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Generalized bulk-interface correspondence for non-quantized spin transport

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that for nonconserved spin, the difference of bulk spin conductances across an interface exactly equals the interface spin-drift conductance plus the spin-torque conductance.

desk verdict First rigorous bulk-interface correspondence for non-quantized spin transport, with a plausible central theorem whose proof has a real gap at Eq. (7.13) — worth refereeing, but the advertised equality is not yet fully certified. read the letter →

arxiv 2505.16331 v1 pith:W4V4SDLP submitted 2025-05-22 math-ph math.MP

Jiayu Qiu , Hai Zhang This is my paper
classification math-phmath.MP
keywords bulk-interfacecorrespondencespintransporttorquenonconservedprincipal-valuetracetight-bindingmodelconductancequantumHalleffect
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 establishes a generalized bulk-interface correspondence for spin transport when spin is not conserved. The main equality, Theorem 1.11, says that the difference between the bulk spin conductances on the two sides of an interface is always balanced by two interface quantities: spin carried by in-gap interface modes and spin generated by torque near the interface. This extends the classical bulk-interface correspondence, previously tied to quantized Chern or $\mathbb{Z}_2$ invariants, to systems where the Hamiltonian and spin operator do not commute. The payoff is that an interface between materials with unequal bulk spin conductances must be physically visible: either interface modes appear or a spin torque can be detected near the interface.

What carries the argument

The load-bearing object is the potential-current correlation $\Sigma^{\Lambda_2,\Lambda_1 S}_\pm$ built from resolvents and the almost-analytic extension of $\rho$, whose principal-value trace (Definition 1.3) defines the bulk spin conductance. The essential mechanism that makes the definitions convergent is the mesoscopic vanishing of the bulk spin-torque response, Proposition 3.2: $\operatorname{Tr}_{D_\pm}(1_{\Omega_{n,k a_\pm^1}}\Sigma^{\Lambda_2,S}_\pm)=0$ for every strip. This exact zero removes the nonlocal part of $[H_\pm,S]$ and turns a non-trace-class correlation into one with a well-defined principal-value trace. The proof of Theorem 1.11 then cuts the interface torque correlation into right-bulk, left-bulk, and box pieces, each converging to the corresponding bulk or interface conductance as the box radius tends to infinity.

What would settle it

Compute, for a tight-binding model with $[H_\pm,S]\neq 0$, the per-strip trace $\operatorname{Tr}_{D_\pm}(1_{\Omega_{n,k a_\pm^1}}\Sigma^{\Lambda_2,S}_\pm)$; a single nonzero value would invalidate the principal-value trace definition and Theorem 1.11. A less direct check is to build an interface with $\sigma^{\Lambda_2}_+ \neq \sigma^{\Lambda_2}_-$ and look for a case where neither $\rho'(H_e)$ nor $\sigma^{\mathrm{torque}}_e$ is nonzero.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.11, is the identity $\sigma^{\mathrm{drift},\Lambda_2,\rho}_e + \sigma^{\mathrm{torque},\Lambda_2,\rho}_e = \sigma^{\Lambda_2}_+ - \sigma^{\Lambda_2}_-$ for any switch function $\Lambda_2$ and density $\rho$ satisfying (1.7). The bulk spin conductance $\sigma^{\Lambda_2}_\pm$ is defined as a principal-value trace of a potential-current correlation and is generally not quantized when $[H,S]\neq 0$. The interface spin-drift conductance $\sigma^{\mathrm{drift},\Lambda_2,\rho}_e$ measures spin transported by interface modes, while the spin-torque conductance $\sigma^{\mathrm{torque},\Lambda_2,\rho}_e$ measures spin created near the interface because the spin axis is not preserved. When $[H,S]=0$ the torque term vanishes and the identity reduces to the known bulk-interface correspondence based on the spin Chern number or the Fu-Kane-Mele $\mathbb{Z}_2$ index. The proof is a conservation-law bookkeeping in a box around the interface, with each term identified as a separately convergent contribution.

