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Topological nature of edge states for one-dimensional systems without symmetry protection

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Pith's one-line read A winding number computed from bulk data counts edge states in one-dimensional two-band chains with arbitrary complex couplings.

desk verdict A serious generalization of edge-state invariants to asymmetric 1D two-band models, but the numerics verify the invariant against the same analytic condition it is built on rather than against independent OBC spectra, and the proof excludes degenerate-root limits. read the letter →

arxiv 2412.10526 v2 pith:I4KB6UXJ submitted 2024-12-13 cond-mat.mes-hall quant-ph

Janet Zhong , Heming Wang , Alexander N Poddubny , Shanhui Fan This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords bulk-boundarycorrespondenceedgestatesnon-HermitiantopologywindingnumberRiemannsurfacegeneralizedBrillouinzonetwo-bandmodelbulkeigenvectordegeneracy
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

This paper proves a bulk-boundary correspondence for one-dimensional, nearest-neighbor two-band tight-binding chains with arbitrary complex couplings: a winding number built from bulk data predicts exactly how many edge states appear when the chain is cut. The invariant uses two special "bulk eigenvector degeneracy" points—pairs of distinct wavevectors at one energy that share the same eigenvector ratio—and winds the image of the (generalized) Brillouin zone around them on the M-Riemann sphere. No symmetry assumption is needed, so the result covers Hermitian and non-Hermitian models, including edge states at nonzero energy and transitions that do not coincide with gap closing. If correct, the criterion replaces symmetry-protected classification for this model class with a purely geometric count that reduces to the familiar SSH invariant when symmetry is present.

What carries the argument

The carrying object is the bulk eigenvector degeneracy point $M_{\rm deg}$: a value of the eigenvector ratio $M = a/b$ shared by two distinct generalized Bloch wavevectors $z$ at the same energy. There are exactly two such points for the nearest-neighbor model, given by the quadratic in Eq. (8), and an edge state appears when the two preimages of $M_{\rm deg}$ on the band-structure torus both lie inside ($M_1=M_2$) or both outside ($M_3=M_4$) the generalized Brillouin zone. The invariant winds the GBZ image $M(C_{\rm GBZ})$ on the M-Riemann sphere around each $M_{\rm deg}$, subtracting the winding around a branch point $M_{\rm branch}$ to fix the parity; Eq. (9) is the Hermitian version integrated over the ordinary Brillouin zone. The Riemann-surface proof shows the band structure is a torus when $z^2\Delta(z)$ has no repeated roots and uses the elliptic-function parametrization to turn the edge-state condition into an argument-principle count.

What would settle it

Compute $W$ from Eq. (10) for a large random sample of complex couplings of the form in Eq. (1) with non-repeated roots, diagonalize the open-boundary chain with thousands of unit cells, and count the discrete eigenvalues lying away from the GBZ continuum; a single parameter point with $W$ different from that count would falsify the claimed correspondence.

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Extended reading notes

Core claim

The paper's claim is that for the model $H(z)=h_- z^{-1}+h_0+h_+ z$, the total winding number $W = W_1 + W_2$ from Eq. (10) (or Eq. (9) for Hermitian models) equals the number of open-boundary edge states, which can only be 0, 1, or 2. An edge state forms exactly when two of the four generalized Bloch solutions at an energy have equal eigenvector ratio $M(z,E)=a/b$; the two bulk eigenvector degeneracy points $M_{\rm deg}$ are where two distinct $z$ values at the same energy share that ratio. The invariant winds the image of the generalized Brillouin zone, $M(C_{\rm GBZ})$, on the M-Riemann sphere around each $M_{\rm deg}$, subtracting a branch-point winding $M_{\rm branch}$ to fix parity, and the mod-two structure tracks whether the two preimages of $M_{\rm deg}$ lie on the same side of the GBZ. Under sublattice or chiral symmetry the invariant reduces to the standard SSH winding number, and it is unchanged by $z$-independent unitary or similarity transformations. The proof relies on the smooth band structure being a genus-one Riemann surface, parametrized by the Weierstrass elliptic function, on which the argument principle converts the edge-state condition into a winding count.

Load-bearing premise

The proof assumes the characteristic polynomial $z^2\Delta(z)$ has no repeated roots, so the band structure is a smooth genus-one torus that can be parametrized by the Weierstrass elliptic function; exactly degenerate models such as the SSH chain are excluded and only recovered as limits.

