See report

See report

The authors have addressed my previous comments in detail, making a substantial change to the text of their revised version and including further discussions. I would to thank the authors for the substantial work in addressing the points I mentioned. I believe that the current version of the manuscript has a much better clarity than their initial submission, and that nicely highlights the findings in their model and the relation with the toric code. I also believe that their manuscript is much easier to follow for the community of non-Hermitian physics in its current form, and that it will motivate further research in this direction. Given all the points above, I recommend publication in its current form.

No changes

Publish (meets expectations and criteria for this Journal)

see report

see report

I apologize for the delayed report.

The presentation in the revised manuscript has been improved, in particular, by adding the mathematical expressions corresponding to the figures discussed. However, several further points require corrections (see below): (i) Several definitions (Fermi arcs, Fermi cuts, spectral “Riemann surface”) are ambiguous or mathematically imprecise; (ii) Some figures contain conceptual and labeling inconsistencies that can mislead readers; (iii) The topological invariant can be given in an explicit, gauge-invariant algebraic form directly in terms of $H(k)$; (iv) The analogy with the toric code requires stronger disclaimers to avoid confusion with topological order. Below, I discuss them point-by-point.

1. Confusion regarding Fig. 1 and the definition of the Fermi arc. Before discussing the details of the new version, I would like to clarify the source of my confusion concerning Fig. 1, as expressed in the first report. The authors defined the Fermi arcs in the sentence “(imaginary-) Fermi arcs are lines on which the real (imaginary) part of the eigenenergies is degenerate.” This “compactified” form of the definition [(A)B=C(D)] can be interpreted ambiguously. The ambiguity arises because the sentence admits two grammatically valid but physically inequivalent parsings. Specifically, the correspondence between the first and second parts of the sentence can be established either by the order in which terms appear or by bracketing. In the first case, the sentence is understood as follows: “imaginary-Fermi arcs are lines on which the real part of the eigenenergies is degenerate” + “Fermi arcs are lines on which the imaginary part of the eigenenergies is degenerate”, while the second (bracketing) option yields: “Fermi arcs are lines on which the real part of the eigenenergies is degenerate” + “imaginary-Fermi arcs are lines on which the imaginary part of the eigenenergies is degenerate”. When looking at Fig. 1 for the first time, I applied the natural ordering convention (i.e., the shown Fermi arc is where Im$\epsilon$ is degenerate, and hence zero for the typical symmetric case) and was confused by the fact that the real part and the absolute value did not agree along this line. I then realized that the proper convention is the “bracketing correspondence”, but I have left this comment in my report to emphasize that the definitions in the manuscript are not presented in a clear way, which might confuse innocent readers. I don’t understand why not to give unambiguous definitions for Fermi arcs and imaginary Fermi arcs separately (A=D and B=C), without the “braiding-bracketing” trick (which only saved half a line).

2. Negative absolute values of energies. My second reason to comment on Fig. 1b was to attract the authors’ attention to a possible inconsistency in panels b and d of Fig. 1, which I noticed during the first reading. According to the axis labels (and the caption in the new version), these panels show the absolute value of the energy eigenvalues. The appearance of the graph in these panels, however, suggested that the exceptional points corresponded to zero energy. Then, the lower parts of the graphs in panels b and d would correspond to negative values of $|\epsilon|$, which is nonsensical. An overall shift of the real part of the energies by a constant would place the exceptional points at a positive energy, but this would definitely destroy the up-down symmetry of the graphs for $|\epsilon|$, whereas such an asymmetry was not seen there. Therefore, in my first report, I asked the authors to show the explicit formulas for the spectrum used to plot the graphs, and I thank the authors for providing this information in the revised version (I guess the authors agree with me that equations help the reader understand the essence of the work).

According to the added expression [Eq. (2)], the exceptional points in Figs. 1a and 1b are indeed at $\epsilon=0$. This means that the authors plotted negative absolute values of energy in Fig. 1b. Following the authors’ style in their response, it is time to put multiple exclamation marks here. Some referees would just stop reviewing a manuscript after reading about $|\epsilon|<0$. It is, of course, clear that the authors meant a “signed absolute value”, i.e., they plotted the product $\mu|\epsilon_\mu|$ instead of $|\epsilon|$. This is a nice visualization tool calling for the reader’s experience with spectra of Hermitian Hamiltonians, but this should be clearly stated in the manuscript, the figure caption, and the axis label. This seemingly trivial but misleading point must be corrected in the published version.

