Here is a short excerpt which can lend assistance to FR readers of the current paper & thread...
"Topological edge states in topological materials are robust against weak perturbations, an ability originating from the global geometry of eigen wave functions in the Hilbert space1,2. Such an intrinsic geometric feature is captured by global topological invariants that are related to edge states through the bulk-boundary correspondence.
However, this conventional paradigm is challenged by localization under disorder3,4,5,6 or non-Hermiticity7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24, which have become the focus of study of late, particularly in light of recent experimental progress in synthetic topological systems25,26,27,28,29,30,31,32,33,34,35. On one hand, despite its gap-closing tendency, the disorder can induce topology from a trivial insulator. In the resulting topological Anderson insulator, the global topology emerges in a bulk with localized states, in the absence of translational symmetry3,4,5,6. On the other hand, in a broad class of non-Hermitian topological systems, the nominal bulk states are exponentially localized toward boundaries under the non-Hermitian skin effect8,9,10,11,12,13,14,15,16,17,18,19,20,21,22.
The deviation of the bulk-state wave functions from the extended Bloch waves invalidates the conventional bulk-boundary correspondence, necessitating the introduction of non-Bloch topological invariants8,9,10,11. While the two localization mechanisms differ in origin and manifestation, the topology of the underlying system gets fundamentally modified in either case.
Remarkably, in the recently proposed non-Hermitian topological Anderson insulator36,37,38,39, the two distinct localization mechanisms are pitted against each other, wherein the interplay of disorder, non-Hermiticity, and topology leads to exotic phenomena such as the non-monotonous localization, disorder-induced non-Bloch topological phase transitions, and biorthogonal critical behaviors.
In this work, we report the experimental observation of non-Hermitian topological Anderson insulators in single-photon quantum-walk dynamics. Driven by a non-unitary topological Floquet operator, the quantum walk undergoes polarization-dependent photon loss and acquires the non-Hermitian skin effect. In contrast to previously implemented quantum walks with the non-Hermitian skin effect30,35, our current experiment resorts to the time-multiplexed configuration, with the spatial degrees of freedom encoded in the discrete arrival time of photons at the detector40. This enables us to implement quantum walks with a larger number of time steps, which is pivotal for the current experiment. We introduce static random disorder through parameters of the optical elements41, which would result in a complete localization of bulk states in the large-disorder limit. In the intermediate regime with moderate loss and disorder, the competition between the non-Hermitian skin effect and Anderson localization yields non-monotonic localization features which we characterize by measuring the Lyapunov exponent20. Using the biorthogonal chiral displacement, we then probe the topological phase transition, which is in qualitatively agreement with theoretical predictions. At the measured topological phase boundary, the biorthogonal localization length diverges, consistent with the biorthogonal critical nature of the phase transition36,37,38. We further measure topological edge states from dynamics close to the boundary of the non-Hermitian topological Anderson insulator."
By: Quan Lin, Tianyu Li, Lei Xiao, Kunkun Wang, Wei Yi & Peng Xue
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Can you give a three sentence max explanation of what hermiticity is—to the ordinary rubber necking gawker.
Then maybe a sentence or two more on why hermiticity is important, how fast the field progresses and what are its implications for the future?