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arxiv: 2604.21562 · v1 · submitted 2026-04-23 · ❄️ cond-mat.mes-hall

Magnetic-flux tunable electronic transport through domain walls in a three-dimensional second-order topological insulator

Zhe Hou , Ai-Min Guo This is my paper

Pith reviewed 2026-05-09 21:08 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords second-order topological insulatordomain wall transportAharonov-Bohm oscillationtopological hinge statesnanowiremagnetic fluxconductanceinterference
0
0 comments X

The pith

Magnetic flux through a domain wall produces sinusoidal Aharonov-Bohm oscillations in topological hinge state conductance.

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

This paper establishes that in a three-dimensional second-order topological insulator nanowire, a magnetic domain wall creates a loop of boundary states that connect the counterpropagating hinge states. A uniform magnetic field parallel to the wire threads flux through this loop, leading to perfect sinusoidal oscillations in the two-terminal conductance according to a specific formula involving the flux quantum. A sympathetic reader would care because this offers an experimentally accessible way to control and potentially detect these protected one-dimensional states using magnetic fields rather than more invasive methods. The work extends the idea by considering multiple domain walls and observing flux-tunable Fabry-Perot oscillations.

Core claim

Due to the sign reversal of magnetization across the domain wall, four one-dimensional topological boundary states form an enclosed loop that mediates the counterpropagating topological hinge states. Applying a uniform magnetic field parallel to the nanowire threads magnetic flux through the domain wall, producing a perfect sinusoidal Aharonov-Bohm oscillation in the two-terminal conductance G formulated as G equals e squared over 2h times one minus cosine of pi times Phi over Phi zero. This interference is attributed to the pi-spin rotation of the hinge states traveling through the domain wall, as captured by a phenomenological scattering matrix approach.

What carries the argument

The enclosed loop of four 1D topological boundary states at the domain wall, which enables interference of counterpropagating hinge states under threaded magnetic flux.

If this is right

  • Conductance reaches maximum at half flux quantum and minimum at zero flux.
  • The oscillation period corresponds to a flux quantum of h/2e rather than h/e.
  • In a double domain wall setup, the conductance shows Fabry-Perot resonances whose minima shift with applied flux.
  • This setup allows fine control of quantum transport in topological systems via external magnetic flux.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The domain wall loop could serve as a basic building block for more complex interferometric devices in topological materials.
  • Similar flux-tunable effects might appear in other second-order topological systems with engineered domain structures.
  • Experimental detection of these oscillations would confirm the presence and coherence of the hinge states.

Load-bearing premise

The domain wall's magnetization reversal produces exactly four one-dimensional topological boundary states that form a perfect closed loop without additional scattering channels.

What would settle it

Observing the two-terminal conductance versus parallel magnetic field at low temperature and finding that it does not follow the exact sinusoidal dependence G = e²/2h [1 - cos(π Φ/Φ₀)] would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.21562 by Ai-Min Guo, Zhe Hou.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic diagram of the magnetic SOTI trans [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Conductance [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Schematic illustration of the propagation paths [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Schematic diagram of the double-DW junction [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Length [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) and (b) Schematic diagrams of the Bloch wall [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Distribution of the LDOSs near the DW for an [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Ensemble-averaged conductance [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original abstract

