Quenching in Chern insulators with satellite Dirac points: The fate of edge states

Abstract

We study the behavior of Fisher Zeros (FZs) and dynamical quantum phase transitions (DQPTs) for a linearly ramped Haldane model occurring in the subsequent temporal evolution of the same and probe the intimate connection with the equilibrium topology of the model}. Here, we investigate the temporal evolution of the final state of the Haldane Hamiltonian (evolving with time-independent final Hamiltonian) reached following a linear ramping of the staggered (Semenoff) mass term from an initial to a final value, {first selecting a specific protocol}, so chosen that the system is ramped from one non-topological phase to the other through a topological phase. {We establish the existence of three possible behaviour of areas of FZs corresponding to a given sector: (i) no-DQPT, (ii) one-DQPT (intermediate) and (iii)two-DQPTs (re-entrant), depending on the inverse quenching rate τ. Our study also reveals that the appearance of the areas of FZs is an artefact of the non-zero (quasi-momentum dependent) Haldane mass (MH), whose absence leads to an emergent one-dimensional behaviour indicated by the shrinking of the areas FZs to lines and the non-analyticity in the dynamical "free energy" itself. Moreover, the characteristic rates of crossover between the three behaviour of FZs are determined by the time-reversal invariant quasi-momentum points of the Brillouin zone where MH vanishes. Thus, we observe that through the presence or absence of MH, there exists an intimate relation to the topological properties of the equilibrium model even when the ramp drives the system far away from equilibrium.