Editorial extensions

If this is right

  • For any nearest-neighbor two-band chain with complex couplings, the number of open-boundary edge states is fixed by bulk data and is always 0, 1, or 2.
  • Edge states may sit at nonzero energy, and their appearance or disappearance is marked by $M_{\rm deg}$ crossing the GBZ image on the M-Riemann sphere, not by the bulk gap closing.
  • When sublattice or chiral symmetry is present, the new invariant reduces to the familiar SSH-type winding number, so the generalized criterion contains the symmetry-protected criterion as a special case.
  • The invariant is invariant under $z$-independent unitary and similarity transformations, matching the physical expectation that such basis changes do not alter the existence of edge states.
  • The two $M_{\rm deg}$ points cannot be created or destroyed by continuous parameter changes, so edge states tied to them are stable against symmetry-preserving, nearest-neighbor, periodic perturbations; the distance between $M_{\rm deg}$ and the GBZ image measures how much perturbation is required to create or remove an edge state.

Reading between the lines

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

  • The same bulk-geometry picture suggests a testable route to longer-range two-band chains: the paper shows the edge-state condition becomes a vanishing of determinants $D_{\rm left}$ and $D_{\rm right}$, and if a finite algebraic set of degeneracy points can be defined there, an analogous winding invariant should count edge states; the paper leaves that extension open.
  • The distance between $M_{\rm deg}$ and $M(C_{\rm GBZ})$ could serve as a design parameter: one could engineer a lattice so that an edge state switches on at a chosen coupling value by tuning that distance to zero.
  • Because the edge-state energy is fixed by bulk quantities alone, an experiment on a photonic or electrical lattice of the form in Eq. (1) could predict edge-state frequency without diagonalizing the finite structure, complementing approaches that require a truncated-system Green's function.
  • One might expect an analogous invariant to exist for continuum models with polynomial band equations, but for models with transcendental band structures, such as photonic crystals, the algebraic degeneracy-point construction would need a different formulation.
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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 manuscript introduces a winding-number invariant for one-dimensional nearest-neighbor two-band tight-binding models with arbitrary complex couplings. The invariant is built from the image of the (generalized) Brillouin zone on a Riemann surface of the eigenvector ratio M, using the two bulk eigenvector degeneracy points Mdeg and, for the general non-Hermitian case, the M-plane branch points Mbranch. The authors prove analytically for generic models, via a Weierstrass elliptic parametrization of the band-structure torus, that the parity of the winding counts whether Mdeg corresponds to an edge state satisfying M1=M2 or M3=M4, and they verify the invariant numerically against that same analytic condition in phase diagrams for Hermitian and non-Hermitian random models. The paper claims a bulk-boundary correspondence that does not rely on symmetry protection and reduces to known invariants when symmetries are imposed.

Significance. If the proof is fully correct for the claimed domain, this is a significant contribution: it provides a symmetry-independent bulk invariant for edge states in a large class of one-dimensional two-band models, covering nonzero-energy edge states and non-Hermitian systems, and it explicitly connects the bulk-boundary correspondence to the Riemann-surface structure of the Bloch Hamiltonian. The analytic proof is substantial, including a torus parametrization and an argument-principle counting argument, and the paper gives a falsifiable prediction (W equals the number of OBC edge states) that can be checked by independent numerical diagonalization. However, the current numerical verification is not independent of the analytic edge-state condition and does not exercise the braided GBZ regimes where the proof's geometry is most delicate, so the claimed universality is not yet fully supported.

