Anomalous Localized Topological Phases

There is a well established principle and formalism for classifying ground state phases of gapped Hamiltonian systems––Hamiltonians which can be deformed into one another without closing the gap belong to the same topological phase. In the nonequilibrium context, topological phases of localized systems may be identified as sets of Hamiltonians which can be deformed into one another without going through a delocalization transition. For Floquet systems, it is known how to map the classification of such Hamiltonians to a classification of locality preserving unitaries––called quantum cellular automata (QCA). We show how to classify localized topological phases using QCA beyond the context of periodic driving, including static and (quasi)periodically driven systems, with or without symmetry. Further, we adapt many tools from the study of gapped ground states to the localized context, allowing for significant progress in the general classification. Some of the localized topological phases so discovered are characterized by eigenstate order––eigenstates of the model are ordered like ground states of nontrivial gapped Hamiltonians. However, there are also anomalous localized topological phases (ALT phases), for which each eigenstate is trivial when regarded individually, but the system as a whole still cannot be deformed into an atomic insulator.