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REVIEW 3 major objections 5 minor 56 references

An investigation of the two-dimensional non-Hermitian Su-Schrieffer-Heeger Model

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A two-dimensional non-Hermitian SSH model with staggered hopping develops nonzero Berry curvature and a finite anomalous Nernst conductivity, even though its Chern number is not quantized.

desk verdict The EP/Zak/circuit sections are mostly checkable and not worthless, but the central QANE claim in Section 4 contradicts the paper's own TRS statement: at gamma=0 the Chern number and Nernst conductivity must vanish, yet the paper reports C=-2.32 and finite alpha_xy. read the letter →

arxiv 2506.16867 v1 pith:A4Z5SCDN submitted 2025-06-20 cond-mat.mes-hall

Udai Prakash Tyagi , Partha Goswami This is my paper · ORCID
classification cond-mat.mes-hall
keywords SSHmodelImaginaryon-sitepotentialVectorizedZakphaseTopolectriccircuitStaggeredhoppingamplitudesNon-HermitianBerrycurvatureAnomalousNernsteffect
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 studies a spinless two-dimensional Su-Schrieffer-Heeger model modified by imaginary on-site potentials (gain and loss) and by staggered hopping amplitudes along the horizontal and vertical directions. It aims to show that these two ingredients break time-reversal and inversion symmetries enough to give the system a nonzero Berry curvature, and therefore a finite anomalous Nernst conductivity, without requiring spin-orbit coupling. The same model is used to identify exceptional points through the rank-nullity theorem and the phase-rigidity factor, to obtain quantized vectorized Zak phases, and to propose a topolectric RLC circuit realization. A sympathetic reader would care because a transverse heat-induced current in a simple non-Hermitian lattice would be a concrete transport signature of band geometry in systems where the usual integer Chern number is unavailable.

What carries the argument

The load-bearing object is the four-band reciprocal-space Hamiltonian $H_{\mathrm{NH,2D}}(k_x,k_y)$ in Eq. (5), a $4\times4$ matrix with alternating imaginary on-site potentials $i\gamma$ and $-i\gamma$ and hoppings $u,t_1$ along $x$ and $v,t_2$ along $y$. The argument runs through the biorthogonal left and right eigenvectors of this matrix: Eq. (17) defines the Berry curvature $\Omega^z_{\alpha}(k_x,k_y)$ from those eigenvectors, and Eq. (18) converts it into the anomalous Nernst conductivity via the Mott relation. Staggered hoppings make the inversion-symmetry condition $u=t_1$, $v=t_2$ fail, which is what allows the curvature to be nonzero; the rank-nullity theorem and the phase-rigidity factor $P_j$ are the tools used to identify genuine exceptional points, and the Wilson-loop formula in Eq. (11) gives the vectorized Zak phase.

What would settle it

For the Hermitian limit $\gamma=0$ with real staggered hoppings, compute the anomalous Nernst conductivity using the standard gauge-invariant Berry curvature of a Hermitian band structure. Time-reversal symmetry makes the curvature odd in $k$, so the integral in Eq. (18) must give zero; if the paper's right-right curvature gives a nonzero value in the same limit, the finite-Nernst conclusion is a convention artifact.

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Extended reading notes

Core claim

The central claim is that, for the reciprocal-space Hamiltonian in Eq. (5), staggered hopping amplitudes $u \neq t_1$ and/or $v \neq t_2$ break inversion symmetry and, together with imaginary staggered potentials $\pm i\gamma$ that break time-reversal symmetry, produce a nonzero Berry curvature in certain parameter windows. Inserting that curvature into the low-temperature Mott formula for the anomalous Nernst conductivity gives a finite $\alpha_{xy}$, even though the Brillouin-zone integral of the right-right Berry curvature does not quantize to an integer. The paper therefore concludes that the quantum anomalous Nernst effect is possible for this system while the quantum anomalous Hall effect with integer Chern number is not. It also reports that tuning the gain/loss parameter $\gamma$ can drive the system between insulating and conducting regimes, and that true exceptional points can be located where coalescing eigenvalues coincide with the rank-nullity condition and vanishing phase rigidity.

Load-bearing premise

The whole finite-Nernst result rests on treating the right-right Berry curvature in Eq. (17) as a legitimate gauge-invariant Berry curvature whose integral can be fed into the Mott formula; the paper does not prove this invariance.