Load-bearing premise

The proof needs the average spin-torque response in every bulk strip to be exactly zero, not merely small; if this mesoscopic trace vanishes only approximately, the definitions of the conductances and the main equality would break.

Editorial extensions

If this is right

  • If the bulk spin conductances differ, $\sigma^{\Lambda_2}_+ \neq \sigma^{\Lambda_2}_-$, the interface is necessarily nontrivial: either in-gap interface modes exist ($\rho'(H_e)\neq 0$) or spin torque appears ($\sigma^{\mathrm{torque}}_e\neq 0$).
  • The bulk spin conductance is a legitimate bulk character even without a quantized invariant, and its difference across an interface determines the interface spin transport exactly.
  • When spin is conserved, the torque term vanishes and the theorem reproduces the established spin-Chern and $\mathbb{Z}_2$ bulk-interface correspondence.
  • The bulk spin conductance is independent of the switch function $\Lambda_1$, the density $\rho$, and the strip size $k$, although it may depend on $\Lambda_2$ when spin is nonconserved.
  • The interface spin-torque term gives a concrete, calculable spin response that could serve as a signature of the bulk mismatch in spin-orbit-coupled materials.

Reading between the lines

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

  • This suggests a concrete numerical test: in a spin-orbit-coupled tight-binding model, the computed bulk mismatch $\sigma^{\Lambda_2}_+ - \sigma^{\Lambda_2}_-$ should show up as either interface modes or a nonzero torque, with no silent interface.
  • The exact vanishing in Proposition 3.2 might be relaxable to exponentially decaying per-strip averages, giving an approximate bulk-interface correspondence with controlled error; the paper does not claim this.
  • Porting the bookkeeping to orbital angular momentum transport is a natural next step, but the position dependence of $O$ blocks the vanishing-torque step; the paper suggests reflection symmetry as a plausible remedy.
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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 / 4 minor

Summary. The paper develops a rigorous bulk-interface correspondence for spin transport in two-dimensional tight-binding systems where the spin operator does not commute with the Hamiltonian. It defines a bulk spin conductance as a principal-value trace of a potential-current correlation, an interface spin-drift conductance using rho'(H_e), and an interface spin-torque conductance from a correlation involving [H_e,S]. The main theorem, Theorem 1.11, states that the difference of bulk spin conductances across an interface equals the sum of the interface drift and torque conductances. The paper also shows that for conserved spin the construction reduces to the standard spin Chern number or Fu-Kane-Mele Z2 correspondence. The proof is organized around a conservation-law identity, with bulk-to-interface replacement estimates and an exact vanishing of mesoscopic spin-torque response in the bulk.

Significance. If correct, the theorem provides a meaningful generalization of the bulk-interface correspondence beyond quantized invariants: a nonzero difference of bulk spin conductances forces either in-gap interface transport or spin-torque response near the interface. The paper contains a genuine analytic framework, including principal-value traces, conditional cyclicity, detailed kernel estimates, and explicit proofs of well-definedness of the interface torque conductance. The authors are transparent about the places where technical estimates are skipped in Section 7. However, the unproved and possibly mis-signed Step 4 of Theorem 1.11 is central, so the paper cannot be accepted without a complete and corrected proof of Eq. (7.13).

