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arxiv: 2606.28809 · v1 · pith:5HOGAFFGnew · submitted 2026-06-27 · ❄️ cond-mat.mes-hall

Inter-band coherence effects in disordered crystals: beyond the non-crossing approximation

Zhanning Wang , James H. Cullen , Roberto Raimondi , Dimitrie Culcer This is my paper

Pith reviewed 2026-06-30 08:56 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords anomalous Hall effectdisordered crystalsinter-band coherencenon-crossing approximationquantum kinetic theorymassive Dirac fermionscrossed impurity processesdensity-matrix formalism
0
0 comments X

The pith

Crossed impurity processes produce an extrinsic anomalous Hall conductivity of order τ^0 that coexists with the intrinsic Berry-curvature term.

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

The paper develops a quantum kinetic theory for Bloch electrons in a uniform electric field that extends the nonequilibrium density-matrix approach beyond the non-crossing approximation. This captures steady-state responses that are nominally independent of disorder strength and compete with band-geometric effects in transport. Working to fourth order in the disorder potential with Gaussian white-noise impurities, an iterative solution for density-matrix fluctuations yields a connected collision integral after removal of disconnected pairings. Applied to the two-dimensional massive Dirac fermion model, the calculation produces analytical expressions for lifetimes and conductivity at the Born level and evaluates the anomalous Hall response including crossed processes.

Core claim

In the two-dimensional massive Dirac fermion model with Gaussian white-noise disorder, fourth-order crossed impurity processes generate an extrinsic contribution of order τ^0 to the anomalous Hall conductivity that coexists with the intrinsic Berry-curvature term. For this model the Ψ-type crossed contribution cancels while the X-type term remains finite. The response is obtained from a connected V^4 collision integral derived by iterative solution of impurity-induced density-matrix fluctuations after subtracting disconnected impurity pairings, and the formalism separates self-energy corrections, ladder vertex renormalization, and crossed quantum-interference contributions while clarifying t

What carries the argument

Connected V^4 collision integral from iterative solution of impurity-induced density-matrix fluctuations after subtracting disconnected impurity pairings.

If this is right

  • The approach decomposes transport responses into Fermi-surface and Fermi-sea components while including both intrinsic geometry and extrinsic crossed-disorder effects.
  • It yields explicit expressions for the single-particle lifetime, transport relaxation time, and longitudinal conductivity at the Born level.
  • The density-matrix kinetic equation maintains a consistent correspondence with the Keldysh formalism.
  • The same machinery applies to spin, pseudospin, orbital, and valley transport phenomena in multiband systems.

Where Pith is reading between the lines

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

  • The method could be used to compute disorder corrections to valley or spin Hall conductivities in other Dirac or Weyl systems where interband coherence is strong.
  • If the surviving X-type term is generic, it implies that certain impurity configurations can produce disorder-independent extrinsic contributions that add to rather than subtract from geometric responses.
  • Higher-order extensions might reveal whether crossed processes modify the scaling of conductivity with disorder strength in topological materials beyond the Born approximation.

Load-bearing premise

An iterative solution for the impurity-induced density-matrix fluctuations produces a connected V^4 collision integral after subtracting disconnected impurity pairings without introducing double counting.

What would settle it

A direct numerical evaluation of the anomalous Hall conductivity in the massive Dirac model at fourth order in white-noise disorder strength that checks whether the X-type crossed term survives as a finite τ^0 contribution while the Ψ-type term vanishes.

Figures

Figures reproduced from arXiv: 2606.28809 by Dimitrie Culcer, James H. Cullen, Roberto Raimondi, Zhanning Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Ladder fourth order contribution on the Keldysh con [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. X-type crossed impurity contribution on the Keldysh [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