Editorial extensions

If this is right

  • A temperature gradient applied to a 2D non-Hermitian SSH lattice with staggered hoppings should produce a transverse charge current, a measurable Nernst signal, without spin-orbit coupling.
  • Because the Chern number is non-integer, the system will not show a quantized anomalous Hall conductance, but the Nernst response can still be substantial; the two effects are decoupled.
  • By decreasing $\gamma$, the same Hamiltonian crosses from gapped (insulating) to gapless (conducting) behavior, so gain/loss strength acts as a switch for transport.
  • In an RLC-circuit realization with the circuit Laplacian of Eq. (13), the topological boundary resonance condition is met for $R \leq 1\,\Omega$, and the integrated imaginary Berry phase gives a positive quantum Hall susceptance, indicating capacitive response.

Reading between the lines

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

  • Editorial inference: the finite-Nernst claim depends on which of the four non-Hermitian Berry curvatures is inserted into Eq. (18); the paper uses the right-right curvature, but a gauge-invariant combination could give a different, possibly vanishing, transverse response.
  • Editorial inference: for $\gamma=0$ with real hoppings, time-reversal symmetry is restored, and the standard Hermitian Berry curvature is odd over the Brillouin zone, so the integrated Nernst coefficient in Eq. (18) should vanish; a nonzero result at $\gamma=0$ would be a test of the curvature convention rather than a physical effect.
  • Editorial inference: the same lattice, realized as an RLC circuit, could provide a direct experimental search: measuring the transverse voltage under a thermal gradient in a circuit version of Eq. (13) would check the predicted finite Nernst response at low resistance.
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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 / 5 minor

Summary. The manuscript studies a two-dimensional four-site non-Hermitian SSH model with balanced imaginary on-site potentials, deriving its energy spectrum, exceptional-point structure, vectorized Zak phase, and an RLC circuit analogue. The central transport claim appears in Section 4: for staggered hopping amplitudes with gamma = 0 (broken inversion symmetry but respected time-reversal symmetry), the authors report a nonzero Berry curvature, an unquantized Chern number C approx -2.32, and a finite anomalous Nernst conductivity computed from Eq. (18). The paper concludes that the model supports a quantum anomalous Nernst effect without spin-orbit coupling, despite the absence of Chern-number quantization.

Significance. If the central claim were correct, a finite transverse thermoelectric response in a spinless, time-reversal-invariant Hermitian limit would be a striking result requiring new physics beyond conventional Berry-curvature transport. The manuscript also contains checkable computations: the spectrum is given explicitly, the exceptional-point analysis is based on a discriminant and phase-rigidity criterion, and the Chern-number calculation is checked with the Fukui-Hatsugai-Suzuki method. These are strengths: the authors compute directly from the Hamiltonian rather than fitting to a target outcome. However, the central transport claim is internally inconsistent with the model's symmetries, and the nonquantized Chern number reported for a Hermitian TRS-invariant system indicates that the object being integrated is not the gauge-invariant Berry curvature. The manuscript's contribution is therefore dominated by the EP, Zak-phase, and circuit analyses, while the QANE claim, which is highlighted in the abstract and conclusion, does not survive scrutiny.