major comments (3)
  1. [Section 7, Step 4, Eq. (7.13)] The printed derivation of Eq. (7.13) is not a proof. The text states 'we will avoid detailed technical arguments' and then applies cyclicity to products whose individual factors, e.g. R_e(z)[H_e,Λ2]R_e(z) and [H_e,S1_{Ω_R}], are not separately trace-class; only the full products with both x1- and x2-localizations are trace-class. The final displayed identity before Eq. (7.13) gives Tr_pv(...) = iπ Tr({{H_e,Λ2}, S1_{Ω_R}}ρ'(H_e)). For R→∞ this equals iπ Tr({{H_e,Λ2},S}ρ'(H_e)), while Eq. (7.13) requires -2πσ_drift = 2π i Tr({{H_e,Λ2},S}ρ'(H_e)) by Definition 1.8. Thus the factor 2, and possibly the sign depending on the normalization in Eq. (2.2), cannot be certified from the text. Since Eq. (1.19) is an equality of three separately defined conductances, a factor or sign error here invalidates the central claim.
  2. [Section 7, Steps 2.2 and 3, Eqs. (7.5), (7.9), (7.11)] The limit identities that replace H_e by H_+ or H_- in principal-value traces are asserted with 'details are skipped' or 'we only sketch the main steps.' These are not routine glosses: they require uniform trace-norm bounds with polynomial dependence on |Im z|^{-1}, exponential decay in R or N, and justification of the order of limits in Step 3, where N is fixed before taking N→∞. Because Theorem 1.11 is the sum of these limits, the proof should contain these estimates explicitly.
  3. [Section 3 and Eq. (1.14)] The exact vanishing of the mesoscopic spin-torque response is load-bearing: it is used to define the principal-value trace in Definition 1.5, to prove Proposition 1.6, and to eliminate the bulk contributions in Steps 2 and 3 of Theorem 1.11. Proposition 3.1 is delegated to [37, Theorem 2.8]. Since this is an exact zero and the whole framework would break if it held only approximately, the paper should either state the cited result as a lemma with a proof or clearly identify it as an imported theorem, and verify that the cited statement indeed covers the present noncommuting Hamiltonian and spin operator.
minor comments (4)
  1. [Throughout] The name 'Helffer-Sjöstrand' is misspelled as 'Hellfer-Sjöstrand' in Section 2.2 and elsewhere; please correct the spelling.
  2. [Proposition 1.10] The phrase 'as the the limit' is a typo and should read 'as the limit'.
  3. [Eq. (7.13)] The notation σ^{drift,Λ2/ρ}_e uses a slash between Λ2 and ρ; this should be a comma for consistency with Definition 1.8.
  4. [Eq. (2.2)] The Helffer-Sjöstrand formula should specify explicitly whether the derivative is ∂_z or ∂_{z̄}; the coefficient computed in Step 4 of Section 7 depends on this convention, and the current notation is ambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the bulk-interface equality is derived from operator identities and independent trace estimates, not from fitting or definitional identity.

full rationale

The paper's central claim, Theorem 1.11, equates three independently defined quantities: the bulk spin conductances σ_±^{Λ2}, the interface spin-drift conductance σ_e^{drift,Λ2,ρ}, and the interface spin-torque conductance σ_e^{torque,Λ2,ρ}. These are not defined in terms of one another; the bulk conductance is a principal-value trace of a potential-current correlation (Definition 1.5), the drift conductance is a trace involving {[H_e,Λ_2], S} ρ'(H_e) (Definition 1.8), and the torque conductance is a principal-value trace of a different correlation involving [H_e, S] (Definition 1.9). The equality (1.19) is proved in Section 7 by decomposing the torque correlation with localization factors, comparing bulk and interface terms through the resolvent identity, and using the vanishing mesoscopic bulk torque response (Proposition 3.2). The vanishing result is imported from the cited reference [37], which is external to the present authors and is not a self-citation. The proof does contain acknowledged gaps: Section 7 Step 4 states 'we will avoid detailed technical arguments' and 'the details are skipped' for the identification (7.13), and similar skipped details appear in (7.5) and (7.11). These are rigor/completeness concerns, not circularity: the asserted equality is not enforced by fitting, normalization, or definitional equivalence, and no parameter is fitted to a subset of data and then renamed as a prediction. The derivation chain is therefore self-contained in the relevant sense, and no circularity score is warranted.

