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REVIEW 2 major objections 8 minor 61 references

Topological classification and edge states of magnons in honeycomb ferromagnets

T0 review · 2 major / 8 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Honeycomb-ferromagnet magnon phases are captured by non-Hermitian SSH chains.

desk verdict Useful bilayer extension of magnon topology, but the AFM-interlayer cases need a stability check before their Chern numbers can be taken physically. read the letter →

arxiv 2504.21534 v1 pith:F32MJ3MN submitted 2025-04-30 cond-mat.mes-hall quant-ph

Youwen wang , Qiutong Wang , Qingjun Tong , Ci Li This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords magnonshoneycombferromagnetBogoliubov-deGennesnon-HermitianSSHchainChernnumberDzyaloshinskii-Moriyainteractionbilayerstackingedgestates
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

The paper argues that the topological phases of magnons in honeycomb ferromagnets, including the existence, number, and location of edge states, can be read off from effective one-dimensional non-Hermitian Su-Schrieffer-Heeger (SSH) chain models labelled by momentum $k_y$. For monolayers, the conventional bulk-edge correspondence remains usable when Dzyaloshinskii-Moriya interaction (DMI) is absent, and edge states can even be the lowest-energy states; with DMI, the edge states tie to a nonzero Chern number of the bulk magnon bands. In bilayers, the stacking (AA versus AB), the sign of the interlayer coupling (ferromagnetic or antiferromagnetic), and the edge termination together determine the symmetry class, the Chern numbers, and the edge-state distribution. The point of the reduction is that a two-dimensional magnon problem becomes a set of tractable one-dimensional chains whose topology is already understood. If true, this gives a practical classification recipe for hexagonal ferromagnet magnonics.

What carries the argument

The load-bearing object is the effective one-dimensional SSH chain $H(k_y)$, obtained by Fourier-transforming only along the edge and keeping open boundary conditions in the perpendicular direction; for bilayers it becomes a coupled SSH ladder that can often be decomposed into decoupled chains with a detuned on-site energy $\pm J_{\mathrm{in}}$. When pairing terms from antiferromagnetic interlayer couplings are present, the relevant operator $\eta h(k)$ is non-Hermitian with respect to the bosonic metric $\eta=\sigma_z\otimes I$, and its eigenproblem is equivalent to the original bosonic BdG problem. This machinery transfers topology from the two-dimensional bulk to the boundary bands: Berry curvature and Chern numbers computed on the pseudo-Hermitian $H_{\sigma\mathrm{BdG}}(k)$ predict the existence and winding of edge states, while the sign of the DMI sets the sign of the Chern number. For monolayer and ferromagnetic-bilayer cases the chain is Hermitian and conventional SSH winding arguments apply; for antiferromagnetic-bilayer cases the paper notes nonzero imaginary parts in the energies and plots $|E|$, so the chain's topology is presented as a partial, not fully justified, bulk-edge correspondence.

What would settle it

Compute the full complex spectrum of $\eta h(k_y)$ for the AA- and AB-stacked antiferromagnetic-interlayer bilayers at the parameters used in the paper's Figure 10 and scan over $k_y$ and DMI strength; if any imaginary part becomes comparable to the real part, the $|E|$ plot would hide a spectral instability. Separately, diagonalize the real-space bosonic BdG Hamiltonian with open boundaries and check whether any excitation energies become negative or complex, which would settle whether the predicted edge states exist in a stable ground state.

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Extended reading notes

Core claim

The central discovery is that the single-particle bosonic Bogoliubov-de Gennes (BdG) Hamiltonians of honeycomb ferromagnets reduce, under open boundary conditions in one direction, to effective one-dimensional SSH-type chains $H(k_y)$ parameterized by the remaining momentum, and the topology of these chains explains the edge states of the original magnon system. In the monolayer case without DMI the BdG Hamiltonian has no pairing block and is Hermitian, so ordinary bulk-edge correspondence applies, producing four edge states for bearded boundaries and two for zigzag boundaries, and these states can sit at the lowest or highest magnon energies rather than near zero energy. Turning on DMI gives the bulk bands a nonzero Chern number (up to magnitude 1 for monolayers, up to magnitude 2 for AB-stacked bilayers), and the edge states are then connected to this Chern number through their winding, with the actual arrangements depending on the boundary. For bilayers, AA versus AB stacking and ferromagnetic versus antiferromagnetic interlayer coupling change the symmetry class and hence the Chern numbers and edge-state counts reported in the paper's classification table.