The three-dimensional (3D) topological insulators (TIs), hosting topologically protected helical surface states, can be promoted into second-order TIs when a diagonal Zeeman term, typical of magnetic doping, is introduced. The latter hosts exotic chiral one-dimensional (1D) topological hinge states (THSs). In this paper, we investigate the electronic transport of THSs through a magnetic domain wall (DW) in a 3D TI nanowire. Due to the sign reversal of the out-of-plane magnetization across the DW, four 1D topological boundary states, residing on the edge of the DW, arise and form an enclosed loop mediating the counterpropagating THSs. By applying a uniform magnetic field parallel to the nanowire, we obtain a perfect sinusoidal Aharonov-Bohm oscillation in the two-terminal conductance $G$, formulated by $G=\frac{e^2}{2h} \left[ 1- \cos(\pi \Phi/\Phi_0) \right]$, with $\Phi$ the magnetic flux through the DW and $\Phi_0 = h/2e$ the flux quantum. Applying a phenomenological scattering matrix approach, we explain this novel Aharonov-Bohm oscillation perfectly, and attribute the constructive (destructive) interference of transmission at $\Phi = \Phi_0$ (0) to the $\pi$-spin rotation of the THSs traveling through the DW. Extending our study to a double-DW junction, where the central region has antiparallel magnetization to the leads, we observe Fabry-P{\'e}rot oscillations, in which the conductance minima are tuned by the magnetic flux. Our findings open a new avenue for finely controlling the quantum transport of THSs in magnetic systems using magnetic flux, and provide a faithful way for detecting THSs in experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies two-terminal electronic transport of topological hinge states (THSs) through a magnetic domain wall (DW) in a 3D second-order topological insulator nanowire. It posits that the sign reversal of out-of-plane magnetization across the DW generates four 1D topological boundary states that close into a loop; a uniform axial magnetic field then produces perfect sinusoidal Aharonov-Bohm oscillations in the conductance, given by G = (e²/2h)[1 − cos(π Φ/Φ₀)], which the authors attribute to a π-spin rotation of the THSs. The result is obtained via a phenomenological scattering-matrix construction; the study is extended to a double-DW junction that exhibits flux-tunable Fabry-Pérot oscillations.

Significance. If the functional form and the underlying interference mechanism can be placed on a microscopic footing, the work would supply a concrete, flux-tunable signature for THSs and a practical route to control their transport in magnetic nanostructures, which is of clear interest to the mesoscopic-topology community.

major comments (2)
  1. [scattering-matrix construction and single-DW conductance formula] The central conductance formula (abstract and § on single-DW transport) is obtained by inserting a phenomenological scattering matrix whose off-diagonal blocks encode a fixed π-spin rotation. No explicit projection of the 3D Hamiltonian onto the four boundary modes, no recursive Green’s-function calculation, and no lattice-model benchmark that includes the Zeeman term, the DW profile, and the Peierls phases for the applied flux are provided. Consequently the precise location of the conductance zeros, the absence of higher harmonics, and the perfect sinusoid rest on an unverified modeling choice rather than a controlled limit of the microscopic theory.
  2. [model definition and DW boundary-state counting] The claim that the sign reversal of the out-of-plane magnetization produces exactly four 1D topological boundary states that form a closed loop mediating the counter-propagating THSs (abstract and introductory model section) is stated without a derivation of the mode spectrum or an explicit count of the protected states from the bulk-boundary correspondence of the second-order TI. This assumption is load-bearing for the subsequent interference picture.
minor comments (2)
  1. [abstract] The abstract contains a LaTeX rendering artifact in “Fabry-Pérot” that should be corrected for the published version.
  2. [conductance formula] Notation for the flux quantum is introduced as Φ₀ = h/2e; consistency with the conventional superconducting flux quantum should be checked or explicitly justified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

  1. Referee: [scattering-matrix construction and single-DW conductance formula] The central conductance formula (abstract and § on single-DW transport) is obtained by inserting a phenomenological scattering matrix whose off-diagonal blocks encode a fixed π-spin rotation. No explicit projection of the 3D Hamiltonian onto the four boundary modes, no recursive Green’s-function calculation, and no lattice-model benchmark that includes the Zeeman term, the DW profile, and the Peierls phases for the applied flux are provided. Consequently the precise location of the conductance zeros, the absence of higher harmonics, and the perfect sinusoid rest on an unverified modeling choice rather than a controlled limit of the microscopic theory.

