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REVIEW 3 major objections 3 minor 65 references

Generalized bulk-edge correspondence for non-hermitian topological systems

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Pith's one-line read This paper restores bulk-edge correspondence in non-Hermitian chains by replacing the ordinary periodic boundary condition with a modified one parameterized by a skin-effect scale $b$, and proves the correspondence in the enlarged…

desk verdict A fresh boundary-condition trick that makes non-Hermitian bulk-edge correspondence intuitive for a chiral SSH model, but the general proof is explicitly restricted to the perturbative regime and the role of the free parameter b remains under-examined. read the letter →

arxiv 1908.09438 v2 pith:VZLZZEFL submitted 2019-08-26 cond-mat.mes-hall

Ken-Ichiro Imura , Yositake Takane This is my paper · ORCID
classification cond-mat.mes-hall
keywords non-Hermitiantopologybulk-edgecorrespondenceskineffectmodifiedperiodicboundaryconditionchiralwindingnumberSSHmodelgeneralizedBrillouinzone
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Non-Hermitian systems with asymmetric hopping pile their bulk wave functions against one boundary, so ordinary periodic boundary conditions do not reproduce the open-boundary physics and the standard bulk-edge correspondence appears to fail. This paper restores it by a modified periodic boundary condition (mpbc) in which wave functions obey $\psi(j+L)=b^L\psi(j)$, introducing a real parameter $b$ that encodes the non-Hermitian skin effect. On this boundary condition the system has a chiral Bloch-like Hamiltonian in $\beta = b e^{ik}$, and the paper proves that the pair of winding numbers $w_\pm$ computed in the enlarged parameter space $\tilde{\tau}=\{\tau,b\}$ controls the existence of protected zero-energy edge states under open boundary. A reader should care because the result turns the apparent failure of bulk-edge correspondence in non-Hermitian systems into a bookkeeping problem with a single extra parameter, and because the same construction exposes genuinely non-Hermitian topological phases with half-integral winding.

What carries the argument

The load-bearing object is the modified periodic boundary condition (mpbc), defined by boundary hopping terms $\Delta H = b^{-L} t |1\rangle\langle L| + b^L t |L\rangle\langle 1|$ for the single-band chain and by the analogous terms for the SSH-type chain; it imposes $\psi(j+L)=b^L\psi(j)$. A similarity transformation $S=\mathrm{diag}[1,1,b,b,b^2,b^2,\dots]$ turns the mpbc Hamiltonian into an ordinary periodic Hamiltonian with rescaled hoppings, leaving the spectrum unchanged. This machinery replaces the ordinary Brillouin zone by the circle $\beta=b e^{ik}$, on which the chiral winding numbers $w_\pm = (1/2\pi)[\arg R_\pm(\beta)]_{k=0}^{2\pi}$ are evaluated. Its essential work is to make the bulk reference geometry compatible with exponential localization of skin modes, so that the deformation argument of the Hermitian proof can be carried over to the enlarged parameter space $\tilde{\tau}=\{\tau,b\}$.

What would settle it

Numerically diagonalize the open-boundary SSH chain at a point where the mpbc phase diagram gives $(w_+,w_-)=(1,-1)$ but which is not smoothly connected to the Hermitian line; if no zero-energy edge state appears, the deformation argument is insufficient. Alternatively, compute the actual generalized Brillouin zone by the method of Ref. 12 and compare its winding number with the mpbc winding number at $b=|\beta(E_{\mathrm{bot}})|$; a disagreement marks the boundary of the proof's validity.