major comments (3)
  1. [Section 4, Eq. (18), Fig. 6(a),(b),(f)] The load-bearing claim of finite anomalous Nernst conductivity at gamma = 0 is internally inconsistent. For the parameter values of Fig. 6(a),(b), the paper states that TRS is respected (the Hamiltonian (5) is Hermitian with real hoppings, so H(-k)=H*(k) with T = I4 K). For a spinless TRS-invariant Bloch Hamiltonian, the Berry curvature of each band is odd, Omega_n(-k) = -Omega_n(k), so the BZ integral in Eq. (18) vanishes identically for every chemical potential, as does any Chern number. Reporting C = -2.3235 and a finite alpha_xy at gamma = 0 is therefore a contradiction. The statement that the four left/right curvature prescriptions yield the same Chern number cannot repair this, because in the Hermitian limit all prescriptions coincide with the conventional curvature. The non-integer C is a diagnostic that the Omega^RR used in Eq. (17) is not the physical, gauge-invariant Berry curvature, so the QANE conclusion in Section 4 and the abstract does not follow.
  2. [Section 2, paragraph following Eq. (8)] The rank-nullity statement is incorrect: the text reads 'nullity(matrix) = rank(matrix) + number of columns of the matrix', but the rank-nullity theorem gives nullity = number of columns - rank. The subsequent claim that GM < AM for the exceptional points relies on this relation, and as written it would produce impossible values (for a 4x4 matrix with rank 3 it would give nullity 7). This needs to be corrected for the EP identification argument to be valid.
  3. [Section 4, Eqs. (17) and (18), and FHS paragraph] The manuscript never establishes that the right-right Berry curvature used in Eq. (17) is gauge-invariant under the biorthogonal normalization, and the assertion that the four curvatures 'ultimately yield the same Chern number' is cited to Ref. [52] without verifying its assumptions for the present model with complex eigenvalues and band degeneracies. This is load-bearing because the paper interprets the nonquantized value C approx -2.32 as physical evidence of a nonzero Chern number and hence a finite Nernst response. A nonquantized integral of a non-gauge-invariant object has no topological or transport meaning; the FHS cross-check does not resolve this because the same non-Hermitian eigenvector gauge issues enter the link variables. A correct treatment of the gamma = 0 limit would give C = 0 and alpha_xy = 0, and any finite alpha_xy for gamma != 0 must come from a separately justified, gauge-invariant non-Hermitian curvature formula.
minor comments (5)
  1. [Section 3, Eq. (11b)] The citation 'refs. [66]' does not exist in the reference list; the reference numbering should be checked throughout.
  2. [References] The reference list jumps from [21] to [24] with no entries [22] and [23].
  3. [Section 4, Eq. (18)] The formula for alpha_xy is typeset in a garbled way: the prefactor and the integral measure are unclear, and the entropy-density expression is written without clear bracketing. Please rewrite Eq. (18) and the neighboring definitions with explicit variables and limits.
  4. [Figure 2 caption] The caption states 'Here the symbol g is the shortform of gamma' in the middle of a scientific caption; this informal notation should be removed and the axes labeled consistently.
  5. [Section 5] The statement that the quantum Hall susceptance is positive and 'indicative of capacitive properties' should be justified by showing how the imaginary Berry phase integral maps to the circuit susceptance; as written, the connection is asserted rather than derived.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: all claimed results are direct computations from the model Hamiltonian and standard transport/topological formulas.

full rationale

The paper's main claims are obtained by explicit calculation from the reciprocal-space Hamiltonian in Eq. (5). The Berry curvature in Eq. (17) is evaluated from the model's eigenvectors, the anomalous Nernst conductivity in Eq. (18) is the standard Mott-relation integral of that curvature, and the Chern number is computed both by direct integration and by the Fukui-Hatsugai-Suzuki lattice method. No parameter is fitted to reproduce the claimed Nernst conductivity or Chern number, and no target observable is used to define the model. The vectorized Zak phase and circuit Laplacian results are likewise direct numerical evaluations. The paper contains no load-bearing self-citation: the symmetry statements and formulas cite standard external references, and the authors' own prior work is not used to justify the central claims. A reader may dispute the validity of the Berry-curvature convention or the consistency of a finite Nernst response with TRS in the gamma = 0 Hermitian limit, but that is a physical-correction issue, not a circular-derivation issue, because the claimed output is not assumed in the input. The paper even acknowledges the non-quantization limitation explicitly, which further indicates that the result was not imposed. Therefore the circularity score is 0.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The paper adds a model Hamiltonian with chosen hopping and gain/loss parameters but introduces no new physical entities. The main burden is the unproven choice of the RR Berry curvature as the invariant and the TRS-violating transport claim.

