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

by Matteo Brunelli, Clara C. Wanjura, Andreas Nunnenkamp

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Submission summary

Authors (as registered SciPost users): Andreas Nunnenkamp
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-05-31 10:28
Submitted by: Nunnenkamp, Andreas
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approach: Theoretical


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.

Author comments upon resubmission

Dear Editor,

Thank you for handling our submission and for sending us the referee reports.

Referee 1 writes that our manuscript “provides an interesting and convincing novel perspective on the role of topology in non-Hermitian systems”, that it will have “significant impact on a wide community of researchers”, and that “SciPost will eventually be an ideal venue for its publication”. Referee 2 finds our paper is “well written and includes interesting results” and they can therefore, after their questions have been addressed, “recommend it to be published on SciPost”.

In particular, Referee 1 asks us to clarify the dependence of non-hermitian topology, i.e. the number of zero singular modes and the winding number, on frequency or detuning of the probe which we address in a new Appendix C in the revised manuscript. We are grateful that Referee 1 has brought the literature on absolute vs. convective instabilities to our attention which we included in the revised manuscript.

We thank both Referees for their careful review and we hope that they can now recommend our work for publication in SciPost.

Matteo Brunelli, Clara C. Wanjura, and Andreas Nunnenkamp

List of changes

Please see our reply to the referee reports.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-7-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2207.12427v3, delivered 2023-07-06, doi: 10.21468/SciPost.Report.7460


I am very satisfied of the authors' revisions in response to my first report. They have duly taken into account all my remarks, I am (almost) ready to recommend publication.

Before moving into production, the authors may anyway wish considering the following remarks and suggestions:

1/ If I understand correctly, the bottom-rightmost panel of fig.1 (and fig.6(b)), if plotted on a restricted vertical axis [0,1], should display a diagonal feature similar to the one of the central-right of fig.1, with a marked north-east/south-west asymmetry due to non-reciprocity. For the sake of completeness, the authors may include such images in the manuscript, e.g. as insets, and add some corresponding discussion in the text.

2/ In the 3rd column (from left) of fig.1 as well as in the text at the top of the LHS column of pag.4 and in Fig.3(right), the authors use the concept of k vector and of BZ also for the OBC case. They should explain how they are defining k in the case of OBC where translational symmetry is absent.

3/ At the beginning of Sec.IV, the authors speak of an "open quantum system". To avoid any risk of misunderstanding, I would refrain from using the word "quantum" in this work that only deals with classical systems.

4/ A few lines before the beginning of Sec.III, the authors speak of "unit gain" when dealing with the non-topological \nu=0 case. While I agree that in this case the response is way smaller than the one of the topological case \nu\neq 0, I don't see why it should be exactly "unit". And, in fact, in the plots on the rightmost column of Fig.1 the susceptibility is close but not exactly 1. The authors may wish to amend the text to clarify this point.

5/ In the plots of the rightmost column of fig.1 and of fig.6b, the authors should specify which of the x/y axis correspond to the input/output site.

6/ I am (and most likely many other readers are) curious to know the authors' point of view on the possibility of using of the experimental technique of Wang et al. Science 371, 1240 (2021) to study their winding number. I suspect that this technique to reconstruct the k-space dispersion is intrinsically useless in the authors' case where the non-trivial topology forces the PBC dispersion to be dynamically unstable. The authors may add some comments in the manuscript to clarify this interesting point.

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Anonymous Report 1 on 2023-6-22 (Invited Report)


The authors addressed all the questions I raised and revised the draft accordingly. The current version can be published on SciPost.

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