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Extended reading notes

Core claim

The central claim is that the modified periodic boundary condition (mpbc), in which the wave function is multiplied by $b^L$ across the boundary, is the correct reference bulk geometry for non-Hermitian systems with skin effect. Under mpbc the eigenstates take the generalized Bloch form with $\beta=b e^{ik}$, and the bulk Hamiltonian $H_{\mathrm{mpbc}}(\beta)$ is explicitly chiral. The winding numbers $w_\pm$ of the off-diagonal entries $R_\pm(\beta)$ around the origin are well defined, and the paper proves a bulk-edge correspondence in $\tilde{\tau}=\{\tau,b\}$: the region with $(w_+,w_-)=(1,-1)$ signals a zero-energy edge state under open boundary, and this region is smoothly connected to the Hermitian topological-insulator region. The proof adapts the chiral-symmetry bulk-edge proof of Ref. 61, with the additional parameter $b$ absorbing the skin effect. The paper also exhibits an optimal path $b=|\beta(E_{\mathrm{bot}})|$ along which the mpbc spectrum reproduces the bottom of the open-boundary bulk spectrum, and shows that switching the mpbc off continuously evolves a detached spectral branch into the zero-energy edge state.

Load-bearing premise

The proof assumes that a single number $b$ captures the whole non-Hermitian skin effect, i.e. that the generalized Brillouin zone is the circle $|\beta|=b$; in general it is not.

Editorial extensions

If this is right

  • On any path in the $(t_1,b)$ phase diagram that is smoothly connected to the Hermitian line $b=1$, the $w=1$ region guarantees a zero-energy edge state under open boundary, whereas the ordinary periodic-boundary spectrum misplaces the gap closings.
  • The central $w=1$ region is smoothly connected to the Hermitian topological-insulator phase, while the $w=1/2$ regions with $(w_+,w_-)=(1,0)$ or $(0,-1)$ are new topological phases with no Hermitian analogue.
  • Along the optimal path $b=|\beta(E_{\mathrm{bot}})|$, the mpbc spectrum reproduces the bottom of the open-boundary bulk spectrum, and gradually switching off the mpbc turns a detached spectral branch into the zero-energy edge state.
  • The generalized bulk-edge correspondence is at present proven only in the perturbative non-Hermitian regime, namely for phases smoothly connected to a Hermitian topological phase; the authors state that a more general proof is left to future work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Read $b$ as the radial scale of the generalized Brillouin zone; where that zone is genuinely non-circular, the natural extension is to let $b$ depend on $k$, making the present construction the constant-$b$ slice of a more general contour formulation.
  • The same strategy should transfer to other symmetry classes and higher dimensions, replacing the scalar $b$ by a boundary-condition object matched to the anisotropic skin effect; a testable consequence is that topological edge-state existence is governed by winding numbers computed on such skin-effect-matched bulk geometries.
  • In engineered directional-gain or lossy lattices, the claim predicts that closing the ring with an amplitude-amplifying link $b^L$ should make the measured bulk spectrum follow the open-boundary gap closings, and the $w=1/2$ windows should appear as parameter ranges with mid-gap states and anomalous winding.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper proposes a modified periodic boundary condition (mpbc) parameterized by b for non-Hermitian one-dimensional tight-binding models with skin effect. For a non-Hermitian SSH model with third-nearest-neighbor hopping and chiral symmetry, the authors define chiral winding numbers w± on the circular contour β = b e^{ik} (Eq. (19)), compute an analytic phase diagram in the enlarged parameter space (t1, b), and claim a generalized bulk-edge correspondence: points with (w+, w−) = (1, −1) in the region smoothly connected to the Hermitian topological phase host zero-energy edge states under open boundary conditions. The paper also introduces an optimal path b = |β(Ebot)|, chosen so that the mpbc spectrum reproduces the bottom of the OBC band, and numerically follows the evolution from mpbc to OBC.

Significance. If fully established, the mpbc construction would be a useful and practical way to restore the bulk-edge correspondence for non-Hermitian systems with skin effect, and the explicit analytic phase diagram for a concrete model is a valuable pedagogical and methodological contribution. The winding-number definitions, the similarity transformation in Sec. III, and the phase-boundary equations in Sec. IV are explicit and correct as far as they go, and the numerical spectra along the optimal path support the existence of protected zero modes in that specific setting. However, the central proof is only a deformation sketch, the authors explicitly restrict it to the perturbative regime, and they defer the general proof to Ref. 64, so the abstract's finite-region claim is considerably stronger than what is demonstrated.