Load-bearing premise

The load-bearing premise is that the effective non-Hermitian Hamiltonians for the antiferromagnetic-interlayer bilayer cases describe a stable bosonic system, even though their spectra acquire small nonzero imaginary parts; the paper plots $|E|$ and asserts the imaginary parts are small compared with the real parts, but does not show that the complex energies correspond to a physically admissible bosonic ground state or that the Chern number remains well-defined in their presence.

Editorial extensions

If this is right

  • The topological phase of a honeycomb-ferromagnet magnon system is fixed by a small parameter set: intralayer exchange, DMI, stacking form, interlayer coupling sign, and boundary termination, with the paper's table giving the resulting class, Chern number, and edge-state count.
  • Edge states in these systems need not be mid-gap: for bearded and zigzag terminations with zero DMI they can appear at the lowest or highest magnon energies, meaning they could be occupied ground states rather than zero-energy modes.
  • With nonzero DMI the edge-state count is tied to the bulk Chern number: monolayer cases show four edge states with $|C|=1$, AA ferromagnetic bilayers show eight with $|C|=1$, and AB ferromagnetic bilayers show eight with $|C|=2$, so edge states act as a boundary probe of the bulk invariant.
  • When DMI is zero, edge states can still exist for certain boundary conditions even though the Chern number vanishes, so the effective one-dimensional SSH or ladder topology, rather than the two-dimensional Chern number, controls the boundary physics.
  • In AB-stacked ferromagnetic bilayers, swapping the boundary types of the two layers leaves the bulk Hamiltonian and Chern number unchanged but moves edge states to different regions in $k_y$, showing that bulk topology alone does not fix the edge-state location.

Reading between the lines

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

  • A testable extension would be to perturb a specific edge in a way that breaks the effective chain symmetry: zero-DMI edge states should disappear under such a perturbation, while nonzero-DMI Chern-connected states should remain, because the latter are protected by the bulk invariant rather than by chain fine-tuning.
  • The results suggest a material-search strategy: honeycomb ferromagnets with DMI, or with suitable stacking and boundary combinations, should show the predicted number of low- or high-energy magnon edge states in spin-resolved or thermal transport, and the sign of the DMI could be inferred from the chirality of those edge bands.
  • Because the paper works in linear spin-wave theory, a natural next step is to test whether magnon-magnon interactions alter the claim that some edge states can appear as ground states at zero DMI.
  • For the antiferromagnetic-interlayer bilayers, a cautious follow-up would be to define physical observables biorthogonally before assigning Chern numbers to the complex-energy states; if the imaginary parts grow with parameters, the classification would need revision.
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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

2 major / 8 minor

Summary. The paper studies magnon excitations in monolayer and bilayer honeycomb ferromagnets using linear spin-wave theory and a single-particle bosonic Bogoliubov-de Gennes (BdG) description. It derives the bulk BdG Hamiltonians for monolayer systems with bearded, zigzag, and armchair edge geometries, and for AA- and AB-stacked bilayers with ferromagnetic (FM) or antiferromagnetic (AFM) interlayer coupling, with or without next-nearest-neighbor Dzyaloshinskii-Moriya interaction (DMI). The effective one-dimensional edge Hamiltonians H(ky) are mapped to Su-Schrieffer-Heeger (SSH) chains or SSH ladders, and edge states are analyzed as functions of ky. The main claims are: (i) for DMI-free systems the conventional bulk-edge correspondence based on the 2D Chern number is only partially valid, with edge states still appearing because of the SSH-like topology of H(ky); (ii) when DMI is present the nonzero Chern number is connected to the existence of edge states; and (iii) for AFM-interlayer bilayers the relevant effective Hamiltonian is non-Hermitian, yet a non-Hermitian Chern number can still be defined. The paper summarizes its results in Table I and illustrates the spectra in Figs. 3-10.