    Authors: We acknowledge that our scattering-matrix construction is phenomenological. It is motivated by the spin-momentum locking of the hinge states together with the π spin rotation that must accompany traversal of the magnetization-reversal domain wall; this rotation, combined with the Aharonov-Bohm phase, produces the exact sinusoidal form and the absence of higher harmonics. In the revised manuscript we have added an expanded paragraph in the single-DW section that derives the matrix elements from symmetry and spin-texture considerations, and we have included a brief appendix sketching how the same interference arises in a minimal two-mode effective model. A full lattice Green’s-function calculation with explicit Peierls phases remains outside the present scope but is noted as desirable future work. revision: partial

  2. Referee: [model definition and DW boundary-state counting] The claim that the sign reversal of the out-of-plane magnetization produces exactly four 1D topological boundary states that form a closed loop mediating the counter-propagating THSs (abstract and introductory model section) is stated without a derivation of the mode spectrum or an explicit count of the protected states from the bulk-boundary correspondence of the second-order TI. This assumption is load-bearing for the subsequent interference picture.

    Authors: The four domain-wall states follow from the change in the second-order topological invariant across the magnetization reversal, which, by the bulk-boundary correspondence, binds one-dimensional modes along the perimeter of the wall. In the revised manuscript we have inserted a concise derivation in the model section that starts from the effective hinge Hamiltonian, shows how the sign flip of the Zeeman term generates the additional pair of counter-propagating modes, and counts the four states that close into the loop. We also cite the standard literature on second-order TIs to anchor the correspondence argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central result follows from explicit phenomenological model without reduction to inputs by construction.

full rationale

The paper introduces a phenomenological scattering matrix to model transmission through the domain wall, motivated by the sign reversal of magnetization creating four 1D boundary states that form a loop. The conductance formula G = (e²/2h)[1 - cos(π Φ/Φ₀)] is obtained directly from this matrix by encoding a π spin rotation in its off-diagonal elements, leading to the sinusoidal Aharonov-Bohm oscillation. No step equates a prediction to a fitted parameter, self-defines quantities, or relies on a load-bearing self-citation whose validity reduces to the present work. The model assumptions are stated separately from the output formula, and the derivation remains self-contained within those assumptions rather than tautological. External benchmarks or microscopic checks are not required for the circularity assessment, which finds none.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard models of 3D topological insulators with Zeeman terms and assumptions about domain-wall-induced states; no explicit free parameters, new entities, or ad-hoc axioms are detailed in the abstract.

axioms (2)
  • domain assumption Introduction of a diagonal Zeeman term promotes 3D TI to second-order TI hosting chiral 1D THSs.
    Standard assumption in magnetic doping literature for TIs, invoked to set up the hinge states.
  • domain assumption Sign reversal of out-of-plane magnetization across DW creates four 1D topological boundary states forming an enclosed loop.
    Central mechanism stated in abstract for mediating counterpropagating THSs.

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Reference graph

Works this paper leans on

68 extracted references · 68 canonical work pages

  1. [1]

    M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, and F. Petroff, Phys. Rev. Lett.61, 2472 (1988), Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic Superlattices

    work page 1988

  2. [2]

    Binasch, P

    G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B39, 4828(R) (1989),Enhanced magnetore- sistance in layered magnetic structures with antiferro- magnetic interlayer exchange

    work page 1989

  3. [3]

    A. E. Berkowitz, J. R. Mitchell, M. J. Carey, A. P. Young, S. Zhang, F. E. Spada, F. T. Parker, A. Hutten, and G. Thomas, Phys. Rev. Lett.68, 3745 (1992),Giant Mag- netoresistance in Heterogeneous Cu-Co Alloys

    work page 1992

  4. [4]

    John Q. Xiao, J. Samuel Jiang, and C. L. Chien, Phys. Rev. Lett.68, 3746 (1992),Giant magnetoresistance in nonmultilayer magnetic systems

    work page 1992

  5. [5]

    A. Fert, A. Barth´ el´ emy, P. Galtier, P. Holody, R. Loloee, R. Morel, F. P´ etroff, P. Schroeder, L. B. Steren, and T. Valet, Mater. Sci. and Engineering B31, 1-9 (1995),Gi- ant magnetoresistance in magnetic nanostructures. Re- cent developments

    work page 1995

  6. [6]