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

The central theorem is a mathematical identity depending on the listed geometric and analytic assumptions. No parameters are fitted to data. The most physically substantive input is the exact vanishing of the mesoscopic bulk spin-torque response, Propositions 3.1-3.2, proved from [S,Λ]=0 and conditional cyclicity.

assumptions (5)
  • domain assumption Bulk and interface Hamiltonians are tight-binding with exponential off-diagonal decay (Definition 1.1, Assumption 1.2(i)).
    All localization and trace-class estimates in Sections 2 through 7 use this exponential decay.
  • domain assumption The two bulk media are periodic on rectangular lattices and have a common spectral gap (Assumption 1.2(ii)-(iii)).
    Periodicity is needed for conditional cyclicity, and the common gap ensures ρ'(H±) vanishes in the bulk.
  • domain assumption The interface perturbation Hδ is exponentially localized near the interface (1.5)-(1.6).
    This gives the needed decay of H_e - H_± away from the interface throughout Section 7.
  • domain assumption The spin operator S = Id ⊗ (1/2)σ_z commutes with position and switch functions, so [S, Λ_i] = 0.
    Used to prove the vanishing mesoscopic torque and standard commutator identities in Sections 3 and 4.
  • standard math Helffer-Sjöstrand functional calculus and almost-analytic extensions are valid (Section 2.2).
    This is the standard tool used to express all correlation functions and to pass to ρ'(H) in Step 4.

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Cite this review

Pith. "Pith review of Generalized bulk-interface correspondence for non-quantized spin transport." pith.science (2026). https://pith.science/paper/W4V4SDLP

@misc{pith2026250516331,
  author       = {Pith},
  title        = {Pith review of: Generalized bulk-interface correspondence for non-quantized spin transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4V4SDLP}},
  note         = {Machine review of arXiv:2505.16331}
}
abstract

This paper establishes a rigorous mathematical framework for a generalized bulk-interface correspondence (BIC) in electronic systems with possibly nonconserved spin charge, where the Hamiltonian and spin operator do not commute. We first introduce the bulk spin conductance as a character of the bulk medium, which is defined as a potential-current correlation function and is not quantized if the spin charge is nonconserved. Then we establish the principle of BIC, which states that the difference of bulk spin conductances across an interface equals the sum of two quantities associated with the spin transport along the interface: the spin-drift conductance, which captures spin transport carried by interface modes, and the spin-torque conductance, which accounts for spin generation near the interface due to the non-conservation of spin. Furthermore, when the spin charge is conserved, our result recovers the existing BIC based on the spin Chern number or Fu-Kane-Mele $\mathbb{Z}_2$ index. Our findings demonstrate that the principle of BIC is not restricted to systems with quantized characters and provides new insights into spin transport phenomena.

Figures

Figures reproduced from arXiv: 2505.16331 by the authors.

Figure 1
Figure 1. An interface model: two bulk mediums are joined along an interface. The bulk [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) Transverse (Hall) Current in a periodic medium induced by an external potential [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

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Reference graph

Works this paper leans on

55 extracted references · 51 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bulk-boundary correspondence for disordered free-fermion topological phases.Communications in Mathematical Physics, 377(3):1761–1821, 2020

    Alexander Alldridge, Christopher Max, and Martin R Zirnbauer. Bulk-boundary correspondence for disordered free-fermion topological phases.Communications in Mathematical Physics, 377(3):1761–1821, 2020

    work page 2020

  2. [2]

    Topological invariants of edge states for periodic two-dimensional models.Mathematical Physics, Analysis and Geometry, 16(2):137–170, 2013

    Julio Cesar Avila, Hermann Schulz-Baldes, and Carlos Villegas-Blas. Topological invariants of edge states for periodic two-dimensional models.Mathematical Physics, Analysis and Geometry, 16(2):137–170, 2013

    work page 2013

  3. [3]