1. Definition of Riemann surfaces. Now, let’s discuss the definition of the spectral Riemann surfaces. In the new version, this object is implicitly defined above Eq. (2): according to this sentence, the spectral Riemann surface in the manuscript is the multivalued complex-valued function of two real variables $k_x$ and $k_y$. This should be clearly stated at the beginning of the manuscript. In their response, the authors correctly quote the mathematical definition of Riemann surfaces (“connected 1D complex manifolds”). However, ironically, what they call a spectral Riemann surface in 2D non-Hermitian band theory is NOT a genuine Riemann surface in the mathematically strict sense, following exactly this definition. Rather, it is a finite ramified (multi-sheeted) complex-valued covering of the Brillouin torus, equipped with nontrivial monodromy. This object lacks the proper complex-analytical structure encoded in the words “complex manifold” in the definition of the Riemann surface (holomorphicity, proper complex coordinate, etc.; see, e.g., the Wikipedia article). Crucially, the existence of a multi-sheeted structure with monodromy does not by itself imply a complex-manifold structure. What is required for a Riemann surface is a single complex coordinate $z$ such that the spectrum is holomorphic in $z$.

Instead of further going into mathematical details and referring to, e.g., the Newlander–Nirenberg theorem (not to be confused with the Nielsen–Ninomiya theorem) or the Nijenhuis tensor, I will describe when the true Riemann surface is encountered in similar problems that are well-known to a “random regular” condensed-matter theorist (for whom, perhaps, the main application of Riemann surfaces is in the areas involving analytical continuation to complex frequency), and show relevant examples demonstrating the problem with the term in 2D Bloch theory.

For the first time, the spectral Riemann surface appeared in the context of band theory, perhaps, in the work by Walter Kohn, Analytic Properties of Bloch Waves and Wannier Functions, Physical Review 15, 809 (1959), https://journals.aps.org/pr/abstract/10.1103/PhysRev.115.809. There, in 1D Bloch theory, it was a genuine Riemann surface, obtained by “complexifying” the quasi-momentum according to $z=\exp(i k)$ and analytically continuing to C. The spectrum is then a holomorphic function of a single complex coordinate $z$. A similar procedure was discussed in non-Hermitian 1D band theory in, e.g., Heming Wang, Lingling Fan, and Shanhui Fan, One-dimensional non-Hermitian band structures as Riemann surfaces, Phys. Rev. A 110, 012209 (2024), https://journals.aps.org/pra/abstract/10.1103/PhysRevA.110.012209. In one spatial dimension, the spectral Riemann surface is a true Riemann surface.

The problem arises in two spatial dimensions for the Brillouin torus (note that one can also consider energies depending on two parameters without Bloch periodicity, as was done in some recent works on non-Hermitian models, see, e.g., the works by Sen Zhang et al. or Zhong and El-Ganainy cited below). The point is that, if one performs the same complexification of two real momenta as in 1D, one gets a nice complex manifold, which is, however, not a 1-complex (2-real) manifold but a 2-complex (4-real) one, and thus not a Riemann surface (the function will depend on two complex variables). Naively, having in mind that $\mathbb{R}^2$ is similar to $\mathbb{C}$, one can try to find a map $(k_x,k_y) \to z(k_x,k_y)$, thus defining a single coordinate for the spectrum as a function of z. The “natural” choice $z(k_x,k_y) = k_x+i k_y$, does not, however, work for the Bloch bands: the resulting complex energy function still depends on two complex coordinates, $z$ and $\bar{z}$, and hence is not holomorphic.

Another possible attempt is to choose $z(k_x,k_y)$ as, e.g., the argument of the square root in Eq. (2):

so that $\epsilon(z)=\sqrt{z}$. This looks like a nice textbook realization of a Riemann surface. The caveat here is that the mapping $(k_x,k_y) \to z(k_x,k_y)$ is not globally invertible: it gives the same z for different neighboring combinations of $(k_x,k_y)$. The Jacobian of the transformation from $(k_x,k_y)$ to Re$z$ and Im$z$ vanishes along extended lines in the Brillouin zone, thus violating the requirement for a proper coordinate. This obstruction is a manifestation of a general property: “on a compact Riemann surface X, every holomorphic function with values in $\mathbb{C}$ is constant due to the maximum principle” (a quote from Wikipedia; in our case, X is the Brillouin torus).