    Ennen, D

    I. Ennen, D. Kappe, T. Rempel, C. Glenske, and A. H¨ utten, Sensors16, 904 (2016),Giant Magnetoresis- tance: Basic Concepts, Microstructure, Magnetic Inter- actions and Applications. 10

    work page 2016

  7. [7]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 226801 (2005),Quantum Spin Hall Effect in Graphene

    work page 2005

  8. [8]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 146802 (2005),Z 2 Topological Order and the Quantum Spin Hall Effect

    work page 2005

  9. [9]

    L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett.98, 106803 (2007),Topological Insulators in Three Dimen- sions

    work page 2007

  10. [10]

    Andrei Bernevig, T

    B. Andrei Bernevig, T. L. Hughes, and S.-C. Zhang, Sci- ence314, 1757 (2006),Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

    work page 2006

  11. [11]

    K¨ onig, S

    M. K¨ onig, S. Wiedmann, C. Br¨ une, A. Roth, H. Buh- mann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science318, 766 (2007),Quantum spin hall insulator state in HgTe quantum wells

    work page 2007

  12. [12]

    Liu, X.-L

    C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. Lett.101, 146802 (2008),Quantum Anoma- lous Hall Effect inHg 1−yMnyTeQuantum Wells

    work page 2008

  13. [13]

    R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Science329, 61-64 (2010),Quantized anomalous Hall effect in magnetic topological insulators

    work page 2010

  14. [14]

    Chang, J

    C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang,et al., Science 340, 167-170 (2013),Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topolog- ical Insulator

    work page 2013

  15. [15]

    A. J. Bestwick, E. J. Fox, X. Kou, L. Pan, K.-L. Wang, and D. Goldhaber-Gordon, Phys. Rev. Lett.114, 187201 (2015),Precise Quantization of the Anomalous Hall Ef- fect near Zero Magnetic Field

    work page 2015

  16. [16]

    Shamim, W

    S. Shamim, W. Beugeling, P. Shekhar, K. Bendias, L. Lunczer, J. Kleinlein, H. Buhmann, and L. W. Molenkamp, Nat. Commun.12, 3193 (2021),Quantized spin Hall conductance in a magnetically doped two di- mensional topological insulator

    work page 2021

  17. [17]

    Y. S. Hor, P. Roushan, H. Beidenkopf, J. Seo, D. Qu, J. G. Checkelsky, L. A. Wray, D. Hsieh, Y. Xia, S.-Y. Xu,et al., Phys. Rev. B81, 195203 (2010),Develop- ment of ferromagnetism in the doped topological insulator Bi2−xMnxTe3

    work page 2010

  18. [18]

    S.-Y. Xu, M. Neupane, C. Liu, D. Zhang, A. Richardella, L. A. Wray, N. Alidoust, M. Leandersson, T. Balasub- ramanian, Jaime S´ anchez-Barriga,et al., Nat. Phys.8, 616-622 (2012),Hedgehog spin texture and Berry’s phase tuning in a magnetic topological insulator

    work page 2012

  19. [19]

    E. D. L. Rienks, S. Wimmer, J. S´ anchez-Barriga, O. Caha, P. S. Mandal, J. R˚ uˇ ziˇ cka, A. Ney, H. Steiner, V. V. Volobuev, H. Groiss,et al., Nature576, 423- 428 (2019),Large magnetic gap at the Dirac point in Bi2Te3/MnBi2Te4 heterostructures

    work page 2019

  20. [20]

    Nomura and N

    K. Nomura and N. Nagaosa, Phys. Rev. B82, 161401(R) (2010),Electric charging of magnetic textures on the sur- face of a topological insulator

    work page 2010

  21. [21]

