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 polarization (Resta polarization) of the CI. This Chern invariant defined from observable correlations is also established to topologically classify many-body Chern states in equilibrium. The nonequilibrium 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 nonlinear 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 counterdiabatic 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 semiperiodic boundary condition, and we clearly establish an adiabatic bulk-boundary correspondence that firmly ensconces the validity of the many-body invariant.