Model non-Hermitian topological operators without skin effect: A general principle of construction

The authors propose a method to construct the non-Hermitian systems in the manuscrip. The constructed non-Hermitian operator has an energy spectrum that can be only purely real or purely imaginary under periodic boundary conditions (PBC). Given such an energy spectrum, the imaginary magnetic flux is zero [Phys. Rev. B 99, 081103(R)], therefore, the conventional bulk-boundary correspondence is valid and the skin effect is necessarily absent.

This method is ingeniously designed so that the imaginary parts inside the square roots of the eigenvalues either cancel out or do not exist. As a result, the value inside the square root can only be real, and thus the eigenvalues can only be purely real or purely imaginary after taking the square roots.

To conclude, this is an interest work and I would like to recommend its publication.

I would like to list a few minor points and invite the authors to fix them in the final version.
1. For \alpha = 0, the Hermitian Hamiltonian in Figure 1 is H(k) = (sink)\sigma_{x} + (2cosk-1)\sigma_{y}. As far as I know, the form of the Su-Schrieffer-Heeger model is H(k) = (w + v*cosk)\sigma_{x} + (v*sink)\sigma_{y}. It is clear that the Hamiltonian in Figure 1 is not the Su-Schrieffer-Heeger model.

2. The energy spectrum in Figure 2(a) is incomplete. According to equation (6), the spectrum should be symmetric around zero.

3. This method has significant limitations, as it constructs a special class of non-Hermitian models whose energy spectrum is restricted to being purely real or purely imaginary. In general, however, the energy spectrum of non-Hermitian systems is typically complex.

Ask for minor revision

1. The authors have discussed the absence of the non-Hermitian skin effect (NHSE) using a general NH Hamiltonian in arbitrary d-dimensions.
2. Have addressed the issue in dimensions greater than 1.
3. From a single Hamiltonian (Eq[5]) they have constructed different models by changing the parameters.
4. Sufficient illustrations are provided to showcase their results.
5. Have proposed sufficient experimental setups where their observations can be realized

1. The general principle of construction of NH operators claimed is not rigorously provided.
2. NHSE is a novel phenomenon, therefore NH operators analyzed before the discovery of NHSE did not show this effect. Hence motivation is not clear.
3. Future direction, stated in the discussion and outlook section is not explained properly. The calculation of topological invariants is not a complete research topic and the relevance of this construction for NH superconductors should be elaborated by a couple of sentences.

In their manuscript, the authors address the phenomenon of the non-Hermitian skin effect and illustrate its non-existence in various arbitrary dimensional lattice models. For this purpose, they review a general Bloch Hamiltonian composed of Hamiltonians from different lattice models. They plot the energy eigenvalues and eigenstate to illustrate the absence of NHSE in different models.
They have got good results regarding the NH-SSH model and are consistent with recent literature. They have also addressed this issue in Higher-order topological models by performing the same analysis. They also have calculated the topological invariant for the 2-D Chern insulator.
They have got interesting results for both the eigenenergies and eigenstates of the system with periodic boundary conditions (PBC) as well as open boundary conditions (OBC). The presentation of the results is decent.
Before making any recommendations, I would like to present some major and minor suggestions to the authors, which I kindly request them to address.

Major:

• In the abstract they claim to propose a general principle of construction of NH operators devoid of any skin effect. But in Sec.[3], I find they have defined an NH Hamiltonian using the universal Bloch Hamiltonian, but have not provided any general principle or formalism for construction. I think Sec.[3], should be more comprehensive and elaborate, the authors can clarify why such NH Hamiltonian is devoid of NHSE in terms of symmetry classification in Sec.[3] itself. If different symmetries are responsible for such effects in different dimensions, they can point them out separately (similar to Table[1]) which will convey their findings in a much clearer way.
• In the introduction they wrote ‘typically NH operators display skin effect’ but this sentence is not clear as NHSE is shown only by particular operators. The word ‘typically’ should be clarified.
• In the introduction they also wrote ‘Therefore, construction of NH
topological operators, featuring the BBC in terms of their left or right eigenvectors and thus
generically devoid of the NH skin effect, is of pressing and urgent theoretical and more crucially,
experimental importance.’ People were aware of NH operators without NHSE in photonics and electronics long before even NHSE was discovered. Therefore, the above sentence seems to be misleading. I want to provide some references in support of my point.