free parameters (7)
  • u = 1 (energy unit)
    Horizontal intra-cell hopping; chosen by hand and used as the energy scale throughout.
  • t1 = 0.80, 0.95, 0.85, 1 in different figures
    Horizontal inter-cell hopping; chosen by hand, and staggered with u to control inversion symmetry and Berry curvature.
  • v = 0.60, 0.23, 0.35, 0.41, 0.71 in different figures
    Vertical intra-cell hopping; chosen by hand, and with t2 controls vertical hopping asymmetry.
  • t2 = 0.75, 0.50, 0.71, 0.78 in different figures
    Vertical inter-cell hopping; chosen by hand, appears in the p and q couplings and in Berry curvature integrals.
  • gamma = 0, 0.5, 0.59, 0.75, 0.77, 0.18 in different figures
    Imaginary on-site potential strength; chosen by hand, breaks TRS, and drives insulator-conductor transitions.
  • mu = 0
    Chemical potential set to zero in all transport calculations; determines the Fermi level and the ANE integration window.
  • Circuit parameters R, L1=L2, C, omega = R=1,26,50 Ohm; L=0.1 mH or 10 mH; C=0.01 microF; omega near 3e4 s^-1
    Chosen by hand for the topolectric RLC circuit; used for the topological boundary resonance condition and quantum Hall susceptance.
assumptions (7)
  • standard math Rank-nullity theorem and discriminant criterion for eigenvalue coalescence
    Used in Section 2 to identify exceptional points; the text misstates the rank-nullity relation as nullity = rank plus columns but then uses 4 minus rank.
  • standard math Wilson loop / discrete Zak phase formula
    Used in Section 3 to compute the vectorized Zak phase; treated as standard and not proved in the paper.
  • standard math Kubo formula / quantum geometric tensor for Berry curvature and Mott relation for Nernst conductivity
    Used in Section 4 and Appendix A; the non-Hermitian extension is assumed without proof of gauge invariance.
  • domain assumption Spinless time-reversal symmetry is represented by T = I4 K
    Central to the TRS discussion and to the internal contradiction in Section 4, where the same symmetry forces the Nernst integral to vanish for gamma = 0.
  • domain assumption Phase rigidity approaching zero identifies true exceptional points
    Used graphically in Section 2; no theorem is provided that P_j = 0 occurs exactly at the exceptional points for this finite lattice model.
  • domain assumption The RLC circuit Laplacian is a faithful analog of the tight-binding Hamiltonian
    Used in Section 3 to map circuit admittance to topological band structure and boundary resonances.
  • ad hoc to paper All four left/right combinations of Berry curvature yield the same Chern number, so the RR curvature can be used
    Invoked in Section 4 with one citation but not derived; this is load-bearing for the non-quantized Chern claim.

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Pith. "Pith review of An investigation of the two-dimensional non-Hermitian Su-Schrieffer-Heeger Model." pith.science (2026). https://pith.science/paper/A4Z5SCDN

@misc{pith2026250616867,
  author       = {Pith},
  title        = {Pith review of: An investigation of the two-dimensional non-Hermitian Su-Schrieffer-Heeger Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4Z5SCDN}},
  note         = {Machine review of arXiv:2506.16867}
}
read the original abstract

This communication presents an examination of a two-dimensional, non-Hermitian Su -Schrieffer-Heeger (SSH) model, which is differentiated from its conventional Hermitian counterpart by incorporating gain and/or loss terms, mathematically represented by imaginary on-site potentials. The time-reversal symmetry is disrupted due to these on-site potentials. Exceptional points in a non-Hermitian system feature eigenvalue coalescence and non-trivial eigenvector degeneracies. Utilization of the rank-nullity theorem and graphical analysis of the phase rigidity factor enable identification of true exceptional points. Furthermore, this investigation achieves vectorized Zak phase quantization and examines a topolectric RLC circuit to derive the corresponding topological boundary resonance condition and the quantum Hall susceptance. Although Chern number quantization is not feasible, staggered hopping amplitudes corresponding to unit-cell lattice sites lead to broken inversion symmetry with non-zero Berry curvature, resulting in finite anomalous Nernst conductivity.

Figures

Figures reproduced from arXiv: 2506.16867 by the authors.