    J. G. Checkelsky, J. Ye, Y. Onose, Y. Iwasa, and Y. Tokura, Nat. Phys.8, 729-733 (2012),Dirac-fermion- mediated ferromagnetism in a topological insulator

    work page 2012

  22. [22]

    Wakatsuki, M

    R. Wakatsuki, M. Ezawa, and N. Nagaosa, Sci. Rep.5, 13638 (2015),Domain wall of a ferromagnet on a three- dimensional topological insulator

    work page 2015

  23. [23]

    Yasuda, M

    K. Yasuda, M. Mogi, R. Yoshimi, A. Tsukazaki, K. S. Takahashi, M. Kawasaki, F. Kagawa, and Y. Tokura, Sci- ence358, 1311-1314 (2017),Quantized chiral edge con- duction on domain walls of a magnetic topological insu- lator

    work page 2017

  24. [24]

    Y.-F. Zhou, Z. Hou, and Q.-F. Sun, Phys. Rev. B98, 165433 (2018),Configuration-sensitive transport at the domain walls of a magnetic topological insulator

    work page 2018

  25. [25]

    Sedlmayr, N

    M. Sedlmayr, N. Sedlmayr, J. Barna´ s, and V. K. Dugaev, Phys. Rev. B101, 155420 (2020),Chiral Hall effect in the kink states in topological insulators with magnetic domain walls

    work page 2020

  26. [26]

    N.-X. Yang, Q. Yan, and Q.-F. Sun, Phys. Rev. B 102, 245412 (2020),Linear and nonlinear thermoelectric transport in a magnetic topological insulator nanoribbon with a domain wall

    work page 2020

  27. [27]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science357, 61-66 (2017),Quantized electric multipole insulators

    work page 2017

  28. [28]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B96, 245115 (2017),Electric multipole mo- ments, topological multipole moment pumping, and chiral hinge states in crystalline insulators

    work page 2017

  29. [29]

    Z.-X. Li, Z. Wang, Z. Zhang, Y. Cao, and P. Yan, Phys. Rev. B103, 214442 (2021),Third-order topological insu- lator in three-dimensional lattice of magnetic vortices

    work page 2021

  30. [30]

    Schindler, A

    F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Sci. Adv. 4, eaat0346 (2018),Higher-order topological insulators

    work page 2018

  31. [31]

    Schindler, Z

    F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook, A. Murani, S. Sengupta, A. Yu. Kasumov, R. Deblock, S. Jeon,et al., Nat. Phys.14, 918–924 (2018),Higher-order topology in bismuth

    work page 2018

  32. [32]

    Noguchi, M

    R. Noguchi, M. Kobayashi, Z. Jiang, K. Kuroda, T. Takahashi, Z. Xu, D. Lee, M. Hirayama, M. Ochi, T. Shirasawa,et al., Nat. Mater.20, 473–479 (2021),Evi- dence for a higher-order topological insulator in a three- dimensional material built from van der Waals stacking of bismuth-halide chains

    work page 2021

  33. [33]

    Langbehn, Y

    J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Phys. Rev. Lett.119, 246401 (2017), Reflection-Symmetric Second-Order Topological Insula- tors and Superconductors

    work page 2017

  34. [34]

    Y. Ren, Z. Qiao, and Q. Niu, Phys. Rev. Lett.124, 166804 (2020),Engineering Corner States from Two- Dimensional Topological Insulators

    work page 2020

  35. [35]

    Z. Hou, C. S. Weber, D. M. Kennes, D. Loss, H. Schoeller, J. Klinovaja, and M. Pletyukhov, Phys. Rev. B107, 075437 (2023),Realization of a three-dimensional quan- tum Hall effect in a Zeeman-induced second-order topo- logical insulator on a torus

    work page 2023

  36. [36]

    Geier, L

    M. Geier, L. Trifunovic, M. Hoskam, and P. W. Brouwer, Phys. Rev. B97, 205135 (2018),Second-order topological insulators and superconductors with an order-two crys- talline symmetry

    work page 2018

  37. [37]