We develop a quantum kinetic theory for Bloch electrons driven by a uniform dc electric field, extending the nonequilibrium density-matrix formalism beyond the non-crossing approximation. This extension is required to capture steady-state terms that are nominally zeroth order in disorder strength and compete with intrinsic band-geometric responses, as in anomalous Hall and related spin, orbital, and valley transport. Working in the length gauge with Gaussian white-noise disorder, we include impurity scattering to fourth order in the disorder potential. An iterative solution for the impurity-induced density-matrix fluctuations yields a connected $V^4$ collision integral after subtracting disconnected impurity pairings, thereby avoiding double counting. The resulting terms separate into self-energy corrections, ladder-type vertex renormalization, and crossed quantum-interference contributions. We clarify the correspondence between this density-matrix kinetic equation and the Keldysh formalism, and decompose the response into Fermi-surface and Fermi-sea components. As an application, we study the two-dimensional massive Dirac fermion model. We obtain analytical expressions for the single-particle lifetime, transport relaxation time, and longitudinal conductivity at the Born level, and then evaluate the anomalous Hall conductivity including crossed impurity processes. These processes generate an extrinsic contribution of order $\tau^0$ that coexists with the intrinsic Berry-curvature term; for Gaussian white-noise disorder in this model, the $\Psi$-type contribution cancels while the $X$-type term remains finite. The formalism provides a consistent route for incorporating band geometry and crossed-disorder corrections into multiband transport, with applications to spin, pseudospin, orbital, and valley phenomena.

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

1 major / 2 minor

Summary. The paper develops a quantum kinetic theory extending the nonequilibrium density-matrix formalism for Bloch electrons in a uniform dc electric field beyond the non-crossing approximation. Working to fourth order in Gaussian white-noise disorder, an iterative solution yields a connected V^4 collision integral after subtracting disconnected impurity pairings. The resulting self-energy, ladder, and crossed (Ψ/X) contributions are applied to the 2D massive Dirac model, producing analytical expressions for lifetimes and longitudinal conductivity at Born level, plus an extrinsic τ^0 anomalous Hall conductivity that coexists with the intrinsic Berry-curvature term, with Ψ canceling and X remaining finite.

Significance. If the connected-integral construction holds, the work supplies a systematic route to include crossed-disorder corrections alongside band geometry in multiband transport, directly relevant to anomalous Hall, spin, orbital, and valley responses. The analytical cancellation result for the Dirac model is a concrete, falsifiable prediction that strengthens the formalism's utility.

major comments (1)
  1. [section deriving the collision integral] The iterative solution for impurity-induced density-matrix fluctuations and the subsequent subtraction of disconnected pairings to obtain the connected V^4 collision integral (abstract and the section deriving the kinetic equation): this step is load-bearing for the claimed Ψ cancellation and finite X term, yet the manuscript supplies no explicit diagrammatic enumeration or independent cross-check confirming that all connected crossed diagrams are retained and that the subtraction does not inadvertently remove part of the X channel. Because the τ^0 extrinsic contribution is precisely the difference between these channels, incompleteness here directly affects the central coexistence statement.
minor comments (2)
  1. [abstract] Clarify the precise correspondence between the density-matrix kinetic equation and the Keldysh formalism (mentioned in the abstract) by adding a short table or diagram mapping the terms.
  2. [application to Dirac model] The decomposition into Fermi-surface and Fermi-sea components is stated but not shown explicitly for the crossed terms; a brief appendix equation would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of the connected V^4 construction. We address the single major comment below.

read point-by-point responses

  1. Referee: [section deriving the collision integral] The iterative solution for impurity-induced density-matrix fluctuations and the subsequent subtraction of disconnected pairings to obtain the connected V^4 collision integral (abstract and the section deriving the kinetic equation): this step is load-bearing for the claimed Ψ cancellation and finite X term, yet the manuscript supplies no explicit diagrammatic enumeration or independent cross-check confirming that all connected crossed diagrams are retained and that the subtraction does not inadvertently remove part of the X channel. Because the τ^0 extrinsic contribution is precisely the difference between these channels, incompleteness here directly affects the central coexistence statement.

    Authors: The iterative solution for the impurity-induced density-matrix fluctuations is constructed to generate every fourth-order scattering sequence exactly once. Subtracting the disconnected pairings (products of two independent second-order processes) isolates the connected V^4 collision integral by definition; this subtraction removes only the factorizable terms and leaves all topologically connected crossed diagrams intact. The resulting X channel is therefore the genuine crossed contribution. The explicit analytic evaluation for the 2D massive Dirac model then serves as an internal consistency check: the Ψ term cancels while the X term survives, yielding a finite τ^0 extrinsic AHC that coexists with the intrinsic term. The manuscript already maps the kinetic equation onto the Keldysh diagrammatic expansion, providing the independent cross-check requested. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard diagrammatic subtraction without reduction to inputs or self-citations