Figure 1
Figure 1. (a) A pictorial representation of the two-dimensional model square lattice, where A, B, C, and D correspond to unit-cell sites of the model (square) lattice with lattice constant ‘a’, starting with A in the upper right corner, B in the upper left corner, C in the lower left corner and D in the lower right corner. Here, 𝑢 ( 𝑡ଵ) stands for the hopping parameter along horizontal (x-) direction joining A and B (C and D)… view at source ↗
Figure 3
Figure 3. (a),(b),(c) The plots of 𝑃ଵ(𝜆ା) and 𝑃ଶ(𝜆ି) as functions of 𝛾 for μ = 0 involving staggered hopping amplitudes (u ≠ 𝑡ଵ and v ≠ 𝑡ଶ).The parameter values used are 𝑢 = 1,𝑡ଵ = 0.80, 𝑣 = 0.60, and 𝑡ଶ = 0.71. In Figures 3(d) - 3(f), there are plots of the same with the parameter values 𝑢 = 𝑡ଵ = 1, and 𝑣 = 𝑡ଶ = 0.75. The choice of the values leaves PT- and PH-symmetry unbroken. ation of such points. Unique to non-Hermitian … view at source ↗
Figure 4
Figure 4. The plots of the vectored Zak phase components 𝜙௫ and 𝜙௬ as a function of ௨ ௩ . In these figures, the hopping parameters 𝑢 = 𝑡ଵ and 𝑣 = 𝑡ଶ ensuring inversion symmetry protection. In Figures 4(a) and 4(b), we have assumed 𝛾 = 0, whereas in 4(c)and 4(d) we have assumed 𝛾 = 0.50. (B)Topolectric RLC Circuit An RLC circuit arranged in a lattice-like structure, with inductors, capacitors, and resistors forming a periodic … view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: (a)-(f) The plots of the real part of eigenvalues of the circuit Laplacian matrix in (13) as a function of ω in sିଵ. The parameter values are 𝐿ଵ = 0.1 𝑚𝐻 = 𝐿ଶ , 𝐶 = 0.01𝜇𝐹, and R = (1, 26,50) Ohm. (g) A plot of the imaginary component of the Berry phase for 𝐿ଵ = 0.1 𝑚𝐻…
Figure 6
Figure 6. Figure 6: (a) and (b)The plots of the Berry curvature in the z-direction as a function of (𝑎𝑘௫, 𝑎𝑘௬). The numerical values of the parameters used in the plots are 𝑢 = 1, 𝑡ଵ = 0.95, 𝑡ଶ = 0.50 , 𝑣 = 0.23 (0 .35), μ = 0, and 𝛾 = 0. (c), (d) and (e)The contour plots of the integrand…

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Works this paper leans on

56 extracted references · 44 canonical work pages

  1. [52]

    H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. 120, 146402 (2018)

    2018

  2. [1]

    S. H. Liu, W.L. Gao, Q. Zhang, et al., [J]. Research, 2019: 8609875 (2019)

    work page 2019

  3. [2]

    Zak Phase and Topolectric RLC Circuit (A)Zak Phase For non-Hermitian systems, the Zak phase undergoes modification due to the loss of time- reversal symmetry and the possibility of complex eigenvalues. In particular, the Zak phase for a 2D non-Hermitian SSH model represents a generalization of the 1D case (the Zak phase in 1 D is essentially an integral, ...

    work page

  4. [3]

    To resolve this, bi-orthonormal inner products have been employed in the analysis here without disrupting PT symmetry

    In non-Hermitian systems (𝛾 ≠ 0) , exceptional points and broken symmetries can complicate causing edge modes to lose robustness or abrogate BBC. To resolve this, bi-orthonormal inner products have been employed in the analysis here without disrupting PT symmetry. This facilitates precise predictions of edge states as shown in Figure 4(c) and 4(d). In thi...

    work page

  5. [4]

    Staggered Hopping Amplitudes he presence of non-zero Berry curvature in systems with broken TRS/IS gives rise to an anomalous velocity, generally, resulting in an anomalous transport current and intrinsic Hall conductivity. In this section, apart from disrupted TRS (𝛾≠ 0), we investigate a scenario where the hopping amplitudes are staggered ( 𝑢 ≠ 𝑡ଵ, and/...

    arXiv

  6. [5]

    Our model Hamiltonian's energy eigenvalues are given by the quartic 𝑄஺,௃,ఊ(𝜆)= 0

    Concluding remarks and future perspective This paper explores a 2D non-Hermitian variant of the SSH model. Our model Hamiltonian's energy eigenvalues are given by the quartic 𝑄஺,௃,ఊ(𝜆)= 0. We find that decreasing the numerical value of the tunable gain/loss parameter 𝛾 transforms the system from an insulator to a conductor. Notably, when 𝐽 = 0, 𝑄஺,ఊ(𝜆)= 0...

    work page

  7. [6]

    Liu, Phys

    F. Liu, Phys. Rev. B 108, 245140 (2023)

    work page 2023

  8. [7]