major comments (3)
  1. [Sec. IV, Eq. (19) and the paragraph after Eq. (23)] The proof of the generalized bulk-edge correspondence is not completed for arbitrary points in the w = 1 region. The winding numbers are evaluated on the circular contour β = b e^{ik}, but the authors themselves note in the discussion of Refs. 11 and 12 that the OBC generalized Brillouin zone is generically non-circular, and they restrict the proof to the perturbative regime, deferring a more general proof to Ref. 64. The smooth deformation of the circular-contour phase diagram as γ1, γ2 → 0 only shows that these circular-contour winding numbers vary continuously; it does not identify the OBC gap-closing condition with the zero set of R±(β) on that circle for every b. Thus the abstract's claim that a finite region in (t1, b) indicates a topologically protected OBC edge state is not established for all points in the region.
  2. [Sec. V, Eq. (24) and Fig. 6] The optimal path ηbar is defined by b = |β(Ebot)|, which is extracted from the OBC generalized Brillouin zone and hence from OBC spectral data. The subsequent numerical evolution from mpbc to OBC along this path therefore partially builds in the presence of the edge state. Because the OBC spectrum is independent of the auxiliary parameter b, the observation of zero modes on ηbar does not prove that every other red-region point with a different value of b hosts an OBC zero mode. A concrete test would be to compute the OBC spectrum for a point in the w = 1 region with b far from |β(Ebot)|; the stated correspondence requires such a point to display the zero mode.
  3. [Sec. IV, paragraph following Eq. (23)] The assertion that 'on any path η smoothly connected to η0, the correspondence ... is guaranteed' is asserted rather than proved. The smooth connection between phase boundaries in the (τ, b) space does not by itself transfer the usual PBC/OBC bulk-edge correspondence to the non-Hermitian setting, because the applicability of Ref. 61 to the mpbc reference geometry is precisely the point at issue. This is a load-bearing gap: the central result of the paper depends on this transfer, and the manuscript's own sentence restricting the proof to the perturbative regime and citing an in-preparation general proof makes the gap explicit.
minor comments (3)
  1. [Eq. (21)] Equation (21) is ambiguous as printed: the text requires w = (w+ − w−)/2, since w = 1 for (w+, w−) = (1, −1) and w = 1/2 for (1, 0) or (0, −1), but the displayed expression appears to be a half-sum with an overall sign. Please correct or clarify.
  2. [Fig. 4 caption] The caption should state explicitly that the plotted |β| is |β(Ebot)| and that the gray bars outline the w = 1 region in the (t1, log b) plane, so that the reader can connect the figure to Eq. (24) without inferring it from the main text.
  3. [Sec. I, paragraph 2] There is a typo in the phrase 'edge geometrygedge'; it should read 'edge geometry g_edge'. Similar spacing artifacts appear elsewhere in the typeset text and should be corrected.

Circularity Check

1 steps flagged · score 3.0 of 10

One fitted-input step in the optimal-path demonstration; the central phase diagram remains independently computed.

  1. fitted input called prediction [Section V, Eq. (24), Figs. 4–6]
    "To reproduce the spectrum at the band bottom Ebot under obc, we need to tune the parameter b to be consistent with the value of β corresponding to Ebot at each t1. Thus, b should be determined as b =|β(Ebot)|. (24) ... Since the gap closing is a property of the band bottom, this ensures that the positions of the gap closing are also reproduced. ... This is the simplest and most direct way to establish the bulk-edge correspondence."

