Bounded lightcone and robust topological order out of equilibrium

The ground state degeneracy of topologically ordered gapped Hamiltonians is the bedrock for a quantum error-correcting code with macroscopic distance, which is unfortunately not stable away from equilibrium. In this work, we show that the presence of a bounded lightcone preserves topological order. As a quantum code, the initial ground space will keep its macroscopic distance during unitary time evolution. We also show how a bounded lightcone can emerge through suitable perturbations in the two-dimensional toric code. Our results suggest that topological quantum memory can be dynamically robust at zero temperature.