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Classification of Static and Driven Topological insulators

Xu Liu

Published Jan 1, 2020• 1 citation
OpenAlex

Abstract

This dissertation focus on the classification of topological matters in static and periodically driven systems.First, we build a complete topological classification of local unitary operators in free fermionic systems. This result can be encoded in a periodic table with a period of eight, similar to the periodic table of topological insulators. In our classification, we define two local unitaries in a certain symmetry class to be topologically equivalent if they can be connected via finite time evolutions (locally generated unitaries) without breaking the symmetries of the symmetry class while maintaining locality. In this way, the classification we derive will allow us to distinguish locally generated unitaries from those that cannot be locally generated.Besides, we also how to find possible generating Hamiltonians for locally generated unitaries. These results can be used to study the edge behaviors of Floquet systems.Second, we propose bulk invariants and edge invariants that are locally computable and improve existing topological invariants by being applicable to systems with the disorder. Afterward, we set up a rigorous connection between bulk and edge invariants for Floquet systems belonging to Class AIII and class AII of the Altland-Zirnbauer symmetry classification.Last, after treating a static system as a Floquet system generated by a constant Hamiltonian, we employ the tools we developed in Floquet systems to classify the edge behaviors in static systems.

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This dissertation focus on the classification of topological matters in static and periodically driven systems.First, we build a complete topological classification of local unitary operators in free fermionic systems. This result can be encoded in a periodic table with a period of eight, similar to the periodic table of topological insulators. In our classification, we define two local unitaries in a certain symmetry class to be topologically equivalent if they can be connected via finite time evolutions (locally generated unitaries) without breaking the symmetries of the symmetry class while maintaining locality. In this way, the classification we derive will allow us to distinguish locally generated unitaries from those that cannot be locally generated.Besides, we also how to find possible generating Hamiltonians for locally generated unitaries. These results can be used to study the edge behaviors of Floquet systems.Second, we propose bulk invariants and edge invariants that are locally computable and improve existing topological invariants by being applicable to systems with the disorder. Afterward, we set up a rigorous connection between bulk and edge invariants for Floquet systems belonging to Class AIII and class AII of the Altland-Zirnbauer symmetry classification.Last, after treating a static system as a Floquet system generated by a constant Hamiltonian, we employ the tools we developed in Floquet systems to classify the edge behaviors in static systems.
This dissertation comprises two parts, each of which details an investigation within topological matter.In the first part, we investigate the real space localization properties of static topological insulators in two dimensions. We develop three numerical procedures each of which removes an extensive fraction of the local degrees of freedom. We probe the real space properties of the residual projection operator, showing that the maximally localized length scale ξ supported in the residual Hilbert space diverges as the fraction of states remaining approaches 0. The power-law exponent ν = 1/2 characterizing the singular behavior of ξ appears universal across class A insulators with non-zero Chern number.In the second part, we explore a class of Floquet topological phases in three dimensions. These phases have translational invariance but no other symmetry, and exhibit anomalous transport at a boundary surface. We show that the boundary behavior of such phases falls into equivalence classes up to local 2D unitary evolution. This provides a classification of the 3D bulk, which we conjecture to be complete. We demonstrate that such phases may be generated by exactly solvable “exchange drives” in the bulk. The edge behavior of a general exchange drive in two or three dimensions is shown to derive from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.
Similar to static systems, periodically driven systems can host a variety of topologically non-trivial phases. Unlike the case of static Hamiltonians, the topological indices of bulk Floquet bands may fail to describe the presence and robustness of edge states, prompting the search for new invariants. We develop a unified description of topological phases and their invariants in driven systems, by using scattering theory. We show that scattering matrix invariants correctly describe the topological phase, even when all bulk Floquet bands are trivial. Additionally, we use scattering theory to introduce and analyze new periodically driven phases, such as weak topological Floquet insulators, for which invariants were previously unknown. We highlight some of their similarities with static systems, including robustness to disorder, as well as some of the features unique to driven systems, showing that the weak phase may be destroyed by breaking translational symmetry not in space, but in time.
Periodically driven (Floquet) systems can exhibit possibilities beyond what can be obtained in equilibrium. Both in Floquet systems and in the related problems of discrete-time quantum walks and quantum cellular automata, a basic distinction arises among unitary time evolution operators: while all physical operators are local, not all are locally generated (i.e., generated by some local Hamiltonian). In this paper, we define the notion of equivalence up to a locally generated unitary in all Altland-Zirnbauer symmetry classes. We then classify noninteracting unitaries in all dimensions on this basis by showing that equivalence up to a locally generated unitary is identical to homotopy equivalence.
Periodically driven systems have recently been shown to host topological phases that are inherently dynamical in character, opening up a new arena in which to explore topological physics. One important group of such phases, known as Floquet topological insulators, arise in systems of free fermions and exhibit protected topological edge modes analogous to the edge modes of static topological insulators. In this work, the authors use methods from K theory to provide a complete topological classification of Floquet topological phases of this kind. The main result is a periodic table for Floquet topological insulators, which may be viewed as a time-dependent extension of the periodic table of topological insulators and superconductors originally introduced by Alexei Kitaev.
Periodically driven Floquet systems can host dynamical phases, including a range of exotic topological phases that have no static analogs. This work presents a homotopy approach to the study of driven systems, which treats unitary evolutions as paths in the space of unitary operators. By considering loop evolutions in this space, the authors obtain a topological classification, free of uncertainties and ambiguities about the long-time robustness of of behavior in specific physical models. Two classes of Floquet symmetry-protected topological phase are identified, characterized by unitary evolutions that act trivially in the bulk but nontrivially at the boundary of an open system. The first class is captured by an explicit group cohomology construction; it has the remarkable property of converting a trivial boundary state into a nontrivial symmetry-protected topological state. The second class exhibits anomalous counter propagating Hilbert-space transport at the boundary; it lies beyond the cohomology construction.
Periodically driven systems with internal and spatial symmetries can exhibit a variety of anomalous boundary behaviors at both the zero and $\ensuremath{\pi}$ quasienergies despite the trivial bulk Floquet bands. These phenomena are called anomalous Floquet topology (AFT) as they are unconnected from their static counterpart, emerging from the winding of the time-evolution unitary rather than the bulk Floquet bands at the end of the driving period. In this paper, we systematically derive the first and inversion-symmetric second-order AFT bulk-boundary correspondence for Altland-Zirnbauer (AZ) classes BDI, D, DIII, and AII. For each AZ class, we start a dimensional hierarchy with a parent dimension having Z\mathbb{Z}Z classification, then use it as an interpolating map to classify the lower-dimensional descendants. From the Atiyah-Hirzebruch spectral sequence, we identify the subspace that contains topological information and faithfully derive the AFT bulk-boundary correspondence for both the parent and descendants. Our theory provides analytic tools for out-of-equilibrium topological phenomena.