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Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification
by Matteo Brunelli, Clara C. Wanjura, Andreas Nunnenkamp
This Submission thread is now published as
|Authors (as registered SciPost users):||Andreas Nunnenkamp|
|Preprint Link:||https://arxiv.org/abs/2207.12427v4 (pdf)|
|Date submitted:||2023-09-06 09:23|
|Submitted by:||Nunnenkamp, Andreas|
|Submitted to:||SciPost Physics|
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.
Published as SciPost Phys. 15, 173 (2023)
Author comments upon resubmission
In the revised manuscript we address the remaining questions and suggestions in Anonymous Report 2.
List of changes
(1) We added the following sentence in the caption of Fig. 6(b): “Note that the diagonal entries are of order 1, although not discernible from the plot.”
(2) We added the following sentence at the end of Sec. VII, before subsection A (page 9, right column): “To obtain the OBC spectrum in (c)-(e), we write the NH Hamiltonian Eqs. (10) and (11) as the PBC Hamiltonian minus the matrix boundary terms, and express it in the plane-wave basis |𝑘 > where the PBC Hamiltonian is diagonal. We then diagonalize the Hamiltonian < 𝑘|𝐻𝑂𝐵𝐶|𝑘' > and label the eigenstates with 𝑘. The same approach is used for computing the singular value spectrum in Fig. 1 (third column from the left).”
(3) We amended this sentence: “We start from the description of the underlying open quantum system (in order to model explicitly both engineered and non-engineered dissipative processes) and study the dynamics of the classical amplitudes.”
(4) A few lines before the beginning of Sec.III, we amended the sentence to “near-unit gain”.
(5) We added a comment and cited the paper Wang et al. Science 371, 1240 (2021) at the top of page 13, right column: “NH topological amplification entails that the ZSMs are directly measurable in a simple transmission experiment and the topological winding number Eq. (19) can be extracted by counting the number and direction of amplified edge modes, without having to measure the momentum-resolved complex energy band .”
Submission & Refereeing History
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