    Avron, Ruedi Seiler, and Barry Simon

    Joseph E. Avron, Ruedi Seiler, and Barry Simon. Charge deficiency, charge transport and comparison of dimensions.Communications in Mathematical Physics, 159(2):399–422, Jan 1994

    work page 1994

  4. [4]

    Continuous bulk and interface description of topological insulators.Journal of Mathe- matical Physics, 60(8), 2019

    Guillaume Bal. Continuous bulk and interface description of topological insulators.Journal of Mathe- matical Physics, 60(8), 2019

    work page 2019

  5. [5]

    Topological invariants for interface modes.Communications in Partial Differential Equa- tions, 47(8):1636–1679, 2022

    Guillaume Bal. Topological invariants for interface modes.Communications in Partial Differential Equa- tions, 47(8):1636–1679, 2022

    work page 2022

  6. [6]

    Andrei Bernevig and Taylor L

    B. Andrei Bernevig and Taylor L. Hughes.Topological Insulators and Topological Superconductors. Prince- ton University Press, 2013

    work page 2013

  7. [7]

    Andrei Bernevig, Taylor L

    B. Andrei Bernevig, Taylor L. Hughes, and Shou-Cheng Zhang. Quantum spin hall effect and topological phase transition in hgte quantum wells.Science, 314(5806):1757–1761, 2006

    work page 2006

  8. [8]

    Orbital hall effect as an alternative to valley hall effect in gapped graphene.Physical Review B, 103(19):195309, 2021

    Sayantika Bhowal and Giovanni Vignale. Orbital hall effect as an alternative to valley hall effect in gapped graphene.Physical Review B, 103(19):195309, 2021

    work page 2021

Show all 55 references
  1. [9]

    The k-theoretic bulk–edge correspondence for topological insulators

    Chris Bourne, Johannes Kellendonk, and Adam Rennie. The k-theoretic bulk–edge correspondence for topological insulators. InAnnales Henri Poincar´ e, volume 18, pages 1833–1866. Springer, 2017. 33

    2017

  2. [10]

    Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases.Mathematical Physics, Analysis and Geometry, 21:1–62, 2018

    Chris Bourne and Adam Rennie. Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases.Mathematical Physics, Analysis and Geometry, 21:1–62, 2018

    2018

  3. [11]

    Spectral flows of toeplitz operators and bulk-edge correspondence.Letters in Math- ematical Physics, 109:2271–2289, 2019

    Maxim Braverman. Spectral flows of toeplitz operators and bulk-edge correspondence.Letters in Math- ematical Physics, 109:2271–2289, 2019

    2019

  4. [12]

    Berry phases in electronic structure theory

    Laurent Chaput. Berry phases in electronic structure theory. electric polarization, orbital magnetization and topological insulators. by david vanderbilt. cambridge university press, 2018. hardback, pp. x+384. price gbp 59.99. isbn 9781107157651.Acta crystallographica. Section...

    2018

  5. [13]

    Edge and impurity effects on quantization of hall currents

    Jean-Michel Combes and Fran¸ cois Germinet. Edge and impurity effects on quantization of hall currents. Communications in mathematical physics, 256:159–180, 2005

    2005

  6. [14]

    General bulk-edge correspondence at positive temperature.arXiv preprint arXiv:2107.13456, 2021

    Horia D Cornean, Massimo Moscolari, and Stefan Teufel. General bulk-edge correspondence at positive temperature.arXiv preprint arXiv:2107.13456, 2021

    2021 arXiv

  7. [15]

    Spectral flows associated to flux tubes

    Giuseppe De Nittis and Hermann Schulz-Baldes. Spectral flows associated to flux tubes. InAnnales Henri Poincar´ e, volume 17, pages 1–35. Springer, 2016

    2016

  8. [16]

    Quantization of edge currents along mag- netic barriers and magnetic guides

    Nicolas Dombrowski, Fran¸ cois Germinet, and Georgi Raikov. Quantization of edge currents along mag- netic barriers and magnetic guides. InAnnales Henri Poincar´ e, volume 12, pages 1169–1197. Springer, 2011