Therefore, the nontrivial coverings of the Brillouin torus cannot become Riemann surfaces, unless the 2D problem is effectively quasi-one-dimensional (say, hopping in the lattice model is allowed only along one primitive vector, or a non-Hermitian Dirac Hamiltonian depends only on $k_x+i k_y$, like in Ref. [26]). Moreover, a similar obstruction seems to be responsible for the topological features discussed in the manuscript, related, in particular, to the presence of non-contractible Fermi cuts extending throughout the entire Brillouin zone.

Thus, although the energy band structure in two-dimensional non-Hermitian Bloch theory is a smooth real two-dimensional manifold, this fact alone does not generically suffice to endow it with the structure of a one-dimensional complex manifold. A Riemann surface requires not only real dimension two, but also the existence of an atlas of local complex coordinates whose transition functions are holomorphic.

Therefore, the term “spectral Riemann surface” as used in the manuscript is a misnomer or, at least, a “poetic metaphor,” possibly contradicting the very essence of the work, which risks obscuring rather than highlighting the central results. I would strongly recommend that the authors put “Riemann surface” in quotation marks, explaining that the term should be interpreted as a physically motivated analogy rather than a literal mathematical identification, or replace this term with “multi-sheeted covering of the Brillouin torus” or with just “Riemann sheets” (as used in Ref. [27]; this term is not equivalent to “Riemann surface”).

It should be emphasized, however, that the absence of a global complex-manifold structure does not diminish the physical and topological relevance of the multi-sheet “Riemann surface” picture presented in the manuscript. It faithfully captures robust and experimentally observable phenomena that are topological in nature and can be formulated in terms of covering spaces and braid-group invariants. The monodromy associated with loops gives rise to braid-group representations, which provide topological invariants for classifying non-Hermitian band structures. This braid-theoretic perspective captures topology without actually requiring the complex-analytic structure of a Riemann surface. Nevertheless, while the term “Riemann surface” serves as a useful and intuitive shorthand to convey topological phenomena, it is mathematically imprecise in the present context. This is not just a matter of linguistics.

To summarize, the terminology “Riemann surface” is mathematically imprecise in 2D Bloch problems. I recommend replacing it with “multi-sheeted spectral covering of the Brillouin zone” or, at least, putting the term “Riemann surface” in quotes, explaining its meaning as used in the manuscript.

1. Definition of the Fermi cuts. In my first report, I also complained that some other terms “are not properly defined or explained in the text”. Let me clarify this more carefully. For example, the Fermi cut is defined in the manuscript only in connection with exceptional points (below Fig. 1): “Here we choose the common identification of the branch cuts with the Fermi arc that always connects the EP2 pair, and refer to the combined object as a Fermi cut.” This definition assumes the presence of EPs. However, later, the authors discuss the objects they also call Fermi cuts, which exist in the gapful spectrum (no EPs; by the way, the definition of a gapped non-Hermitian spectrum, which is based on |epsilon|, appears only in Sec. 3.1, although the term was used already in Sec. 2, where it could have been understood as referring to the gaps present simultaneously in both Re$\epsilon$ and Im$\epsilon$) and which are the primary objects in the subsequent analysis. However, there is no definition of Fermi cuts without EPs. The authors should provide such a definition without invoking EPs. It appears that all Fermi cuts considered in the manuscript always coincide with Fermi arcs. A possible definition would probably combine both real and imaginary Fermi arcs, as was done below Fig. 4 (“the Fermi cut and the associated imaginary Fermi arc”).

In short: as currently written, the definition of Fermi cuts is circular, since it presupposes the existence of EPs, while later sections rely on Fermi cuts in EP-free spectra. This should be corrected.

1. Clarity of figure captions. Regarding the figures, I appreciate the authors’ effort to improve the accessibility of figure captions. However, even with the revised captions, the figures remain somewhat unclear. I already mentioned the problem with Figs. 1b and 1d; here, I would like to return to Fig. 4. In general, figures should be self-explanatory and accessible without reading their description in the text. The caption of Fig. 4 does not fully describe the figure. In particular, it does not explain what red lines are (this is only said in the text) and what the “windows” in the outer sheet are (are the sheets open or is this only a visualization tool); it does not explain how we see from the figures that the topological configurations are distinct; it does not refer to Fig. 3 to make a connection with “braiding”. This is exactly the reason why I wrote in my previous report that the information delivered by this figure is essentially not much higher than that in the caricature of a mug. Without extensive cross-referencing to the main text and other figures in the caption, the figure does not by itself convey the claimed topological distinction.

