Superior robustness of anomalous non-reciprocal topological edge states.

Robustness against disorder and defects is a pivotal advantage of topological systems1, manifested by the absence of electronic backscattering in the quantum-Hall2 and spin-Hall effects3, and by unidirectional waveguiding in their classical analogues4,5. Two-dimensional (2D) topological insulators4-13, in particular, provide unprecedented opportunities in a variety of fields owing to their compact planar geometries, which are compatible with the fabrication technologies used in modern electronics and photonics. Among all 2D topological phases, Chern insulators14-25 are currently the most reliable designs owing to the genuine backscattering immunity of their non-reciprocal edge modes, brought via time-reversal symmetry breaking. Yet such resistance to fabrication tolerances is limited to fluctuations of the same order of magnitude as their bandgap, limiting their resilience to small perturbations only. Here we investigate the robustness problem in a system where edge transmission can survive disorder levels with strengths arbitrarily larger than the bandgap-an anomalous non-reciprocal topological network. We explore the general conditions needed to obtain such an unusual effect in systems made of unitary three-port non-reciprocal scatterers connected by phase links, and establish the superior robustness of anomalous edge transmission modes over Chern ones to phase-link disorder of arbitrarily large values. We confirm experimentally the exceptional resilience of the anomalous phase, and demonstrate its operation in various arbitrarily shaped disordered multi-port prototypes. Our results pave the way to efficient, arbitrary planar energy transport on 2D substrates for wave devices with full protection against large fabrication flaws or imperfections.

Non-Hermitian Spectral Flows and Berry-Chern Monopoles.

We propose a non-Hermitian generalization of the correspondence between the spectral flow and the topological charges of band crossing points (Berry-Chern monopoles). A class of non-Hermitian Hamiltonians that display a complex-valued spectral flow is built by deforming an Hermitian model while preserving its analytical index. We relate those spectral flows to a generalized Chern number that we show to be equal to that of the Hermitian case, provided a line gap exists. We demonstrate the homotopic invariance of both the non-Hermitian Chern number and the spectral flow index, making explicit their topological nature. In the absence of a line gap, our system still displays a spectral flow whose topology can be captured by exploiting an emergent pseudo-Hermitian symmetry.

Topological Properties of Floquet Winding Bands in a Photonic Lattice.

The engineering of synthetic materials characterized by more than one class of topological invariants is one of the current challenges of solid-state based and synthetic materials. Using a synthetic photonic lattice implemented in a two-coupled ring system we engineer an anomalous Floquet metal that is gapless in the bulk and shows simultaneously two different topological properties. On the one hand, this synthetic lattice presents bands characterized by a winding number. The winding emerges from the breakup of inversion symmetry, and it directly relates to the appearance of Bloch suboscillations within its bulk. On the other hand, the Floquet nature of the lattice results in well-known anomalous insulating phases with topological edge states. The combination of broken inversion symmetry and periodic time modulation studied here enriches the variety of topological phases available in lattices subject to Floquet driving and suggests the possible emergence of novel phases when periodic modulation is combined with the breakup of spatial symmetries.

Unidirectional Modes Induced by Nontraditional Coriolis Force in Stratified Fluids.

Using topology, we unveil the existence of new unidirectional modes in compressible rotating stratified fluids. We relate their emergence to the breaking of time-reversal symmetry by rotation and vertical mirror symmetry by stratification and gravity. We stress the role of the Coriolis force's nontraditional part, induced by a rotation field tangent to the surface. In contrast with horizontally trapped equatorial waves induced by the traditional component of the Coriolis force perpendicular to the surface, we find vertically trapped modes that propagate along interfaces between regions with distinct stratification properties. We show that such modes are generalized atmospheric Lamb waves whose direction of propagation can be selected by the nontraditional component of the Coriolis force.

Symmetry-Protected Multifold Exceptional Points and Their Topological Characterization.

