Anomalous topological waves in strongly amorphous scattering networks

Abstract

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.

Fig. 1.. Anomalous edge states can survive any level of amorphism.

(A) Anomalous topological edge states occur in any scattering signal graphs, when a limit can be found in which bulk signals travel in closed loops (in blue), leaving a large signal loop on the edge (in orange). This picture is not only valid for periodic systems (top row, honeycomb case) but should also be true for any level of amorphism (bottom row). We validate this idea by mapping the oriented graphs (first column) to practical scattering networks made of three-port circulators linked with reciprocal connections (second column). We built prototypes (third column) operating around 5.7 GHz and experimentally observed the resilience of anomalous edge states to strong levels of amorphism (fourth column). EPFL, École Polytechnique Fédérale de Lausanne. (B) To characterize the effect of amorphism, we introduce an amorphous factor α that controls the continuous transition from the clean limit to a strongly amorphous phase. We track both the link length variance and the number of sides of each loop in the network [numbers in (A), second column]. As α increases from 0 to 1.75, the links start to deform, but the percentage of loops with N = 6 sides stays at 100% as in the clean honeycomb limit (left panel). Beyond this weakly amorphous regime, a transition occurs during which the percentage of loop 6 drops notably as the one of loops with N ≠ 6 increases. For α above 5, we enter a fully amorphous phase, characterized by a stabilized distribution of loops of various sizes. The networks statistics are computed from 1000 random realizations of networks made of 1000 nodes.

Fig. 2.. Exceptional resilience of anomalous edge states to strong amorphism.

We compare the resilience of the anomalous edge mode to amorphism with the one of a standard Chern edge mode when propagating along a domain wall with a trivial insulator. (A) In the clean limit (α = 0, first row), both anomalous and Chern phases provide a robust channel with unitary transmission. Then, we impart strong amorphism (α = 8, second row). Only the anomalous interface state survives. Conversely, the Chern case undergoes Anderson localization. (B) Experimental demonstration of the topological distinction between anomalous (left) and trivialized Chern (right) phases in the strong amorphous limit (α = 6). The trivialized Chern phase is obtained by adding amorphism to a Chern crystal, which differs from an anomalous phase only by the presence of extra scatterers between the circulators. (C) Measured field map when exciting the interface from the top, confirming the existence of a topological state at the interface.

Fig. 3.. Amorphism-enhanced edge transmission and bulk insulation of the anomalous phase.

We consider the evolution of the anomalous edge and bulk transmissions when increasing the level of amorphism, for any value of the phase delay ϕ ∈ [0,2π] of the reciprocal links, defined in the clean limit. Each point corresponds to an average over 200 realizations of randomly generated scattering networks, with some examples shown in (B). (A) In the weakly amorphous regime, the edge (top) and bulk (bottom) transmissions are consistent with the clean-limit band structures, with large edge transmission only in the topological gaps and nonzero bulk transmission only in the bulk bands. After the transition stage, the edge transmission is enhanced to 1, and the bulk transmission is pinned to zero regardless of the value of ϕ. This confirms the nucleation of a single amorphism-enhanced anomalous topological phase, which now spans the full 2π range. (B) Examples of amorphous networks and simulated fields of edge and bulk transmissions at ϕ = 1.1 [dashed line marked with stars in (A)], which corresponds to a bulk band in the clean limit. The panels demonstrate how the bulk modes localize as the amorphous factor is increased from the clean limit (first row) to the fully amorphous case (last row), creating an amorphous topological phase with large edge transmission.

Fig. 4.. Direct measurement of topological indices in the strongly amorphous regime.

(A) Scheme for measuring the topological index of amorphous scattering networks. We impart a twisted boundary condition to the scattering network, which links the left and right boundaries with a nonreciprocal phase Φ. The topological index W is the winding of the reflection coefficient R measured at the external probe, when Φ is varied over all angles. (B) Picture of the experimental setup, with the microwave nonreciprocal phase shifter shown in the inset. (C and D) Measured winding numbers and corresponding field maps when starting from different situations in the clean limit. In (C), we start with anomalous and Chern networks in a topological gap, whereas in (D), we start inside bulk bands. The measurements show that regardless of the starting point in the clean limit, the anomalous network is always topological under strong amorphism. Conversely, the Chern network always becomes a trivial insulator. (E) Amorphism-induced topological phase transitions. For each amorphous level α and phase delay ϕ, we calculate the winding number W of a randomly generated network. In the weakly amorphous regime, the bulk bands of the anomalous network persist with trivial windings, while the Chern networks exhibit a robust nontrivial topology only near the clean limit. Under moderate levels of amorphism, opposite transitions occur for the two phases. The occurrence of Chern networks with nonzero winding decreases to zero, consistent with a trivial Anderson insulator. Very differently, anomalous networks undergo a topologically nontrivial Anderson transition, as the entire spectrum becomes topological. (F) Statistical study of the proportion of realizations with nontrivial winding versus α performed on 200 random realizations of amorphous networks with phase delay ϕ = 1.1.