Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials

Dear Editor,

We thank the reviewer for their report with suggestions and criticisms that allowed us to improve our paper. However, we strongly disagree with the main conclusion of the report, namely that our paper does not contain enough new results to warrant publication in SciPost Physics. Indeed, this is simply stated without any supporting arguments - not even a single reference is provided. Here we argue the opposite by listing the new results of our work followed by a list of possible lines of future research motivated by them:

• A first demonstration of a proper bulk-boundary correspondence for a physically relevant non-Hermitian model:

  • Even though the bulk-boundary correspondence for non-Hermitian systems has been considered in the literature before, most of the available results lack a proper mathematical foundation and are, in fact, often simply incorrect. These theories are, in particular, often based on translational invariance, however,''topological features must be properties of a translationally not invariant Hamiltonian'' (see e.g.~Ryu, Schnyder {\it et al.} NJP (2010)) and thus should be stable against perturbations which break translational invariance. In the theories mentioned above this is not the case as has been recently demonstrated [21].

  • A rigorous approach based on Toeplitz theory has just very recently been introduced by one of us in Ref. [21]. Here we show that topological invariants are related to the properties of the singular value spectrum and not the eigenspectrum. Topologically protected singular values belong to states which are exact zero-energy eigenstates for the semi-infinite chain, however, they are not exact eigenstates for a finite chain with open boundaries but rather long-lived metastable states. We call these states 'hidden zero modes'. The Hermitian case is special because the singular values are then just the absolute values of the eigenvalues and there are then eigenstates with exponentially small energies for the finite chain which do converge to the zero-energy eigenstates for the semi-infinite chain. I.e., the 'standard' bulk-boundary correspondence is recovered in this case.

  • Since the proper bulk-boundary correspondence has just been formulated in Ref.~[21] we are absolutely certain, in contrast to what the referee implies, that the results presented here are novel.

  • Our goal here has been to demonstrate this bulk-boundary correspondence in detail for a physically relevant model and to do so based on analytical solutions and not just numerical data.

  • In particular, the exact solution for the singular values given by Equations (V.4-V.7) and (V.8-V.11) has not been previously obtained. Together with the exact solution for the zero-energy eigenstates of a semi-infinite chain based on recurrence relations, we explicitly demonstrate the aforementioned proper bulk-boundary correspondence for the dimerized Hatano-Nelson (DHN) model. Also, we observe analytically the decoupling of the singular value spectrum in terms of the variable pairs ${V_L,W_R}$ and ${V_R,W_L}$, each satisfying a transcendental equation. This result is directly related to the existence of two independent topological winding numbers in the sublattice symmetric case and would have been impossible to obtain numerically.

• New results on exceptional points:

  • For the $\cal{PT}$-symmetric case, we find a gapless phase in which exceptional points become dense in the thermodynamic limit. This phase seems to have been overlooked so far in the literature and has important consequences for our understanding of criticality in non-Hermitian systems.

  • To the best of our knowledge, the loci of exceptional points for the DHN model have not been investigated in full generality. Most of the previous papers on the subject only considered the exceptional points that occur for real coupling parameters (usually at $k=\pi$ or $k=2\pi$ in our notation) and also only for periodic boundary conditions. For example, equation (III.17), for every value of $k$, although very simple, does not seem to have appeared in the literature so far.

  • For open boundary conditions, we are not aware of any paper in which the exceptional points were considered for the DHN model. In particular, this part of our work was motivated by the recent paper [53], which was the first to determine the loci of exceptional points in a family of open free (para)fermionic models. In our paper, the transcendental equation (IV.16) and the equation (IV.20) for the exceptional points, valid for odd length $N$ of the chain, are new. For chains with even length, equations (IV.24) and (IV.26) appeared before in [53] for another model (Baxter $Z_N$ chain). However, equations (IV.22) and (IV.23), which are field dependent, are new. Also, we remark that for the case given by Eqs. (IV.24) and (IV.26) we find novel non-trivial curves along which the exceptional energies lie.

• New results on entanglement entropy:

  • In a topological phase, one expects a topologically protected non-zero amount of entanglement which cannot be removed. This is our main motivation to study the entanglement entropy here as well. We indeed find topological protected entanglement consistent with the topological invariants.

  • We provide a detailed analysis of the correlation matrix eigenvalues and of the entanglement spectrum in all the different phases of the model, using a consistent filling rule. We acknowledge that some related results have appeared in Refs. [27-33] but none of these references offers a complete study of the entanglement entropy across the entire phase diagram.

  • For the sublattice symmetric case (zero field) in the point-gapped $(0,-1)$ and $(1,0)$ phases, we find that the entanglement entropy becomes real in the thermodynamic limit and that the real part scales as $S\sim \frac{c}{3}\ln N$ consistent with a $c=1$ conformal field theory. We find this somewhat surprising and this result seem to warrant further studies.

  • To the best of our knowledge, within the $\cal{PT}$-symmetric regime, the entanglement entropy was not previously analyzed in the complex phase as well as in the exceptional phase. In particular, we showed that the entanglement entropy does detect the presence of real exceptional points.

• Possible future directions:

  • It would be important to go back to models studied previously in the literature where incorrect bulk-boundary correspondences were proposed and to correct those. For one specific model this has been done in Ref.~[21] and in the manuscript here we have provided a more detailed account how a full bulk-boundary correspondence can be established based on analytical results. Looking further ahead, it would be of interest to understand what happens once interactions are included.