Significance. If the results hold, the paper provides a systematic catalog of magnon edge-state behavior and topological invariants for a widely studied family of honeycomb magnetic systems, and it highlights an instructive mapping from bosonic BdG problems to non-Hermitian SSH chains. The explicit analytic derivation of h(k) for the monolayer and FM-interlayer bilayer cases is mostly standard, the internal relations among the Chern numbers in Table I are consistent, and the SSH-chain interpretation gives a useful physical picture. The treatment of the AFM-interlayer cases is the most interesting part but also the least secure: the paper itself admits the presence of imaginary energies, yet it does not establish that the corresponding bosonic ground state is stable or that the quoted non-Hermitian Chern numbers have a well-defined physical meaning.

major comments (2)
  1. [Sec. IV.A.2 and IV.B.2, Eqs. (47)-(51) and (61)-(63), Fig. 10] The AFM-interlayer bilayer cases are central to the paper's claim of a 'straight connection' between edge states and nonzero Chern numbers, but their physical validity is not established. The text explicitly states that 'imaginary energies appear' for the effective Hamiltonian ηh(ky) and that |E| is plotted in Fig. 10 for this reason. In a bosonic BdG problem, the physical magnon frequencies are the eigenvalues of Pz HBdG, and non-real eigenvalues are a standard signature that the Hermitian matrix HBdG is not positive definite, meaning the assumed collinear magnetic state is dynamically unstable. The paper never checks positive definiteness, never identifies a parameter regime in which the spectrum of ηh(ky) is real, and does not discuss whether the biorthogonal Chern numbers C1=±1 listed in Table I for these rows remain well defined when complex eigenvalues are present. Without such a check, the Chern numbers and the edge-state spectra in Fig. 10 for the AFM-interlayer bilayers cannot be interpreted as physical magnon properties, and the corresponding claims in the abstract are unsupported.
  2. [Abstract and Sec. III.B, Table I, Eq. (31)] The phrase 'conventional bulk-edge correspondence is partially valid' is never made precise. For D=0 the 2D Chern number is zero (Eq. (31)), yet edge states exist; the paper attributes these edge states to the SSH-like 1D winding of H(ky), which is a different invariant. For D≠0, the paper claims a 'straight connection' to the nonzero Chern number, but Table I lists NES=4 for Chern number ±1, while the text in Sec. III.B indicates that only one in-gap chiral branch is the Chern-related feature, the other three being boundary-condition-dependent nonchiral states. The manuscript should state explicitly which invariant (Chern number of the 2D bulk, 1D winding of H(ky), or neither) protects each class of edge states and how the entry NES in Table I is computed.
minor comments (8)
  1. [Sec. II.D] The word 'pesudo-Hermiticity' should be 'pseudo-Hermiticity' in Eq. (22) and the surrounding text.
  2. [Sec. II.D, around Eq. (24)] The phrase 'two boardly suitable conclusions' should read 'two broadly suitable conclusions'.
  3. [Abstract and Sec. III.B] The statement that 'edge states can appear as the ground state' is confusing; the edge states shown in Figs. 3-6 are lowest-energy eigenstates of the single-magnon Hamiltonian, not many-body ground states of the spin system. Please rephrase to avoid the implication of a Bose-condensed or zero-energy ground state.
  4. [Sec. IV.A.2, Eq. (48)] The quantities A, B±, λ1±, and λ2± are used in the expressions for the eigenstates and eigenvalues before they are defined. Please move the definitions ahead of their first use.
  5. [Table I and Sec. IV.A.2] For the rows involving ηh(k) with AFM interlayer coupling, the table and the text should state explicitly whether the quoted Chern number is computed in a regime where the eigenvalues of ηh(k) are real; otherwise the classification is incomplete.
  6. [Fig. 10 caption and Sec. IV.B.2] The caption says the figure shows |E| for the effective chain, but it does not clarify whether the imaginary energies appear in the bulk spectrum, the edge-state branches, or both. This point should be stated in the main text and the caption.
  7. [Footnote [54]] The self-referential footnote about the difference between ky and k should be incorporated into the main text or removed, as it reads like an editorial note rather than a regular reference.
  8. [Abstract] The phrase 'straight connection' is likely meant as 'direct connection' and would be less overclaiming if reworded, especially given the additional nonchiral edge states discussed in the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SSH-chain and non-Hermitian mappings are derived explicitly from the bosonic BdG Hamiltonians, and the cited topological classification is external.