major comments (3)
  1. [Fig. 3, SM Sec. IX.A-B] The phase diagrams in Fig. 3 compare W from Eq. (9) or Eq. (10) to Nedge obtained from the analytic condition |M1/M2-1|<epsilon or |M3/M4-1|<epsilon, which is the same leading-term criterion (Eq. (11)) used to define edge states in the proof. This is an internal consistency check, not an independent verification against the open-boundary spectrum; direct OBC comparisons appear only for individual examples in Fig. 1 and SM Sec. IV. Please add a systematic comparison of W to an independently counted number of discrete OBC eigenvalues, for example using the signed indicator Iedge defined in SM Sec. IX.C or an IPR-based counting of finite-N spectra.
  2. [SM Sec. V.A] The proof of Eq. (10) explicitly assumes that z^2 Delta(z) has no repeated roots (SM Sec. V A, text after Eq. (S24)), so that the band structure is a smooth genus-1 curve. The abstract and main text nevertheless claim validity for 'any complex couplings' and 'regardless of the underlying GBZ eigenvalue topology.' Degenerate cases such as the SSH model are excluded by this assumption and are only reached as limits; the proof as written does not establish the invariant for them. Please either prove the degenerate cases by a limiting argument or restrict the universality claims to generic models.
  3. [SM Sec. IX.B, Fig. 3(b)] The non-Hermitian phase diagram is computed under the approximation that the GBZ is unlinked, and the braiding transitions are stated to occur over parameter ranges smaller than the grid size. The proof's inside/outside GBZ construction and the loop-sorting procedure are precisely the nontrivial parts in braided or linked phases, so the numerical support for the 'regardless of the underlying GBZ eigenvalue topology' claim does not cover those regimes. Please add explicit tests in a parameter region with nontrivial GBZ braiding, or state the domain of validity of the numerical verification.
minor comments (3)
  1. [End Matter, Analytical OBC] The main text defines OBC as setting the wavefunction to zero at n=0 and n=N+1, but the determinant in Eq. (14) and the coefficient expressions in Eqs. (16)-(19) use powers z^N (i.e., a boundary at n=N); please reconcile this notation or clarify the convention.
  2. [Reference [19]] The footnote for the Supplemental Material contains a placeholder text '[todo] will be inserted by publisher' that should be replaced with an actual URL before publication.
  3. [SM Sec. IX.B] The phrase 'the two links of the unlink are the subGBZ loops' should read 'the two loops of the unlink are the subGBZ loops' for clarity.

Circularity Check

2 steps flagged · score 6.0 of 10

The winding number is constructed as an index tracker for the analytic edge-state condition M1=M2/M3=M4, and Fig. 3 compares W to that same condition rather than to an independent OBC edge-state count.

  1. renaming known result [Main text, 'Sketch of the proof' (and SM Sec. V B, Eqs. S31-S36)]
    "The Mdeg point on the sphere will be covered twice by the GBZ inside if M1 = M2, zero times if M3 = M4, and exactly once for all other cases. By defining a winding integral on the band structure Riemann surface and mapping it to the M -Riemann sphere, the number of times the GBZ inside covers a region can be expressed as an M winding number."

    In the proof, W is introduced precisely as the parity of coverage of Mdeg by the inside of the GBZ, and the inside/outside classification is the edge-state criterion M1=M2 or M3=M4 obtained from Eq. (11). Therefore the statement 'W equals the number of edge states' is equivalent, by the paper's own construction, to the analytic condition it was designed to detect; the invariant is an index tracker for that condition rather than an independent first-principles prediction. The bulk-boundary content lies in expressing the condition via Mdeg, the GBZ and Mbranch, but the numerical target and the invariant are two encodings of the same criterion.

  2. fitted input called prediction [Numerical results, Fig. 3 and SM Sec. IX.C]
    "Eq. (11) predicts OBC edge states at energies where |M1/M2 − 1| = 0 and |M3/M4 − 1| = 0 ... we classify a mode as an edge state if and only if |M1/M2 −1| < ε or |M3/M4 −1| < ε ... The analytical edge state condition in Eq. (11) does not contain the winding information in Eq. (9) and Eq. (10) and Fig. 3 therefore verifies the validity of Eq. (9) and Eq. (10). [SM IX.C:] from Edeg or Mdeg, one could have already found the number of edge states by applying the analytical edge state criteria."

    The Nedge compared with W in Fig. 3 is not an independent count of discrete OBC eigenvalues; it is obtained by applying the same |M1/M2−1|<ε / |M3/M4−1|<ε criterion that the proof uses to give W its edge-state meaning. The agreement is therefore an internal consistency check between two ways of evaluating the same analytic condition, not a verification that W predicts actual OBC edge states. The SM explicitly concedes that Eqs. (9)-(10) provide no computational shortcut, and independent OBC eigenvalue checks appear only in single examples (Fig. 1, SM Sec. IV) rather than in the 6400-model phase diagrams.

