Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification

• URL: http://arxiv.org/abs/2207.12427v4
• Date: Sun, 3 Sep 2023 17:13:08 GMT
• Title: Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification
• Authors: Matteo Brunelli, Clara C. Wanjura, Andreas Nunnenkamp
• Abstract summary: Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred. We restore the bulk-boundary correspondence for the most paradigmatic class of NH Hamiltonians.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred and the (conventional) bulk-boundary correspondence is lost. Here, we restore the bulk-boundary correspondence for the most paradigmatic class of NH Hamiltonians, namely those with one complex band and without symmetries. We obtain the desired NH Hamiltonian from the (mean-field) unconditional evolution of driven-dissipative cavity arrays, in which NH terms -- in the form of non-reciprocal hopping amplitudes, gain and loss -- are explicitly modeled via coupling to (engineered and non-engineered) reservoirs. This approach removes the arbitrariness in the definition of the topological invariant, as point-gapped spectra differing by a complex-energy shift are not treated as equivalent; the origin of the complex plane provides a common reference (base point) for the evaluation of the topological invariant. This implies that topologically non-trivial Hamiltonians are only a strict subset of those with a point gap and that the NH skin effect does not have a topological origin. We analyze the NH Hamiltonians so obtained via the singular value decomposition, which allows to express the NH bulk-boundary correspondence in the following simple form: an integer value $\nu$ of the topological invariant defined in the bulk corresponds to $\vert \nu\vert$ singular vectors exponentially localized at the system edge under open boundary conditions, in which the sign of $\nu$ determines which edge. Non-trivial topology manifests as directional amplification of a coherent input with gain exponential in system size. Our work solves an outstanding problem in the theory of NH topological phases and opens up new avenues in topological photonics.

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