full rationale

The paper's derivation chain is self-contained. The Heisenberg spin Hamiltonian is converted by the Holstein-Primakoff transformation into single-particle bosonic BdG form, and the bulk and open-boundary Hamiltonians h(k) and H(ky) are written out explicitly for monolayer, AA-stacked, and AB-stacked bilayer cases (e.g., Eqs. (29), (34), (41), (47), (54), (61)). The 'effective SSH chain' descriptions are not posited as ansatze: they are obtained by rewriting the real-space BdG Hamiltonians under the periodic/ky geometry, with the chain parameters v, w, D1 expressed directly in terms of J, D, and ky, so the classification of edge states is a derived equivalence rather than a fitted input. The topological invariants (Chern numbers, winding numbers, symmetry classes) are computed from the eigenstates of these explicit Hamiltonians using standard formulas, and the classification tables are imported from external references [40,49] that are not authored by the present paper's authors, so no uniqueness theorem is being smuggled in through self-citation. The only self-citation, Ref. [53], is used to interpret the edge states via the known graphene/SSH-chain analogy; that mapping is re-derived in the present work, and the citation is illustrative rather than load-bearing. The admitted non-zero imaginary energies in the AFM-interlayer bilayer cases ('imaginary energies appear (which is pretty small compared to the real part, but non-zero, so we show |E| in Fig. 10 for simplicity)') are a physical ground-state-stability and validity concern that is outside the circularity analysis: the quoted |E| spectra and Chern numbers are computed from the stated Hamiltonian, not assumed as inputs. No parameter is fitted to the data it is later said to predict, and no equation is shown to reduce to its own input by construction. Therefore no circular step is present.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard linear spin-wave theory and on the non-Hermitian topological classification of Ref. [40]; no new physical entities are introduced. The hand-chosen parameters S=1, J=1 eV, D=0.1J, Jin=0.1J, gamma=3J are illustrative; the topological conclusions are claimed to depend only on signs and on the presence or absence of couplings, not on these magnitudes. The most fragile premises are the validity of the non-Hermitian classification when complex eigenvalues appear in AFM-interlayer bilayers, and the truncation of the HP transformation to linear order.