    2011

  9. [17]

    Microlocal analysis of the bulk-edge correspondence.Communications in Mathematical Physics, 383:2069–2112, 2021

    Alexis Drouot. Microlocal analysis of the bulk-edge correspondence.Communications in Mathematical Physics, 383:2069–2112, 2021

    2021

  10. [18]

    The bulk-edge correspondence for curved interfaces, 2024

    Alexis Drouot and Xiaowen Zhu. The bulk-edge correspondence for curved interfaces, 2024

    2024

  11. [19]

    Equality of bulk and edge hall conductance revisited.Communica- tions in mathematical physics, 229:415–432, 2002

    Peter Elbau and Gian-Michele Graf. Equality of bulk and edge hall conductance revisited.Communica- tions in mathematical physics, 229:415–432, 2002

    2002

  12. [20]

    Equality of the bulk and edge hall conductances in a mobility gap.Communications in mathematical physics, 259:185–221, 2005

    Alexander Elgart, Gian M Graf, and Jeffrey H Schenker. Equality of the bulk and edge hall conductances in a mobility gap.Communications in mathematical physics, 259:185–221, 2005

    2005

  13. [21]

    Symmetry protected topological charge in symmetry broken phase: Spin-chern, spin- valley-chern and mirror-chern numbers.Physics Letters A, 378(16-17):1180–1184, 2014

    Motohiko Ezawa. Symmetry protected topological charge in symmetry broken phase: Spin-chern, spin- valley-chern and mirror-chern numbers.Physics Letters A, 378(16-17):1180–1184, 2014

    2014

  14. [22]

    Edge states for second order elliptic operators in a channel.Journal of Spectral Theory, 12(3):1155–1202, 2023

    David Gontier. Edge states for second order elliptic operators in a channel.Journal of Spectral Theory, 12(3):1155–1202, 2023

    2023

  15. [23]

    Bulk-edge correspondence for two-dimensional topological insu- lators.Communications in Mathematical Physics, 324:851–895, 2013

    Gian Michele Graf and Marcello Porta. Bulk-edge correspondence for two-dimensional topological insu- lators.Communications in Mathematical Physics, 324:851–895, 2013

    2013

  16. [24]

    The bulk-edge correspondence for disordered chiral chains.Com- munications in Mathematical Physics, 363:829–846, 2018

    Gian Michele Graf and Jacob Shapiro. The bulk-edge correspondence for disordered chiral chains.Com- munications in Mathematical Physics, 363:829–846, 2018

    2018

  17. [25]

    Spintronics meets orbitronics: Emer- gence of orbital angular momentum in solids.npj Spintronics, 2(1):19, 2024

    Daegeun Jo, Dongwook Go, Gyung-Min Choi, and Hyun-Woo Lee. Spintronics meets orbitronics: Emer- gence of orbital angular momentum in solids.npj Spintronics, 2(1):19, 2024

    2024

  18. [26]

    C. L. Kane and E. J. Mele.Z 2 topological order and the quantum spin hall effect.Phys. Rev. Lett., 95:146802, Sep 2005

    2005

  19. [27]

    Kellendonk, T

    J. Kellendonk, T. Richter, and H. Schulz-Baldes. Edge current channels and chern numbers in the integer quantum hall effect.Reviews in Mathematical Physics, 14(01):87–119, 2002

    2002

  20. [28]

    Boundary maps for c*-crossed products with with an application to the quantum hall effect.Communications in Mathematical Physics, 249:611–637, 2004

    Johannes Kellendonk and Hermann Schulz-Baldes. Boundary maps for c*-crossed products with with an application to the quantum hall effect.Communications in Mathematical Physics, 249:611–637, 2004

    2004

  21. [29]

    K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine- structure constant based on quantized hall resistance.Phys. Rev. Lett., 45:494–497, Aug 1980