    Z. Song, Z. Fang, and C. Fang, Phys. Rev. Lett.119, 246402 (2017),(d-2)-Dimensional Edge States of Rota- tion Symmetry Protected Topological States

    work page 2017

  38. [38]

    C. S. Weber, M. Pletyukhov, Z. Hou, D. M. Kennes, J. Klinovaja, D. Loss, and H. Schoeller, Phys. Rev. B107, 235402 (2023),Second-order topology and supersymmetry in two-dimensional topological insulators

    work page 2023

  39. [39]

    Wang, Y.-B

    J.-H. Wang, Y.-B. Yang, N. Dai, and Y. Xu, Phys. Rev. Lett.126, 206404 (2021),Structural-Disorder-Induced Second-Order Topological Insulators in Three Dimen- sions

    work page 2021

  40. [40]

    B. A. Levitan, and T. Pereg-Barnea, Phys. Rev. Res.2, 033327 (2020),Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport. 11

    work page 2020

  41. [41]

    Queiroz and A

    R. Queiroz and A. Stern, Phys. Rev. Lett.123, 036802 (2019),Splitting the Hinge Mode of Higher-Order Topo- logical Insulators

    work page 2019

  42. [42]

    K. Luo, H. Geng, L. Sheng, W. Chen, and D. Y. Xing, Phys. Rev. B104, 085427 (2021),Aharonov-Bohm effect in three-dimensional higher-order topological insulators

    work page 2021

  43. [43]

    Li, S.-B

    C.-A. Li, S.-B. Zhang, J. Li, and B. Trauzettel, Phys. Rev. Lett.127, 026803 (2021),Higher-Order Fabry-P´ erot Interferometer from Topological Hinge States

    work page 2021

  44. [44]

    Brouwer, and Nicholas Sedl- mayr, Phys

    Adam Yanis Chaou, Piet W. Brouwer, and Nicholas Sedl- mayr, Phys. Rev. B107, 035430 (2023),Hinge states of second-order topological insulators as a Mach-Zehnder interferometer

    work page 2023

  45. [45]

    Zhang, C.-X

    H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and S.-C. Zhang, Nat. Phy.5, 438-442 (2009),Topological insula- tors in Bi 2Se3, Bi 2Te3 and Sb 2Te3 with a single Dirac cone on the surface

    work page 2009

  46. [46]

    Liu,1 X.-L

    C.-X. Liu,1 X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.- C. Zhang, Phys. Rev. B82, 045122 (2010),Model Hamil- tonian for topological insulators

    work page 2010

  47. [47]

    Qi, and S.-C

    X.-L. Qi, and S.-C. Zhang, Rev. Mod. Phys.83, 1057 (2011),Topological insulators and superconductors

    work page 2011

  48. [48]

    Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nat. Phys.5, 398 (2009),Observation of a large- gap topological-insulator class with a single Dirac cone on the surface

    work page 2009

  49. [49]

    Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, Z.-X. Shen, Science 325, 178 (2009),Experimental Realization of a Three- Dimensional Topological Insulator, Bi 2Te3

    work page 2009

  50. [50]

    Hosur, S

    P. Hosur, S. Ryu, and A. Vishwanath, Phys. Rev. B81, 045120 (2010),Chiral topological insulators, superconduc- tors, and other competing orders in three dimensions

    work page 2010

  51. [51]

    Hosur, P

    P. Hosur, P. Ghaemi, R. S. K. Mong, and A. Vishwanath, Phys. Rev. Lett.107, 097001 (2011),Majorana Modes at the Ends of Superconductor Vortices in Doped Topological Insulators

    work page 2011

  52. [52]

    Shen, Topological Insulators: Dirac Equation In Condensed Matters, Vol

    S.-Q. Shen, Topological Insulators: Dirac Equation In Condensed Matters, Vol. 174 (Springer Science & Busi- ness Media, 2013)

    work page 2013

  53. [53]

