Much of the recent enthusiasm directed towards topological insulators [1][2][3][4][5][6][7][8][9][10][11][12][13] as a new state of matter is motivated by their hallmark feature of protected chiral edge states. In fermionic systems, Kramers degeneracy gives rise to these entities in the presence of time-reversal symmetry (TRS) [1-3, 14, 15]. In contrast, bosonic systems obeying TRS are generally assumed to be fundamentally precluded from supporting edge states [3,16]. In this work, we dispel this perception and experimentally demonstrate counterpropagating chiral states at the edge of a time-reversal-symmetric photonic waveguide structure. The pivotal step in our approach is encoding the effective spin of the propagating states as a degree of freedom of the underlying waveguide lattice, such that our photonic topological insulator is characterised by a Z 2 -type invariant. Our findings allow for fermionic properties to be harnessed in bosonic systems, thereby opening new avenues for topological physics in photonics as well as acoustics, mechanics and even matter waves. arXiv:1812.07930v1 [physics.optics]

We introduce an efficient algorithm for the computation of the W 3 invariant of general unitary maps, which converges rapidly even on coarse discretization grids. The algorithm does not require extensive manipulation of the unitary maps, identification of the precise positions of degeneracy points, or fixing the gauge of eigenvectors. After construction of the general algorithm, we explain its application to the 2 + 1 dimensional maps that arise in the Floquet-Bloch theory of periodically driven two-dimensional quantum systems. We demonstrate this application by computing the W 3 invariant for an irradiated graphene model with a continuously modulated Hamilton operator, where it predicts the number of anomalous edge states in each gap.

We introduce Z2-valued bulk invariants for symmetry-protected topological phases in 2 + 1 dimensional driven quantum systems. These invariants adapt the W3-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 non-zero 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 KaneMele, 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, combination of the W3 and the Z2-invariants allows us to distinguish between weak and strong topological phases.

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In two-dimensional anomalous Floquet insulators, chiral boundary states can spectrally detach from the bulk bands through non-Hermitian boundary state engineering. We show that this spectral detachment enables spatial detachment: The non-Hermitian boundary can be physically cut off from the bulk while retaining its topological transport properties. The resulting one-dimensional chain is identified as a non-Hermitian Floquet chain with non-zero winding number. Through the spatial detachment, the conventional bulk-boundary correspondence is recovered in the anomalous Floquet insulator. We demonstrate our theoretical findings for the standard model of an anomalous Floquet insulator and discuss their experimental relevance.