    1980

  22. [30]

    W. Kohn. Density functional and density matrix method scaling linearly with the number of atoms.Phys. Rev. Lett., 76:3168–3171, Apr 1996

    1996

  23. [31]

    Controlled topological phases and bulk-edge correspondence.Communications in Math- ematical Physics, 349(2):493–525, 2017

    Yosuke Kubota. Controlled topological phases and bulk-edge correspondence.Communications in Math- ematical Physics, 349(2):493–525, 2017. 34

    2017

  24. [32]

    Quantum valley hall effect in wide-gap semiconductor sic monolayer

    Kyu Won Lee and Cheol Eui Lee. Quantum valley hall effect in wide-gap semiconductor sic monolayer. Scientific reports, 10(1):5044, 2020

    2020

  25. [33]

    Interface modes in honeycomb topological photonic structures with broken reflection symmetry.arXiv preprint arXiv:2405.03238, 2024

    Wei Li, Junshan Lin, Jiayu Qiu, and Hai Zhang. Interface modes in honeycomb topological photonic structures with broken reflection symmetry.arXiv preprint arXiv:2405.03238, 2024

    2024 arXiv

  26. [34]

    Mathematical theory for topological photonic materials in one dimension

    Junshan Lin and Hai Zhang. Mathematical theory for topological photonic materials in one dimension. Journal of Physics A: Mathematical and Theoretical, 55(49):495203, 2022

    2022

  27. [35]

    Cobordism invariance of topological edge-following states

    Matthias Ludewig and Guo Chuan Thiang. Cobordism invariance of topological edge-following states. arXiv preprint arXiv:2001.08339, 2020

    2001 arXiv

  28. [36]

    From charge to spin: Analogies and differences in quantum transport coefficients.Journal of Mathematical Physics, 63(7):072102, 07 2022

    Giovanna Marcelli and Domenico Monaco. From charge to spin: Analogies and differences in quantum transport coefficients.Journal of Mathematical Physics, 63(7):072102, 07 2022

    2022

  29. [37]

    Spin conductance and spin conductivity in topological insulators: Analysis of kubo-like terms.Annales Henri Poincar´ e, 20(6):2071–2099, Jun 2019

    Giovanna Marcelli, Gianluca Panati, and Cl´ ement Tauber. Spin conductance and spin conductivity in topological insulators: Analysis of kubo-like terms.Annales Henri Poincar´ e, 20(6):2071–2099, Jun 2019

    2019

  30. [38]

    A new approach to transport coefficients in the quantum spin hall effect

    Giovanna Marcelli, Gianluca Panati, and Stefan Teufel. A new approach to transport coefficients in the quantum spin hall effect. InAnnales Henri Poincar´ e, volume 22, pages 1069–1111. Springer, 2021

    2021

  31. [39]

    Spin hall conductivity in insulators with nonconserved spin.Physical Review B, 102(12):125138, 2020

    Domenico Monaco and Lara Ulˇ cakar. Spin hall conductivity in insulators with nonconserved spin.Physical Review B, 102(12):125138, 2020

    2020

  32. [40]

    A mathematical theory of integer quantum hall effect in photonics.arXiv preprint arXiv:2405.17200, 2024

    Jiayu Qiu and Hai Zhang. A mathematical theory of integer quantum hall effect in photonics.arXiv preprint arXiv:2405.17200, 2024

    2024 arXiv

  33. [41]

    Bulk-edge correspondence in finite photonic structure.arXiv preprint arXiv:2501.15531, 2025

    Jiayu Qiu and Hai Zhang. Bulk-edge correspondence in finite photonic structure.arXiv preprint arXiv:2501.15531, 2025

    2025 arXiv

  34. [42]

    Asymmetric transport for magnetic dirac equations.Pure and Applied Analysis, 6(2):353–377, 2024