free parameters (5)
  • S (spin length) = 1
    Set to 1 in all figures; it is an overall scale for magnon energies and does not affect the topological classification.
  • J (nearest-neighbor exchange) = 1 eV
    Used as the energy unit in plots; Chern numbers are independent of J, so this choice is illustrative.
  • D/J (DMI strength) = 0.1
    Illustrative nonzero DMI used in figures; topology for D != 0 depends only on sign(D), not magnitude.
  • Jin/J (interlayer coupling) = 0.1
    Illustrative weak interlayer coupling in bilayer figures; the qualitative classification is stated to hold for small Jin.
  • gamma (on-site energy parameter) = 3J
    Set by taking K=0 and Bz=0; affects whether edge states appear at lowest or highest energy, but not the topological class.
assumptions (5)
  • domain assumption Holstein-Primakoff transformation truncated at linear order is valid, meaning magnon-magnon interactions are negligible.
    Used in Section II.A to convert spin operators to bosons; all subsequent Hamiltonians are non-interacting single-particle models.
  • domain assumption The magnetic ground state has collinear order (ferromagnetic or Neel) along z on the honeycomb lattice.
    Assumed in Section II.A-B; the HP transformation is built on this order, so the topology is for magnons above this ordered state.
  • standard math The topological classification of non-Hermitian Hamiltonians from Ref. [40] applies to the bosonic BdG operator sigma_z H_BdG.
    Invoked in Section II.D for classes A, AI, AII and the Chern formulas; the paper cites rather than re-derives this classification.
  • domain assumption For the monolayer, Delta(k)=0, so the -h*(-k) block is 'redundant and unphysical' and the physics is fully described by h(k).
    Stated in Section II.D conclusion 2 and used to treat the monolayer and FM-interlayer bilayers as Hermitian systems.
  • domain assumption For AFM-interlayer bilayers, the eigenstates of eta h(k) with positive and negative norm are characterized by the same topological invariant, so Chern numbers can still be assigned.
    Used in Section IV.A.2 near Eq. (49) to assign Chern numbers despite the non-Hermitian structure and the presence of complex energies.

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Cite this review

Pith. "Pith review of Topological classification and edge states of magnons in honeycomb ferromagnets." pith.science (2026). https://pith.science/paper/F32MJ3MN

@misc{pith2026250421534,
  author       = {Pith},
  title        = {Pith review of: Topological classification and edge states of magnons in honeycomb ferromagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F32MJ3MN}},
  note         = {Machine review of arXiv:2504.21534}
}
read the original abstract

We study the topological classification and related edge states of magnons in ferromagnets on honeycomb that can be described by a class of single-particle bosonic Bogoliubov-de Gennes (BdG) models. Both single layer and bilayer situations are considered. The calculations show that the existence and related topologies of these edge states are well captured by a class of non-Hermitian single or coupled Su-Schrieffer-Heeger chains models H(ky) parameterized by momentum ky, where the edge states can appear as the ground state for some cases. Interestingly, although the eigenproblem of bosonic BdG models is equivalent to the one of non-Hermitian systems, the conventional bulkedge correspondence for Hermitian systems is partially valid. The influence of Dzyaloshinskii-Moriya interactions between next nearest-neighbor spins are also discussed, which break the time-reversal symmetry and lead to a straight connection between edge states for magnonic systems and non-zero Chern number of non-Hermitian bulk two-dimensional systems.

Figures

Figures reproduced from arXiv: 2504.21534 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) (a) Schematics of monolayer ferromag [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) Upper panel: The schematic illustra [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) (a) Top panel: The schematic illustration of 1D effective SSH chain parameterized by [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) (a) Schematic illustration of quasi-1D effective chain parameterized by [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) (a) The change of the band structure [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) (a) The change of the band structure [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) Top panel: The schematic illustra [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (Color online) (a) Top panel: The schematic illus [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (Color online) (a) The schematic illustration of quasi [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (Color online) (a) The schematic illustration of [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

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Reference graph

Works this paper leans on

61 extracted references · 47 canonical work pages

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    Both monolayer and bilayer configurations are consid- ered. For the monolayer case, since there are no double anni- hilation or creation operators in the related BdG models, the conventional bulk-edge correspondence for Hermitian systems remains effective. Non-trivial edge states can emerge even without the Dzyaloshinskii-Moriya interac- tion (DMI) [43, 4...