2. Additional references The caption of Fig. 4 should be contrasted with a much more informative one of Fig. 1 in Jung-Wan Ryu, Jae-Ho Han, Chang-Hwan Yi, Hee Chul Park, and Moon Jip Park, Pseudo-Hermitian topology in multiband non-Hermitian systems, Phys. Rev. A 111, 042205 (2025), https://journals.aps.org/pra/abstract/10.1103/PhysRevA.111.042205. Note the terminology used there: “pseudo-Hermitian line” (can it be used in the context of the manuscript?). By the way, it is quite surprising that this highly relevant paper (which appeared as a preprint arXiv:2405.17749 on 24 May 2024, i.e., well before the first arXiv version of the current manuscript, submitted on 24 Oct 2024) is not quoted by the authors.

I would also suggest that, in addition to the already mentioned work by Wang, Fan, and Fan, the authors quote the following closely related works: Dali Cheng, Heming Wang, Janet Zhong, Eran Lustig, Charles Roques-Carmes, and Shanhui Fan, Experimental observation of energy-band Riemann surface, https://arxiv.org/abs/2510.08819 (it is about 1D, hence a true Riemann surface in the title); Sen Zhang et al., Dynamically encircling an exceptional point through phase-tracked closed-loop control. Commun. Phys. 8, 344 (2025), https://doi.org/10.1038/s42005-025-02275-y (in connection with experimental platforms and the adiabatic theorem discussed in Sec. 5); Qi Zhong and Rami El-Ganainy, Crossing exceptional points without phase transition. Scientific Reports 9, 134 (2019), https://doi.org/10.1038/s41598-018-36701-9 (non-Bloch spectral Riemann surfaces as functions of two parameters).

1. Plots of the imaginary part of complex energies. Figures from these references strongly indicate that it is extremely helpful for the reader to see the plots of both Re$\epsilon$ and Im$\epsilon$. Indeed, in the present manuscript, the illustrations of topology focus on the plots of Re$\epsilon$ in Figs. 2, 3, 4, and 6. This might create the impression that the imaginary part of the eigenenergies is not important for topological analysis.

In fact, what the authors implicitly assume when drawing, e.g., Figs. 2 and 3, is some choice of labeling the eigenvalues. Only then can one talk about the sign of Re$[\Delta\epsilon(\mathbf{k})]$ in the upper-right panel in Fig. 2, where a non-contractible Fermi cut appears. The labeling choice adopted in this figure is dictated by the “continuity” of the deformation of eigenvalues by beta. With the adopted choice, the imaginary part of $\Delta\epsilon$ will jump (change its sign) at this Fermi cut, which is not seen in the plot of $|\Delta\epsilon|$ (the absolute value should always be positive, as we agreed when discussing Fig. 1b).

Importantly, the labeling choice is a gauge-like freedom. In the presence of a Fermi arc in the gapped spectrum, one can choose an arbitrary labeling of Re$\epsilon$ sheets upon crossing the arc. With the choice opposite to that in Fig. 2, the sign of Re$[\Delta\epsilon(\mathbf{k})]$ will change across the Fermi arc, while the imaginary part will be continuous throughout the entire Brillouin zone. In this case, however, the bands of the real part must jump at the boundary of the Brillouin zone.

In general, there exists no continuous labeling of the bands in the entire Brillouin zone for the gapped spectrum with non-contractible Fermi cuts: either Re$\epsilon$ or Im$\epsilon$ must jump. This is exactly the obstruction detected by the nontrivial topological invariant. This important aspect of the band structure is best illustrated by showing plots for both Re$\epsilon$ and Im$\epsilon$ side by side. In particular, the situation of the unprotected Fermi arc in Sec. 3.2 (which, by the way, corresponds to a genuine Riemann surface, as the model is quasi-one-dimensional) differs from that for touching Re$\epsilon$ bands in Fig. 2 by the fact that Im$\epsilon$ does not jump across the Fermi arc. Thus, plotting Im$\epsilon$ is extremely instructive.

