In two dimensions, interacting Floquet topological phases may arise even in the absence of any protecting symmetry, exhibiting chiral edge transport that is robust to local perturbations. We explore a similar class of Floquet topological phases in three dimensions, with translational invariance but no other symmetry, which also exhibit anomalous transport at a boundary surface. By studying the space of local 2D unitary operators, we show that the boundary behavior of such phases falls into equivalence classes, each characterized by an infinite set of reciprocal lattice vectors. In turn, this provides a classification of the 3D bulk, which we conjecture is complete. We demonstrate that such phases may be generated by exactly solvable ``exchange drives'' in the bulk. In the process, we show that the edge behavior of a general exchange drive in two or three dimensions can be deduced from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.In two dimensions, interacting Floquet topological phases may arise even in the absence of any protecting symmetry, exhibiting chiral edge transport that is robust to local perturbations. We explore a similar class of Floquet topological phases in three dimensions, with translational invariance but no other symmetry, which also exhibit anomalous transport at a boundary surface. By studying the space of local 2D unitary operators, we show that the boundary behavior of such phases falls into equivalence classes, each characterized by an infinite set of reciprocal lattice vectors. In turn, this provides a classification of the 3D bulk, which we conjecture is complete. We demonstrate that such phases may be generated by exactly solvable ``exchange drives'' in the bulk. In the process, we show that the edge behavior of a general exchange drive in two or three dimensions can be deduced from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.

arXiv:1802.00818Physical review. B./Physical review. BJul 19, 201811 citations

Disentangling signatures of Floquet topological systems

T. Pereg-Barnea

Bulletin of the American Physical SocietyMar 6, 20180 citations

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

Bastian Höckendorf, Andreas Alvermann, Holger Fehske

We introduce ${\mathbb{Z}}_{2}$ -valued bulk invariants for symmetry-protected topological phases in $2+1$ -dimensional driven quantum systems. These invariants adapt the ${W}_{3}$ invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a nonzero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, the combination of the ${W}_{3}$ and the ${\mathbb{Z}}_{2}$ invariants allows us to distinguish between weak and strong topological phases.We introduce ${\mathbb{Z}}_{2}$-valued bulk invariants for symmetry-protected topological phases in $2+1$-dimensional driven quantum systems. These invariants adapt the ${W}_{3}$ invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a nonzero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, the combination of the ${W}_{3}$ and the ${\mathbb{Z}}_{2}$ invariants allows us to distinguish between weak and strong topological phases.

arXiv:1708.07420Physical review. B./Physical review. BJan 22, 201824 citations

Scattering theory of Floquet topological insulators

Ion Cosma Fulga, Mykola Maksymenko

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.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.

Aug 11, 20151 citation

Strongly Disordered Floquet Topological Systems

Jacob Shapiro, Clément Tauber

Jan 1, 201928 citations

We introduce

{\mathbb{Z}}_{2}

-valued bulk invariants for symmetry-protected topological phases in

2+1

-dimensional driven quantum systems. These invariants adapt the

{W}_{3}

invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a nonzero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, the combination of the

{W}_{3}

and the

{\mathbb{Z}}_{2}

invariants allows us to distinguish between weak and strong topological phases.