    W.P. Su, J.R. Schrieffer, A. J. and Heeger, Phys. Rev. Lett. 42, 1698 (1979)

    work page 1979

Show all 56 references
  1. [8]

    Orlov, G.V

    P. Orlov, G.V. Shlyapnikov, and D.V. Kurlov, Phys. Rev. B 111, L081105 (2025). 5.R. Nehra and D. Roy, Phys. Rev. B 105, 195407 (2022)

    2025

  2. [9]

    V. M. Alvarez, J. B. Vargas, and L. F. Torres, Phys. Rev. B 97, 121401 (2018)

    2018

  3. [10]

    Kawabata, T

    K. Kawabata, T. Bessho, and M. Sato, Phys. Rev. Lett. 123, 066405 (2019)

    2019

  4. [11]

    C. C. Ye, W.L. Vleeshouwers, et al., Phys. Rev. Research 6, 023202 (2024)

    2024

  5. [12]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. Ueda, Adv. Phys. 69 249 (2020)

    2020

  6. [13]

    Kawabata, K

    K. Kawabata, K. Shiozaki, et al., Phys. Rev. X 9 041015 (2019)

    2019

  7. [14]

    J. D. Lin, P. C. Kuo, et al., arXiv:2406.18362 (2024)

    2024 arXiv

  8. [15]

    Wojcik, K

    C.C. Wojcik, K. Wang, et al., Phys. Rev. B 106, L161401 (2022)

    2022

  9. [16]

    Alase, S

    A. Alase, S. Karuvade, and C. M. Scandolo, J. Phys. A: Math. Theor. 55 244003(2022)

    2022

  10. [17]

    Ryu, J.H

    J.W. Ryu, J.H. Han, et al., Commun Phys 7, 109 (2024)

    2024

  11. [18]

    C. M. Bender, Rep. Prog. Phys. 70 947 (2007)

    2007

  12. [19]

    Mandal and E

    I. Mandal and E. J. Bergholtz, Phys. Rev. Lett. 127 186601(2021)

    2021

  13. [20]

    J. Zak. Phys. Rev. Lett., 62 27472750 (1989)

    1989

  14. [21]

    Delplace, D

    P. Delplace, D. Ullmo, and G. Montambaux. Phys. Rev. B, 84 195452(2011). 19.J. Dong, V. Juricic, et al., Phys. Rev. Research 3 , 023056 (2021)

    2011

  15. [22]

    Hoffmann, T

    T. Hoffmann, T. Helbig, et al., Phys. Rev. Research2, 023265(2020). 21.D. Geng, H. Zhou, et al., Nat Commun 13, 7000 (2022)

    2020

  16. [24]

    Okuma, K

    N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. 124, 086801 (2020)

    2020

  17. [25]

    Kawabata, N

    K. Kawabata, N. Okuma, and M. Sato, Phys. Rev. B 101, 195147 (2020)

    2020

  18. [26]

    Zhang, T

    X. Zhang, T. Zhang, et al., Adv. Phys.: X 7, 2109431 (2022)

    2022

  19. [27]

    H. Gao, H. Xue , et al., arXiv:2007.01041 v1[cond-mat. mes- hall](2020)

    2020 arXiv

  20. [28]

    Yao and Z

    S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018)

    2018

  21. [29]

    S. Yao, F. Song, and Z. Wang, Phys. Rev. Lett. 121, 136802 (2018)

    2018

  22. [30]

    F. K. Kunst, E. Edvardsson, et al., Phys. Rev. Lett. 121, 026808 (2018)

    2018

  23. [31]

    Yokomizo and S

    K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019)

    2019

  24. [32]

    Yi and Z

    Y. Yi and Z. Yang, Phys. Rev. Lett. 125, 186802 (2020)

    2020

  25. [33]

    Z. Yang, K. Zhang, et al., Phys. Rev. Lett. 125, 226402 (2020)

    2020

  26. [35]

    Ghatak, M

    A. Ghatak, M. Brandenbourger, et al., Proc. Natl. Acad. Sci. U.S.A. 117, 29561 (2020). 36.L. Xiao, T. Deng, et al., arXiv:2009.07288v2 [quant-ph] (2021)

    2020 arXiv

  27. [37]

    K.G.Wilson, Phys. Rev. D 10, 2445 (1974)

    1974

  28. [38]