full rationale

The paper's central construction—an iterative solution for density-matrix fluctuations that produces a connected V^4 collision integral after subtracting disconnected pairings—is presented as a direct application of standard perturbation theory to avoid double counting at O(V^4). No equations or claims in the abstract or described derivation reduce a target quantity (such as the τ^0 extrinsic AHC or the Ψ/X cancellation) to a fitted parameter, a renamed input, or a load-bearing self-citation. The separation into self-energy, ladder, and crossed terms follows from the subtracted integral by explicit decomposition rather than by definition. The model-specific result for Gaussian white-noise disorder in the 2D Dirac case is obtained analytically from the resulting kinetic equation, with no indication that the outcome is forced by prior author work or by construction. This is the normal case of a self-contained perturbative expansion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the length-gauge formulation, the Gaussian white-noise disorder model, and the validity of the connected-diagram subtraction procedure at fourth order; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Gaussian white-noise disorder model for impurities
    Invoked for all scattering calculations and the specific cancellation result in the Dirac model.
  • domain assumption Length gauge for the uniform dc electric field
    Chosen framework for the quantum kinetic theory derivation.

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Reference graph

Works this paper leans on

81 extracted references · 9 canonical work pages

  1. [1]

    Raimondi, C

    R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, Quasiclassical approach to the spin hall effect in the two-dimensional electron gas, Phys. Rev. B74, 035340 (2006)

    2006

  2. [2]

    Culcer, E

    D. Culcer, E. M. Hankiewicz, G. Vignale, and R. Win- kler, Side jumps in the spin hall effect: Construction of the boltzmann collision integral, Phys. Rev. B81, 125332 (2010)

    2010

  3. [3]

    Y. Gao, S. A. Yang, and Q. Niu, Field induced positional shift of bloch electrons and its dynamical implications, Phys. Rev. Lett.112, 166601 (2014)

    2014

  4. [4]

    Z. Z. Du, C. M. Wang, H.-P. Sun, H.-Z. Lu, and X. C. Xie, Quantum theory of the nonlinear hall effect, Nature Communications12, 5038 (2021)

    2021

  5. [6]

    Valet and R

    T. Valet and R. Raimondi, Semiclassical kinetic theory for systems with non-trivial quantum geometry and the expectationvalueofphysicalquantities,EurophysicsLet- ters143, 26004 (2023)

    2023

  6. [7]

    Raimondi and T

    R. Raimondi and T. Valet, Quantum kinetic the- ory of the spin hall effect for disordered graphene with rashba spin–orbit coupling, Condensed Matter10, 10.3390/condmat10010004 (2025)

    work page doi:10.3390/condmat10010004 2025

  7. [8]

    T.ValetandR.Raimondi,Quantumkinetictheory ofthe linear response for weakly disordered multiband systems, Phys. Rev. B111, L041118 (2025)

    2025

  8. [9]

    Culcer, A

    D. Culcer, A. Sekine, and A. H. MacDonald, Interband coherence response to electric fields in crystals: Berry- phase contributions and disorder effects, Phys. Rev. B 96, 035106 (2017)

    2017

  9. [10]

    R. B. Atencia, Q. Niu, and D. Culcer, Semiclassical re- sponse of disordered conductors: Extrinsic carrier veloc- ity and spin and field-corrected collision integral, Phys. Rev. Res.4, 013001 (2022)

    2022

  10. [12]

    I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, Anomalous hall effect with massive dirac fermions, Eu- rophysics Letters111, 37004 (2015)

    2015

  11. [13]

    I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, Anomalous hall effect in a 2d rashba ferromagnet, Phys. Rev. Lett.117, 046601 (2016)

    2016

  12. [14]

    I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, Sensitivity of the anomalous hall effect to disorder corre- lations, Phys. Rev. B96, 235148 (2017)

    2017

  13. [15]

    Zhang and W

    J.-X. Zhang and W. Chen, Anomalous hall effect in type- i weyl metals beyond the noncrossing approximation, Phys. Rev. B107, 214204 (2023)

    2023

  14. [16]

    Gorini, P

    C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Non-abelian gauge fields in the gradient expansion: Gen- eralized boltzmann and eilenberger equations, Phys. Rev. B82, 195316 (2010)

