramping the paradigmatic Haldane model of graphene from its trivial to the topological phase. We show that a non-linear ramp may work more efficiently in approaching the topological state, thereby establishing the existence of an optimal topological state preparation. Furthermore, to ensure the near adiabatic dynamics across the quantum critical point, we propose a novel counter-diabatic scheme. The topological nature of the prepared state is firmly established by observing an emerging $U(1)$ topological charge. We also compute the edge current in the time evolved state of the system under a semi-periodic boundary condition and clearly establish an adiabatic bulk-boundary correspondence which firmly ensconces the validity of the many-body invariant.