    The mpbc parameter b is an auxiliary degree of freedom not fixed by the Hamiltonian. In the optimal-path construction it is fitted to the OBC spectrum via β(Ebot), and the matching of gap-closing positions between the mpbc bulk and OBC is then guaranteed by construction ('this ensures that the positions ... are reproduced'). The interpolation Hδ (Eq. (25)) returns to the same OBC Hamiltonian, so the zero-energy edge state in Fig. 6 is already an OBC input rather than an independent prediction. This makes the optimal-path demonstration circular in itself. It is not load-bearing for the analytically computed phase diagram of Sec. IV, so the circularity is partial and non-central.

full rationale

The central phase diagram (Fig. 2(a)) and winding numbers w± are computed analytically from H_mpbc(β) alone, with no OBC data entering the calculation; they are not fitted. The generalized BEC is argued in Sec. IV by deforming across γ1,γ2 to the Hermitian limit and invoking Ryu–Hatsugai's proof (Ref. 61), an external result, not a self-citation. The paper explicitly restricts this proof to the perturbative regime and defers a general proof to footnote 64 ('in preparation'); that is a rigor/scope limitation, not circularity. The one genuinely constructed step is the optimal path η̄ in Sec. V, where b is chosen from the OBC spectrum (Eq. (24)); the resulting agreement of gap closings and the edge-state evolution are therefore partly built in. Since this path is presented as an illustrative 'intuitive form' rather than the basis of the central classification, the derivation's main content remains independent. Overall: one non-central fitted-input step, partial circularity, score 3.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central construction rests on three categories: the free parameter b that defines the modified periodic boundary condition; domain assumptions about chiral symmetry and circular generalized Brillouin zone; and ad hoc assumptions that the mpbc is a valid reference geometry and that deformation preserves edge states. No new physical entities are introduced.

free parameters (1)
  • b = b = |β(E_bot)| for the optimal path
    The modified periodic boundary condition parameter in Eqs. (3) and (8). Treated as an independent parameter enlarging the parameter space to (τ,b). For the illustrative optimal path, b is chosen from the open-boundary band bottom via Eq. (24).
assumptions (4)
  • domain assumption Chiral (sublattice) symmetry of the non-Hermitian SSH model
    The bulk Hamiltonian H(β) is off-diagonal (Eq. 17), which justifies the winding-number definition w± and the application of the Hermitian bulk-edge proof.
  • ad hoc to paper The modified periodic boundary condition provides a valid reference bulk geometry for non-Hermitian systems with skin effect
    The paper's central proposal is that mpbc realizes the bulk geometry gbulk for non-Hermitian systems; this is assumed and then argued by smooth deformation to the Hermitian limit.
  • domain assumption The generalized Brillouin zone under open boundary is a circle of radius b
    The mpbc uses a single decay rate b for all boundary hoppings; the authors note this is valid only in the perturbative regime, where the generalized Brillouin zone is circular.
  • ad hoc to paper Smooth deformation in the enlarged parameter space (τ,b) preserves zero-energy edge states under open boundary
    The generalized bulk-edge correspondence is proven by deforming any path η to the Hermitian limit η0 and assuming the correspondence holds throughout; no non-Hermitian-specific proof is given.

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Pith. "Pith review of Generalized bulk-edge correspondence for non-hermitian topological systems." pith.science (2026). https://pith.science/paper/VZLZZEFL

@misc{pith2026190809438,
  author       = {Pith},
  title        = {Pith review of: Generalized bulk-edge correspondence for non-hermitian topological systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VZLZZEFL}},
  note         = {Machine review of arXiv:1908.09438}
}
read the original abstract

A modified periodic boundary condition adequate for non-hermitian topological systems is proposed. Under this boundary condition a topological number characterizing the system is defined in the same way as in the corresponding hermitian system and hence, at the cost of introducing an additional parameter that characterizes the non-hermitian skin effect, the idea of bulk-edge correspondence in the hermitian limit can be applied almost as it is. We develop this framework through the analysis of a non-hermitian SSH model with chiral symmetry, and prove the bulk-edge correspondence in a generalized parameter space. A finite region in this parameter space with a nontrivial pair of chiral winding numbers is identified as topologically nontrivial, indicating the existence of a topologically protected edge state under open boundary.

Figures

Figures reproduced from arXiv: 1908.09438 by the authors.

Figure 1
Figure 1. FIG. 1. A typical energy spectrum of the non-hermitian SSH [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The phase diagrams in the space of parameters ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The spectrum and winding numbers under pbc. In [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Spectrum under mpbc [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Evolution of the spectrum from mpbc to obc. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.