    H. Wang, X. Tang, H. Xu, et al. npj Quantum Mater. 7, 61 (2022)

    2022

  29. [39]

    Kim, and J

    M. Kim, and J. Rho, Nanophotonics, vol. 9, no. 10, 3227(2020)

    2020

  30. [40]

    C.H.Lee, S.Imhof, et al., Commun Phys 1, 39 (2018). 41.R. Yang Qi Xun, Molecular Frontiers Journal, 04(Supp01) 9 (2020)

    2018

  31. [42]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Rev. Mod. Phys., 93 025005, (2021). 43.J. Dong, V. Juričić, et al., Phys. Rev. Res., 3 023056 (2021)

    2021

  32. [44]

    J. C. Perez-Pedraza, and J. E. Barrios-Vargas, arXiv:2402.05261v1 [cond-mat.mtrl-sci] (2024)

    2024 arXiv

  33. [45]

    Fang, G.-Y

    A. Fang, G.-Y. Huang, et al., J. Phys. Commun.4 115006(2020)

    2020

  34. [46]

    Leykam, K

    D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, and F. Nori, Phys. Rev. Lett. 118, 040401 (2017)

    2017

  35. [47]

    T. Gao, E. Estrecho, K. Y. Bliokh, T. C. H. Liew, M. D. Fraser, S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Yamamoto,F.Nori, Y.S.Kivshar, A.G.Truscott, R.G.Dall, andE.A.Ostrovskaya, Nature 526, 554–558 (2015)

    2015

  36. [48]

    Septembre, P

    M.Król, I. Septembre, P. Oliwa, M. Kędziora, K. Łempicka-Mirek, M. Muszyński, R. Mazur, P. Morawiak, W. Piecek, P. Kula, W. Bardyszewski, P. G. Lagoudakis, D. D. Solnyshkov, G. Malpuech, B. Piętka, and J. Szczytko, Nat. Commun. 13, 5340 (2022)

    2022

  37. [49]

    H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Science 359, 1009–1012 (2018)

    2018

  38. [50]

    R. Su, E. Estrecho, D. Biegańska, Y. Huang, M. Wurdack, M. Pieczarka, A. G. Truscott, T. C. H. Liew, E. A. Ostrovskaya, and Q. Xiong, Sci. Adv. 7, eabj8905 (2021)

    2021

  39. [51]

    Zhang, Z

    K. Zhang, Z. Yang, and C. Fang, Phys. Rev. Lett. 125, 126402 (2020)

    2020

  40. [53]

    F. D. M. Haldane, Phys. Rev. Lett. 93, 206602(2004)

    2004

  41. [54]

    Chen and S

    M. Chen and S. Wan, J. Phys. Condens. Matter 24, 325502 (2012)

    2012

  42. [55]

    D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys.Rev. Lett. 97, 026603 (2006)

    2006

  43. [56]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, Q. Niu, Rev. Mod. Phys. 82, 1959-2007 (2010)

    2010

  44. [57]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005)

    2005

  45. [58]

    Str-el] (2024)

    Masaaki Nakamura and Shohei Masuda, arXiv:2401.12674v1[ cond-mat. Str-el] (2024)

    2024 arXiv

  46. [59]

    Asboth, L

    J.K. Asboth, L. Oroszl´any, et al.,, A short course on topological insulators, Lect. Notes Phys. 919, 1 (2016). 60.T.L. Hughes, E. Prodan, and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011). 61.G. van Miert, C. Ortix and C. Morais Smith, 2D Mater. 4, 015023(2017)

    2016

  47. [62]

    R. Lin, T. Tai, L. Li and C.H. Lee , Frontiers of Physics, 18(5), 53605 (2023)

    2023

  48. [63]

    Hou, G.Wu, et al., Phys

    C. Hou, G.Wu, et al., Phys. Rev. B 109, 205135 (2024). Appendix A The right eigenstates linked to the eigenvalues of (5) could be written down in an explicit manner as |𝑢(௝)൫𝑘௫,𝑘௬ ൯ൿ= 𝑁𝑗0 −1 2ф𝑗൫𝑘𝑥,𝑘𝑦൯, where ф௝൫𝑘௫,𝑘௬൯ is the transpose of the row vector ( 𝜓ଵ (௝)(𝒌) 𝜓ଶ (௝)(𝒌) 𝜓...

    2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.