1. Non-Hermitian Topological Theory of Finite-Lifetime Quasiparticles: Prediction of Bulk Fermi Arc Due to Exceptional Point. Vladyslav Kozii, Liang Fu.
2. Parity-Time Symmetry meets Photonics: A New Twist in non-Hermitian Optics. Stefano Longhi
3. Parity–time symmetry and exceptional points in photonics. Ş. K. Özdemir et. al.


• Generally, the phenomenon of NHSE is associated with non-reciprocal nature of the underlying Hamiltonian. But recently it has been shown in a 1D NH model that NHSE is absent though the Hamiltonian is non-reciprocal.

Ref. Circuit realization of a two-orbital non-Hermitian tight-binding chain. Dipendu Halder. et. al.

In the above reference, they claim that pseudo-Hermiticity is responsible for suppressing NHSE in their model.
Taking this as a motivation, I would like to suggest the authors emphasize the symmetries (reciprocity, pseudo-Hermiticity, PT) responsible for suppressing NHSE in their models, especially the 3D ones (which is addressed but should be more clearly presented), with the following questions in mind.
1. Is there a general rule on symmetry consideration for NHSE to occur? What symmetries are needed to predict whether I will observe NHSE or not? (For a particular model)
2. In 2D or 3D NH-material is it possible that some states will be extended and some localized due to NHSE? If yes then what constrain should be there in the Hamiltonian?
3. Is it sufficient to look only at the PBC and OBC spectra to comment on NHSE?

I would also propose, if possible, to use non-Bloch analysis in 3D models to mathematically show the non-existence of NHSE.
With this, I hope that the authors can illuminate me as well as strengthen my understanding of NHSE on a much deeper level.

Minor:

• Fig. 1c. All states are exponentially localized at the boundary, why is there no skin effect? I think NHSE that they are trying to convey should be clearly defined, early in the main text.
Ref. Edge States and Topological Invariants of Non-Hermitian Systems. Shunyu Yao1 and Zhong Wang.
• What is the difference between the two inset plots of Fig1c and Fig1d?
• Fig. 2. The black greyscale color code looks odd, I suggest using some light color instead.
• Fig. 3. The black greyscale color code looks odd, I suggest using some light color instead.
• Fig. 4. In the caption it is written ’Panels (c), (f) and (i) are same as (a), (d) and (g), respectively, but for α = 10….’. They are not the same kind of plots, panels (a), (d) and (g) are index vs eigenenergy, whereas panels (c), (f), and (i) are real vs imaginary eigenspectrum plots.
• What is the relevance of Fig 5? Along the skin effect line of thought?
• 3.1 NH topological insulator: One dimension. It is written ‘This model never shows NH skin effect, as anticipated’. Please be clear about this anticipation.
• Discussion and outlook ‘In order to numerically ensure the bi-orthonormality condition … respectively, we sometimes have to add an extremely small amount of random charge disorder (∼ 10−4−10−6).’ They could have shown this explicitly in the appendix as this is important. This phenomenon happens at an Exceptional point (EP), but the manuscript does not mention EPs.
• In Appendix B., and Appendix C. how \Gamma matrices are redefined to \tau matrices should be specified. Moreover, as a whole, I think the form of the \Gamma matrices should be explicitly given.


I cannot recommend the publication of this manuscript in Scipost in the current version, as the motivation is unclear to me. I would strongly suggest the authors rewrite the motivation part and state the novelty of their work clearly. In addition, the definition of the NHSE that is used should be clearly mentioned in the introduction. However, I will rethink my decision if they answer my queries and implement the suggestions provided.

Ask for major revision