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    It means T+ = I2l is the identity ma- trix and leads toHσBdG in the classAI withη+, a trivial topological classification

    Once ∆ ∗ (k) = ∆ ( −k), which is the situation for most cases, we have H∗ σBdG (k) = HσBdG (−k) if h∗ (k) = h (−k). It means T+ = I2l is the identity ma- trix and leads toHσBdG in the classAI withη+, a trivial topological classification. So only h∗ (k)̸=h (−k) results in a non-trivial class and may correspond to a non-trivial topology

    work page

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    If ∆ (k) is a zero matrix, i.e., there is no coupling betweenβ† k andβ† −k, HσBdG (k) = h (k) 0 0 −h∗ (−k) (25) Bearded edge (1,1) (1,2) (2,1) Ԧ𝑎1 Ԧ𝑎2 Zigzag edge (a) (b) 𝑥 (c) Armchair edge 𝐴 (1,1) (1,1) (2,1) (1,2) Ԧ𝑎1 Ԧ𝑎2 𝐵 Ԧ𝑎1 Ԧ𝑎2 (2,1) (1,2) Zigzag edge 𝑦 𝑎2𝑚−1,𝑛 𝑏2𝑚−1,𝑛 𝑎2𝑚,𝑛 𝑏2𝑚,𝑛 (m,n) (m,n) 𝑎𝑚,𝑛 𝑏𝑚,𝑛 𝑎𝑚,𝑛 𝑏𝑚,𝑛 (m,n) 𝑧 FIG. 2. (Color online) Upper...

    work page

  5. [4]

    The straightfor- ward calculation shows in momentum space, Hint (k) = JinS[ a2 k † a2 k + b2 k † b2 k (40) + a1 k † a1 k + b1 k † b1 k − a1 k † a2 k− b1 k † b2 k +H.c.]

    FM interlayer interaction Combining with the bulk BdG Hamiltonian Hl FM with bearded boundary as we showed in the last section as an example, the FM interlayer interaction can be expressed as Hint = −Jin X i S1 i·S2 i (39) ≈ JinS MX m=1 NX n=1 { h a2 m,n † a2 m,n + b2 m,n † b2 m,n i + h a1 m,n † a1 m,n + b1 m,n † b1 m,n i − h a1 m,n † a2 m,n + b1 m,n † b2...

    work page

  6. [5]

    AFM interlayer interaction The AFM one can be expressed as Hint = Jin X i S1 i·S2 i (44) ≈ JinS MX m=1 NX n=1 { h a2 m,n † a2 m,n + b2 m,n † b2 m,n i + h a1 m,n † a1 m,n + b1 m,n † b1 m,n i − h a1 m,n † a2 m,n † + b1 m,n † b2 m,n †i +H.c.}, with Jin > 0. Similar process as above shows that Hint (k) = JinS{ a2 k † a2 k + b2 k † b2 k (45) + a1 k † a1 k + b1...

    work page

  7. [6]

    The straightforward calculation shows in momentum space, Hint (k) = JinS[ a2 k † a2 k + b1 k † b1 k (53) − b1 k † a2 k +H.c.]

    FM interlayer interaction Still taking the bulk BdG Hamiltonian Hl FM with the bearded boundary condition that we showed in the last section as an example, the FM interlayer interaction in this condition can be expressed as Hint = −Jin X ⟨i,j⟩ S1 i·S2 j (52) ≈ JinS MX m=1 NX n=1 { h a2 m,n † a2 m,n + b1 m,n † b1 m,n i − b1 m,n † a2 m,n +H.c.} in the basis...

    work page

  8. [7]

    in momentum space

    AFM interlayer interaction The AFM one can be expressed as Hint = Jin X i S1 i·S2 i (59) ≈ JinS MX m=1 NX n=1 { h a2 m,n † a2 m,n + b1 m,n † b1 m,n i − b1 m,n † a2 m,n † +H.c.}, with Hint (k) =JinS[ a2 k † a2 k+ b1 k † b1 k− b1 k † a2 −k † +H.c.]. in momentum space. Then H (k) = η†H bea−bea BdG (k)η,η = ξk,ξ† −k T , (60) ξk = h a1 k,b 1 k, a2 −k † , b2 −k...

    work page

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    Here we would like to point out that one should not mis- leadky andk. Although they are both wave vector num- ber, ky is treated as one of parameters of Hamiltonian H bea σBdG (ky), different ky means different H bea σBdG (ky); k is just a wave vector number of 1D infinite (pe...

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