    Jackiw and C

    R. Jackiw and C. Rebbi, Phys. Rev. D13, 3398 (1976), Solitons with fermion number 1/2

    work page 1976

  54. [54]

    M. P. Lopez Sancho, J. M. Lopez Sancho, J. M. L. Sancho and J Rubio, J. Phys. F: Met. Phys.15, 851 (1985), Highly convergent schemes for the calculation of bulk and surface Green functions

    work page 1985

  55. [55]

    Meir and N

    Y. Meir and N. S. Wingreen, Phys. Rev. Lett.68, 2512 (1992),Landauer Formula for the Current through an Interacting Electron Region

    work page 1992

  56. [56]

    Jauho, N

    A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 505528 (1994),Time-dependent Transport in Interacting and Noninteracting Resonant-tunneling Systems

    work page 1994

  57. [57]

    Datta,Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995), pp

    S. Datta,Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995), pp. 133–136

    work page 1995

  58. [58]

    M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Es- tyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. B. Wolter, A. V. Koroleva, A. M. Shikin,et al., Nature576, 416- 422 (2019),Prediction and observation of an antiferro- magnetic topological insulator

    work page 2019

  59. [59]

    Zhang, M

    D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Phys. Rev. Lett.122, 206401 (2019),Topological Axion States in the Magnetic Insulator MnBi 2Te4 with the Quantized Magnetoelectric Effect

    work page 2019

  60. [60]

    Hai-Peng Sun, C. M. Wang, Song-Bo Zhang, Rui Chen, Yue Zhao, Chang Liu, Qihang Liu, Chaoyu Chen, Hai- Zhou Lu, and X. C. Xie, Phys. Rev. B102, 241406(R) (2020),Analytical solution for the surface states of the antiferromagnetic topological insulator MnBi2Te4

    work page 2020

  61. [61]

    Choi, H.-P

    S.-J. Choi, H.-P. Sun, and B. Trauzettel, Phys. Rev. B 107, 235415 (2023),Conductance oscillations of antifer- romagnetic layer tunnel junctions

    work page 2023

  62. [62]

    Sun, C.-A

    H.-P. Sun, C.-A. Li, S.-J. Choi, S.-B. Zhang, H.-Z. Lu, and B. Trauzettel, Phys. Rev. Research5, 013179 (2023), Magnetic topological transistor exploiting layer-selective transport

    work page 2023

  63. [63]

    Chen, H.-P

    R. Chen, H.-P. Sun, M. Gu, C.-B. Hua, Q. Liu, H.-Z. Lu, and X. C. Xie, Natl. Sci. Rev.11, nwac140 (2024), Layer Hall effect induced by hidden Berry curvature in antiferromagnetic insulators

    work page 2024

  64. [64]

    Zhang, K

    Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan,et al., Nat. Phys.6, 584-588 (2010),Crossover of the three-dimensional topological insulator Bi2Se3 to the two- dimensional limit

    work page 2010

  65. [65]

    Lin, H.-P

    H.-J. Lin, H.-P. Sun, T. Liu, and P.-L. Zhao, Phys. Rev. B108, 165427 (2023),Tuning three-dimensional higher- order topological insulators by surface state hybridization

    work page 2023

  66. [66]

    Long, Q.-F

    W. Long, Q.-F. Sun, and J. Wang, Phys. Rev. Lett.101, 166806 (2008),Disorder-Induced Enhancement of Trans- port through Graphenep−nJunctions

    work page 2008

  67. [67]

    Y. Long, M. Wei, F. Xu, and J. Wang, Phys. Rev. B 111, 035428 (2025),Scaling behavior and emergent con- ductance plateau with uniform conductance distribution in disordered topological insulators with a domain wall structure

    work page 2025

  68. [68]

    Hou, Codes for the paper: figshare (2025), https://doi.org/10.6084/m9.figshare.29039711.v1

    Z. Hou, Codes for the paper: figshare (2025), https://doi.org/10.6084/m9.figshare.29039711.v1

    work page doi:10.6084/m9.figshare.29039711.v1 2025