Notably, in Fig. 3, the labeling convention is chosen to be opposite to that in Fig. 2: in Fig. 3, the Re$\epsilon$ bands change their signs at the Fermi arcs, leading to “braiding” and “permutation” (jump) at the boundary of the Brillouin zone. This corresponds to continuous bands of Im$\epsilon$. This different labeling choice adds to the reader’s confusion when comparing Figs. 2 and 3: no “braiding” is explicitly seen in Fig. 2, as the sign of Re$[\Delta\epsilon(\mathbf{k})]$ is always positive. This labeling freedom and the specific choices of labeling must be indicated in the text and figure captions.

1. Algebraic definition of the topological invariants. The manuscript defines the $\mathbb{Z}_2 \times \mathbb{Z}_2$ invariants via permutations of the eigenvalues, which requires diagonalizing $H(k)$ and tracking eigenvalue branches along loops in the Brillouin zone, including a labeling convention for the sheets. The $\mathbb{Z}_2$ invariant proposed in the manuscript is, of course, not affected by the labeling freedom. However, operationally, it might seem to be connected to the specific labeling shown in Fig. 3, where the “permutations” are evident for Re epsilon (the convention of continuous Im$\epsilon$ sheets). Trying to extract the invariant for the convention of Fig. 2, following the permutation prescription, is not a trivial task. A natural question arises: What is the definition of “permutation” of eigenenergies, which does not rely on pictures obtained for a specific labeling convention? Or, more generally, what is the algebraic definition of this invariant? Given that the paper studies symmetry-protected topology of non-Hermitian systems (and not topological order), the answer is inspired by the Fu-Kane $\mathbb{Z}_2$ invariant for time-reversal-symmetric (TRS) topological insulators.

In the gapped, TRS case, there exists an equivalent, fully gauge-invariant formulation of $\mathbb{Z}_2 \times \mathbb{Z}_2$ invariants in terms of the discriminant of the Bloch Hamiltonian. Specifically, for a two-band model, one can define the function (strictly positive in the gapped phase)

This function can be written as $\Delta(k)=(\epsilon_1-\epsilon_2)^2$ in terms of eigenvalues of $H(k)$. Next, we introduce a Fu-Kane-type formula at the four time-reversal invariant momenta $(0,0), (\pi,0), (0,\pi), (\pi,\pi)$, with $\Delta(k)$ replacing the Pfaffian. Writing and one obtains the same $\mathbb{Z}_2 \times \mathbb{Z}_2$ classification as in the permutation-based definition. However, now, the invariants are expressed purely in terms of gauge-invariant combinations of matrix elements of $H(k)$, without any explicit use of eigenvalues or branch choices: no diagonalization, no pictures, no labeling freedom, and even no explicit braiding. Time-reversal symmetry implies $\Delta(k) = \Delta^*(-k)$, so that $Delta(k)$ is real at the time-reversal invariant momenta and its sign is well defined there; this reality condition is essential for the construction and shows that the invariant is symmetry-protected rather than intrinsic.

One can also write equivalent expressions using the other pair of time-reversal invariant momenta,

and and the equivalence of the two definitions modulo 2 follows from the absence of zeros of $\Delta(k) $ in the gapped phase.

It is worth noting that in all models considered in the manuscript, Tr$[H(k)]=0$, so that the discriminant is proportional to the determinant. However, in the general case, the invariant is expressed through the discriminant. Indeed, adding a constant term to the diagonal of $H$ will shift the determinant and can change its sign, whereas the topology of the band structure will not be altered (cf. discussion of Figs. 1a and 1b above).

In my view, mentioning this discriminant-based, Fu-Kane-type formula would strengthen the manuscript by (i) emphasizing the gauge-invariant nature of the invariant, and (ii) providing an operational way to compute it directly from $H(k)$ without diagonalization. It also makes the connection to the first Stiefel-Whitney class of $\sqrt{\Delta}$ (shown in Fig. 2) transparent, highlighting the underlying bundle-theoretic structure. Thus, I consider the inclusion of the algebraic formulation essential for the clarity and completeness of the manuscript.

1. Critical values of beta in Eq. (3). Returning to Fig. 2, it should be noted that the panel labels there (values of $\beta$) do not literally correspond to Eq. (3); the critical values “$\beta=3$” and “$\beta=5$” claimed below Eq. (3) also do not correspond to Eq. (3). In particular, taking $k_x=k_y=0$ in Eq. (3), one gets the matrix
   $$H(0,0)=\left(\begin{array}{cc} 3 & 1/5+2 i\beta \ -1/5+2 i \beta & -3 \end{array} \right),$$
   with the eigenvalues $\pm (2/5) \sqrt{56-25 \beta^2}$, so that $\beta=3$ corresponds to the gap.