  • The simple model we considered shows the existence of a rich variety of exceptional points, particularly if the model parameters are allowed to be complex. This for sure should motivate similar studies for other families of non-Hermitian free-fermionic models. More importantly, it would also be very interesting to start a systematic investigation of exceptional points in non-Hermitian interacting models.

  • The scaling of the entanglement entropy at the various critical lines observed in this work certainly deserves further investigation. The exceptional phase in the $\cal{PT}$-symmetric is particularly intriguing, and its further study might lead to advances in the understanding of criticality in non-Hermitian systems.

In the following, we reply to the requested changes by the referee.

1- While the manuscript's title is Bulk-Boundary Correspondence and Exceptional Points, the paper also extensively discusses entanglement entropy. Although this discussion is well-written, it scatters the central goal of the paper suggested in the title/abstract. This suggests that varying the title to better represent the content of the manuscript might be an option.

Following the referee's suggestion, we changed the title of the paper to: ``Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials". It better reflects the content of the paper and the ordering of the sections.

2- For a non-Hermitian matrix H, the right and left eigenvectors satisfy $H|R\rangle=\epsilon|R\rangle$ and $H^\dagger|L\rangle=\epsilon^*|L\rangle$ with $\dagger$ imposing the conjugate transpose. However, in the manuscript for the (complex) hopping matrix, merely the transpose operation is considered. A clarification on this should be provided.

Equation $(\mathcal{T}_N)^T |\vec{\ell}\rangle= \epsilon~|\vec{\ell}\rangle$ is correct: by taking the complex conjugation of $(\mathcal{T}_N)^T |\vec{\ell}\rangle= \epsilon~|\vec{\ell}\rangle$ one obtains $(\mathcal{T}_N)^\dagger (|\vec{\ell}\rangle)^\ast= \epsilon^\ast~(|\vec{\ell}\rangle)^*\Rightarrow (\mathcal{T}_N)^\dagger |L\rangle= \epsilon^\ast|L\rangle$ with $|L\rangle=(|\vec{\ell}\rangle)^\ast$ which is the form for the left eigenvectors mentioned by the referee. To avoid confusion, we now use the Dirac notation and we write the left eigenvalue equation in Eq.~(II.6) directly for the bra-vector.

3- Summarizing various arising phases, their associated symmetries, and parameter regimes in a table or a subsection may improve the accessibility of different results in this work.

Fig. 2 is a direct representation of all the arising phases. To make this more clear, we have expanded and reformulated the caption.

4- It is stated that `we can observe the merging of the zero field phases $(v1,v2)=(0,1)$ and $(1,-1)$ into a single phase as well as the bending of the phases boundaries." This observation merely occurs for phases close to $|VL|\leq 1$ and $|WL|\leq 1$. Is this related to the positive value of $u=0.5$? Could the authors provide a plot for $u=-0.5$? Is this because of the emergence of a real-line gap as u is increased?

Since the complex gap (III.14) is a function of $u^2$, the plots for $u=\pm 0.5$ are the same. To clarify this point, we modified the third-to-last paragraph of Subsection III B.

5- It would improve the readability of the manuscript if the authors considered transferring the paragraph on the PT-symmetric phases on page 13 (the first paragraph) to section III.

Following the referee's suggestion, we moved the first paragraph of Section VI B to the the penultimate paragraph of Section III B, modifying the text accordingly to fit into this part of the manuscript.

6- Here are some minor typos and comments: * Appendix B is referred to before Appendix A. * On page 7, below Eq.(IV.21) "by by" should be replaced with "by" * On page 12, the second column, the fourth line from the bottom, "then" should be replaced with "than".

The appendix on the discriminant of the characteristic polynomial now comes before the appendix on the details of the open boundary case. The other typos were corrected.

In addition to the points above, we have made the following changes:

• We have modified the abstract to stress the novel results with regard to the bulk-boundary correspondence and the exceptional points which are obtained in this manuscript.

• We discuss the proper bulk-boundary correspondence now in more detail already in the introduction.

• We expanded the description in the introduction about why we study the entanglement entropy and what the main results are.

• Sec. II: Point out that the Hamilton operator for the semi-infinite chain has block Toeplitz form. New Eq. (II.5).

• Eq. (II.6) now written in bra-ket notation.

• Sec. III: Merging of phases for finite field when the number of winding numbers is reduced from 2 to 1 is now explained in much more detail.

• PT case now discussed already in Sec. III.

• Fig. 2: Caption extended summarizing possible phases.

• Sec. V: Introduction into bulk-boundary correspondence completely rewritten to better explain issues with previous attempts and to explain the mathematical foundation a proper correspondence has to rest on.

• Conclusions: Future directions added.

• We added references [12,30,55,57,60,61,62].

• Some minor typos were corrected.

• We removed Fig. 3 in the old version as well as an unnecessary paragraph in Sec. (VI.B) about the $\cal{PT}$-symmetric case with $V=0$ and larger lattices (since the system decouples, it adds nothing to the previously discussed 2-cell case).

A trackable version of the paper is provided with major changes highlighted in blue.

We hope that our reply as well as the changes in the amended manuscript fully clarify the contributions of our work to the field of non-Hermitian physics, in particular by establishing a proper bulk-boundary correspondence and by exhaustively studying exceptional points. With these clarifications and changes we ask for our work to be reconsidered for publication in SciPost Physics.

See above.