full rationale

The derivation of W is mathematically self-contained: W is defined from bulk quantities (Mdeg, M(C_GBZ), Mbranch), and the proof that W equals the analytic edge-state condition M1=M2/M3=M4 follows from the argument principle. This is a legitimate theorem rather than a bare definitional tautology. However, the numerical verification in Fig. 3 is internal: Nedge is computed from exactly the M1=M2/M3=M4 criterion that W was built to detect, so the agreement is consistency between two representations of the same condition, not an independent test against counting discrete OBC eigenvalues. The paper's own SM notes that Eqs. (9)-(10) add no computational shortcut. Independent physical support exists in the OBC spectra of Fig. 1 and SM Sec. IV, and there is no load-bearing self-citation chain: references to the authors' prior work supply standard ingredients (M ratio, chiral invariant, GBZ tutorial) but are not the basis of the central equivalence. The overall structure is therefore partially circular: the claimed general 'prediction' of edge-state number reduces, in the phase diagrams and in the proof, to the analytic criterion that defines edge states in this model. Score 6 reflects this partial reduction, tempered by the genuinely derived bulk-boundary content and the independent single-example checks.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the generalized Bloch ansatz, the smooth-torus assumption for the band structure, the leading-term analysis of the OBC determinant, and standard complex analysis. No physical entities are newly postulated; Mdeg and Mbranch are derived mathematical quantities. The free parameters are numerical choices in the verification, not fitted physical constants.

free parameters (2)
  • epsilon (edge-state tolerance) = 0.001
    Hand-chosen threshold in the numerical verification (main text Fig. 3 and SM Sec. IX A) to classify an Edeg point as an edge state if |M1/M2 - 1| < epsilon or |M3/M4 - 1| < epsilon. Not part of the invariant itself, but affects the reported verification of Eq. (9) and (10).
  • N (chain length) = 32
    Finite chain size used for OBC spectra and GBZ computation in Fig. 3 and SM. Introduces finite-size effects that the authors acknowledge, and is a numerical choice rather than a fitted physical parameter.
assumptions (4)
  • domain assumption Generalized Bloch theorem: bulk equations of motion with arbitrary boundary conditions admit solutions z^n (psi0a, psi0b) with complex z, and a general solution is a superposition of p=4 such terms.
    Invoked in Eq. (3), (4), and the derivation of the OBC determinant in the End Matter and SM Sec. II A, following Ref. [41]. This is standard for non-Hermitian tight-binding models.
  • domain assumption For generic models of Eq. (1), the polynomial z^2 Delta(z) has no repeated roots, so the band structure is a smooth genus-1 curve (torus) and admits a Weierstrass parametrization.
    Assumed in SM Sec. V A to prove Eq. (10). The paper states it is readily satisfied by models with general coefficients and excludes models with duplicate roots (e.g., SSH).
  • domain assumption In the thermodynamic limit N goes to infinity, the OBC eigenvalue equation (determinant of the 4x4 coefficient matrix) requires the leading terms shown in Eq. (11) to vanish, giving either |z2|=|z3| (bulk GBZ states) or M1=M2 or M3=M4 (edge states).
    This leading-term cancellation is the analytical edge-state condition underlying the whole construction (End Matter). It assumes a clean ordering |z1| <= |z2| <= |z3| <= |z4| and neglects subleading terms; the paper notes the analysis changes for degenerate edge states.
  • standard math Argument principle and standard properties of the Weierstrass elliptic function.
    Used in the winding-number proof (SM Sec. V B) to convert contour integrals over the torus into winding numbers on the M-plane.

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Pith. "Pith review of Topological nature of edge states for one-dimensional systems without symmetry protection." pith.science (2026). https://pith.science/paper/I4KB6UXJ

@misc{pith2026241210526,
  author       = {Pith},
  title        = {Pith review of: Topological nature of edge states for one-dimensional systems without symmetry protection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I4KB6UXJ}},
  note         = {Machine review of arXiv:2412.10526}
}
read the original abstract

We numerically verify and analytically prove a winding number invariant that correctly predicts the number of edge states in one-dimensional, nearest-neighbor (between unit cells), two-band models with any complex couplings and open boundaries. Our winding number uses analytical continuation of the wave-vector into the complex plane and involves two special points on the full Riemann surface band structure that correspond to bulk eigenvector degeneracies. Our winding number is invariant under unitary or similarity transforms. We emphasize that the topological criteria we propose here differ from what is traditionally defined as a topological or trivial phase in symmetry-protected classification studies. It is a broader invariant for our model that supports nonzero energy edge states and its transition does not coincide with the gap closing condition. When the relevant symmetries are applied, our invariant reduces to well-known Hermitian and non-Hermitian symmetry-protected topological invariants.

Figures

Figures reproduced from arXiv: 2412.10526 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Two unit cells of the model in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Analytical edge state and topological invariant [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2. General invariant as a winding number resulting from [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.