We investigate the occurrence of n-fold exceptional points (EPs) in non-Hermitian systems, and show that they are stable in n-1 dimensions in the presence of antiunitary symmetries that are local in parameter space, such as, e.g., parity-time (PT) or charge-conjugation parity (CP) symmetries. This implies in particular that threefold and fourfold symmetry-protected EPs are stable, respectively, in two and three dimensions. The stability of such multofold exceptional points (i.e., beyond the usual twofold EPs) is expressed in terms of the homotopy properties of a resultant vector that we introduce. Our framework also allows us to rephrase the previously proposed Z_{2} index of PT and CP symmetric gapped phases beyond the realm of two-band models. We apply this general formalism to a frictional shallow water model that is found to exhibit threefold exceptional points associated with topological numbers ±1. For this model, we also show different non-Hermitian topological transitions associated with these exceptional points, such as their merging and a transition to a regime where propagation is forbidden, but can counterintuitively be recovered when friction is increased furthermore.

Topological index for periodically driven time-reversal invariant 2D systems.

We define a new Z2-valued index to characterize the topological properties of periodically driven two dimensional crystals when the time-reversal symmetry is enforced. This index is associated with a spectral gap of the evolution operator over one period of time. When two such gaps are present, the Kane-Mele index of the eigenstates with eigenvalues between the gaps is recovered as the difference of the gap indices. This leads to an expression for the Kane-Mele invariant in terms of the Wess-Zumino amplitude. We illustrate the relation of the new index to the edge states in finite geometries by numerically solving an explicit model on the square lattice that is periodically driven in a time-reversal invariant way.

Z2 peak of noise correlations in a quantum spin Hall insulator.

We investigate the current noise correlations at a quantum point contact in a quantum spin Hall structure, focusing on the effect of a weak magnetic field in the presence of disorder. For the case of two equally biased terminals we discover a robust peak: the noise correlations vanish at B = 0 and are negative for B ≠ 0. We find that the character of this peak is intimately related to the interplay between time reversal symmetry and the helical nature of the edge states and call it the Z2 peak.

Anomalous topological waves in strongly amorphous scattering networks.

Topological insulators are crystalline materials that have revolutionized our ability to control wave transport. They provide us with unidirectional channels that are immune to obstacles, defects, or local disorder and can even survive some random deformations of their crystalline structures. However, they always break down when the level of disorder or amorphism gets too large, transitioning to a topologically trivial Anderson insulating phase. We demonstrate a two-dimensional amorphous topological regime that survives arbitrarily strong levels of amorphism. We implement it for electromagnetic waves in a nonreciprocal scattering network and experimentally demonstrate the existence of unidirectional edge transport in the strong amorphous limit. This edge transport is shown to be mediated by an anomalous edge state whose topological origin is evidenced by direct topological invariant measurements. Our findings extend the reach of topological physics to a class of systems in which strong amorphism can induce, enhance, and guarantee the topological edge transport instead of impeding it.

Magnetic-field-induced localization in 2D topological insulators.

Localization of the helical edge states in quantum spin Hall insulators requires breaking time-reversal invariance. In experiments, this is naturally implemented by applying a weak magnetic field B. We propose a model based on scattering theory that describes the localization of helical edge states due to coupling to random magnetic fluxes. We find that the localization length is proportional to B^{-2} when B is small and saturates to a constant when B is sufficiently large. We estimate especially the localization length for the HgTe/CdTe quantum wells with known experimental parameters.

Wavefront dislocations reveal the topology of quasi-1D photonic insulators.

Phase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation - a 2D phase singularity - in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.

Topological origin of equatorial waves.

Topology sheds new light on the emergence of unidirectional edge waves in a variety of physical systems, from condensed matter to artificial lattices. Waves observed in geophysical flows are also robust to perturbations, which suggests a role for topology. We show a topological origin for two well-known equatorially trapped waves, the Kelvin and Yanai modes, owing to the breaking of time-reversal symmetry by Earth's rotation. The nontrivial structure of the bulk Poincaré wave modes encoded through the first Chern number of value 2 guarantees the existence of these waves. This invariant demonstrates that ocean and atmospheric waves share fundamental properties with topological insulators and that topology plays an unexpected role in Earth's climate system.