I guess, there appears to be a typo in the off-diagonal term in the model defined in Eq. (3). To approximately reproduce the quoted critical values for the gap-closing transitions ($\beta \approx 3$ and $\beta \approx 5$; more precisely, $\beta = 4 \sqrt{14}/5 \approx 2.993$ and $\beta = 4 \sqrt{39}/5 \approx 4.996$), the off-diagonal element must be scaled by a factor of 1/2. The correct term is then $i (\beta/2)(1+\cos k_x)$, not $i \beta (1+\cos k_x)$. I also believe that, if my guess is correct, one should use the approximate-equality sign "$\approx$" when quoting the values 3 and 5 in the text. This discussion again confirms the importance of presenting equations used for plotting figures.

1. Toric-code analogy. Finally, I would like to comment on the toric-code analogy. Specifically, the manuscript draws a structural analogy between the topology of complex energy spectra in certain gapped, TRS non-Hermitian systems and the ground-state manifold of Kitaev's toric code. The analogy is rooted in the homology of the torus, which appears in both a canonical model of quantum topological order and in the spectral structure of certain gapped non-Hermitian systems. Indeed, both are classified by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ invariant corresponding to the parity of monodromy along the two generators of the first homology group of the torus.

In my first report, I appreciated ‘a nice idea of "mapping" the topological features of the toric code to a general structure of bands in non-Hermitian Bloch Hamiltonians.’ With the better understanding of the essence of the paper (thanks to improvements done in the revised version and, in particular, to the added new equations), I still consider the analogy nice and encouraging. This translation of topological tools and intuition between distinct fields is very useful for exploring symmetry-protected topological phases in classical and photonic systems, while stressing the distinct structures present in fully quantum topological phases.

However, now I believe that some more disclaimers and clarifications should appear in the discussion, in addition to “The energy spectrum behaves as a classical object, so that this analogy does not inherit the full quantum-mechanical properties of the toric code.” While the analogy is mathematically substantial, it is much more constrained physically. A crucial distinction is the difference of "universality classes": symmetry-protected topology vs. topological order. Importantly, in the toric code, the four sectors are degenerate ground states of a single Hamiltonian, whereas in the non-Hermitian analogy, they represent distinct gapped phases of different Hamiltonians (or different parameter regimes), separated by gap-closing transitions.

What does the analogy capture? (i) The $\mathbb{Z}_2 \times \mathbb{Z}_2$ classification from two independent non-contractible cycles on a torus; (ii) The concept of distinct topological sectors defined by lines of defects (spin flips/branch cuts); (iii) The pair-wise creation of excitations (anyons/exceptional points) connected by a defect string (flux line / Fermi cut).

What does the analogy not capture? (i) The quantum many-body nature of the toric code (as the authors also emphasized in the manuscript), (ii) Long-ranged topological entanglement – one of the defining properties of topological order; (iii) Ground-state degeneracy of a single Hamiltonian – again, a defining feature of topological order: the non-Hermitian analog yields distinct phases, not a degenerate ground space; (iv) Anyonic statistics and toric-code fusion rules (the authors mention the mismatch in the number of “quasiparticle” types in Sec. 4.3); (v) The full structure of the $\mathbb{Z}_2$ gauge electromagnetic theory of the toric code, and, more generally, topological quantum field theory for models with topological order.

Without a more extended discussion of the latter (“not captured”) items, the reader might decide that topologically ordered phases can be realized following the authors’ proposal, which is highly misleading. This is not a “topological order" in the many-body quantum sense, but rather a symmetry-protected topological phase of a single-particle spectrum. I noticed that the first arXiv version of this manuscript contained “topological order” in the title. Removing this term from the title and the abstract was already a step in the right direction. Still, a clear distinction between the physical concepts addressed in the manuscript and those involved in topological order should be spelled out.

I recommend expanding this discussion in the manuscript, not resorting to a brief disclaimer in the Conclusion section.

1. Conclusion. To conclude, overall, this is a nice paper deserving of publication. The revised version of the manuscript is substantially improved compared to the previous one. However, the points raised above should be adequately addressed in the manuscript before it can be recommended for publication. In fact, despite the length of my report, further improvements can be made in the manuscript without significantly extending its length.

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