    Solomon Quinn and Guillaume Bal. Asymmetric transport for magnetic dirac equations.Pure and Applied Analysis, 6(2):353–377, 2024

    2024

  35. [43]

    Proper definition of spin current in spin-orbit coupled systems.Physical review letters, 96(7):076604, 2006

    Junren Shi, Ping Zhang, Di Xiao, and Qian Niu. Proper definition of spin current in spin-orbit coupled systems.Physical review letters, 96(7):076604, 2006

    2006

  36. [44]

    Number 120

    Barry Simon.Trace ideals and their applications. Number 120. American Mathematical Soc., 2005

    2005

  37. [45]

    World Scientific, 1992

    Michael Stone.Quantum Hall Effect. World Scientific, 1992

    1992

  38. [46]

    Nonconserved density accu- mulations in orbital hall transport: Insights from linear response theory.arXiv preprint arXiv:2410.20668, 2024

    Hao Sun, Alexander Kazantsev, Alessandro Principi, and Giovanni Vignale. Nonconserved density accu- mulations in orbital hall transport: Insights from linear response theory.arXiv preprint arXiv:2410.20668, 2024

    2024 arXiv

  39. [47]

    Equality of bulk and edge hall conductances for continuous magnetic random schr¨ odinger operators, 2014

    Amal Taarabt. Equality of bulk and edge hall conductances for continuous magnetic random schr¨ odinger operators, 2014

    2014

  40. [48]

    Bulk-interface correspondences for one-dimensional topological ma- terials with inversion symmetry.Proceedings of the Royal Society A, 479(2270):20220675, 2023

    Guo Chuan Thiang and Hai Zhang. Bulk-interface correspondences for one-dimensional topological ma- terials with inversion symmetry.Proceedings of the Royal Society A, 479(2270):20220675, 2023

    2023

  41. [49]

    Lectures on the quantum hall effect, 2016

    David Tong. Lectures on the quantum hall effect, 2016

    2016

  42. [50]

    40 years of the quantum hall effect.Nature Reviews Physics, 2(8):397–401, Aug 2020

    Klaus von Klitzing, Tapash Chakraborty, Philip Kim, Vidya Madhavan, Xi Dai, James McIver, Yoshinori Tokura, Lucile Savary, Daria Smirnova, Ana Maria Rey, Claudia Felser, Johannes Gooth, and Xiaoliang Qi. 40 years of the quantum hall effect.Nature Reviews Physics, 2(8):397–401,...

    2020

  43. [51]

    Direct observation of valley-polarized topological edge states in designer surface plasmon crystals.Nature communications, 8(1):1304, 2017

    Xiaoxiao Wu, Yan Meng, Jingxuan Tian, Yingzhou Huang, Hong Xiang, Dezhuan Han, and Weijia Wen. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals.Nature communications, 8(1):1304, 2017

    2017

  44. [52]

    Conserved current of nonconserved quantities.Physical Review B, 104(24):L241411, 2021

    Cong Xiao and Qian Niu. Conserved current of nonconserved quantities.Physical Review B, 104(24):L241411, 2021

    2021

  45. [53]

    Valley-contrasting physics in graphene: magnetic moment and topological transport.Physical review letters, 99(23):236809, 2007

    Di Xiao, Wang Yao, and Qian Niu. Valley-contrasting physics in graphene: magnetic moment and topological transport.Physical review letters, 99(23):236809, 2007

    2007

  46. [54]

    Spin transport and spin torque in antiferro- magnetic devices.Nature Physics, 14(3):220–228, 2018

    J ˇZelezn` y, P Wadley, K Olejn ´ ık, A Hoffmann, and H Ohno. Spin transport and spin torque in antiferro- magnetic devices.Nature Physics, 14(3):220–228, 2018

    2018

  47. [55]

    Zworski.Semiclassical Analysis

    M. Zworski.Semiclassical Analysis. Graduate studies in mathematics. American Mathematical Society, 2012. 35

    2012

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.