    2010

  15. [17]

    Milletarì and A

    M. Milletarì and A. Ferreira, Quantum diagrammatic theory of the extrinsic spin hall effect in graphene, Phys. Rev. B94, 134202 (2016)

    2016

  16. [18]

    Raimondi, C

    R. Raimondi, C. Gorini, and S. Tölle, Spin-charge cou- pling effects in a two-dimensional electron gas, inSpin Orbitronics and Topological Properties of Nanostruc- tures, pp. 80–109

  17. [19]

    S. Lee, A. Trionfi, and D. Natelson, Quantum coherence in a ferromagnetic metal: Time-dependent conductance fluctuations, Phys. Rev. B70, 212407 (2004)

    2004

  18. [20]

    Miyasato, N

    T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda, Y. Onose, N. Nagaosa, and Y. Tokura, Crossover behav- ior of the anomalous hall effect and anomalous nernst effect in itinerant ferromagnets, Phys. Rev. Lett.99, 086602 (2007)

    2007

  19. [21]

    Y. Tian, L. Ye, and X. Jin, Proper scaling of the anoma- lous hall effect, Phys. Rev. Lett.103, 087206 (2009)

    2009

  20. [22]

    Nagaosa, J

    N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous hall effect, Rev. Mod. Phys.82, 1539 (2010)

    2010

  21. [23]

    Morota, Y

    M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani, T. Kimura, and Y. Otani, Indication of in- trinsic spin hall effect in4dand5dtransition metals, Phys. Rev. B83, 174405 (2011)

    2011

  22. [24]

    Niimi, M

    Y. Niimi, M. Morota, D. H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y. Otani, Extrinsic spin hall effect induced by iridium impurities in copper, Phys. Rev. Lett.106, 126601 (2011)

    2011

  23. [25]

    Sinova, S

    J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin hall effects, Rev. Mod. Phys.87, 1213 (2015)

    2015

  24. [26]

    Musha, Y

    A. Musha, Y. Kanno, and K. Ando, Extrinsic-intrinsic crossover of the spin hall effect induced by alloying, Phys. Rev. Mater.3, 054411 (2019)

    2019

  25. [27]

    J. Chen, H. J. Qin, F. Yang, J. Liu, T. Guan, F. M. Qu, G. H. Zhang, J. R. Shi, X. C. Xie, C. L. Yang, K. H. Wu, Y. Q. Li, and L. Lu, Gate-voltage control of chemical potential and weak antilocalization inbi2se3, Phys. Rev. Lett.105, 176602 (2010)

    2010

  26. [28]

    Culcer, E

    D. Culcer, E. H. Hwang, T. D. Stanescu, and S. Das Sarma, Two-dimensional surface charge transport in topological insulators, Phys. Rev. B82, 155457 (2010)

    2010

  27. [29]

    Y. S. Kim, M. Brahlek, N. Bansal, E. Edrey, G. A. Kapilevich, K. Iida, M. Tanimura, Y. Horibe, S.-W. Cheong, and S. Oh, Thickness-dependent bulk properties and weak antilocalization effect in topological insulator bi2se3, Phys. Rev. B84, 073109 (2011)

    2011

  28. [30]

    Steinberg, J.-B

    H. Steinberg, J.-B. Laloë, V. Fatemi, J. S. Moodera, and P. Jarillo-Herrero, Electrically tunable surface-to- bulk coherent coupling in topological insulator thin films, Phys. Rev. B84, 233101 (2011)

    2011

  29. [31]

    Ma, S.-Y

    Q. Ma, S.-Y. Xu, H. Shen, D. MacNeill, V. Fatemi, T.-R. Chang, A. M. Mier Valdivia, S. Wu, Z. Du, C.-H. Hsu, S. Fang, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, E. Kaxiras, H.-Z. Lu, H. Lin, L. Fu, N. Gedik, and P. Jarillo-Herrero, Observation of the nonlinear hall effect under time-reversal-symmetric conditions, Nature 565, 337 (2019)

    2019

  30. [32]

    D. V. Chichinadze, N. J. Zhang, J.-X. Lin, E. Morissette, X. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. I. A. Li, Observation of giant nonlinear hall conductiv- ity in bernal bilayer graphene (2025), arXiv:2411.11156 [cond-mat.mes-hall]

    work page arXiv 2025

  31. [33]

    L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker, T. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C. Bell, L. Fu, R. Comin, and J. G. Checkelsky, Massive 18 dirac fermions in a ferromagnetic kagome metal, Nature 555, 638 (2018)

    2018

  32. [34]

    D. Chen, C. Le, C. Fu, H. Lin, W. Schnelle, Y. Sun, and C. Felser, Large anomalous hall effect in the kagome ferromagnetlimn 6sn6, Phys. Rev. B103, 144410 (2021)

    2021

  33. [35]

    E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao, S.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. Süß, H. Borrmann, C. Shekhar, Z. Wang, C. Xi, W. Wang, W. Schnelle, S. Wirth, Y. Chen, S. T. B. Goennenwein, and C. Felser, Giant anomalous hall effect in a ferromag- netic kagome-lattice semimetal, Nature Physics14, 1125 (2018)

    2018

  34. [36]

    Jungwirth, Q

    T. Jungwirth, Q. Niu, and A. H. MacDonald, Anomalous hall effect in ferromagnetic semiconductors, Phys. Rev. Lett.88, 207208 (2002)

    2002

  35. [37]

    Culcer, A

    D. Culcer, A. MacDonald, and Q. Niu, Anomalous hall effect in paramagnetic two-dimensional systems, Phys. Rev. B68, 045327 (2003)

    2003

  36. [38]

    J. H. Cullen, P. Bhalla, E. Marcellina, A. R. Hamilton, and D. Culcer, Generating a topological anomalous hall effect in a nonmagnetic conductor: An in-plane magnetic field as a direct probe of the berry curvature, Phys. Rev. Lett.126, 256601 (2021)

    2021

  37. [39]

    R. B. Atencia, D. Xiao, and D. Culcer, Disorder in the nonlinear anomalous hall effect ofPT-symmetric dirac fermions, Phys. Rev. B108, L201115 (2023)

    2023

  38. [40]

    Sinova, D

    J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung- wirth, and A. H. MacDonald, Universal intrinsic spin hall effect, Phys. Rev. Lett.92, 126603 (2004)

    2004

  39. [41]

    Culcer, J

    D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. MacDonald, and Q. Niu, Semiclassical spin transport in spin-orbit-coupled bands, Phys. Rev. Lett.93, 046602 (2004)

    2004

  40. [42]

    Bhalla, M.-X

    P. Bhalla, M.-X. Deng, R.-Q. Wang, L. Wang, and D. Culcer, Nonlinear ballistic response of quantum spin hall edge states, Phys. Rev. Lett.127, 206801 (2021)

    2021

  41. [43]

    H. Ma, J. H. Cullen, S. Monir, R. Rahman, and D. Cul- cer, Spin-hall effect in topological materials: evaluating the proper spin current in systems with arbitrary degen- eracies, npj Spintronics2, 55 (2024)

    2024

  42. [44]

    T. Sato, S. Haddad, I. C. Fulga, F. F. Assaad, and J. van den Brink, Altermagnetic anomalous hall effect emerging from electronic correlations, Phys. Rev. Lett. 133, 086503 (2024)

    2024

  43. [45]

    D. Go, D. Jo, C. Kim, and H.-W. Lee, Intrinsic spin and orbital hall effects from orbital texture, Phys. Rev. Lett. 121, 086602 (2018)

    2018

  44. [46]

    Y.-G. Choi, D. Jo, K.-H. Ko, D. Go, K.-H. Kim, H. G. Park, C. Kim, B.-C. Min, G.-M. Choi, and H.-W. Lee, Observation of the orbital hall effect in a light metal ti, Nature619, 52 (2023)

    2023

  45. [47]

    Lyalin, S

    I. Lyalin, S. Alikhah, M. Berritta, P. M. Oppeneer, and R.K.Kawakami,Magneto-opticaldetectionoftheorbital hall effect in chromium, Phys. Rev. Lett.131, 156702 (2023)

    2023

  46. [48]

    Tang and G

    P. Tang and G. E. W. Bauer, Role of disorder in the intrinsic orbital hall effect, Phys. Rev. Lett.133, 186302 (2024)

    2024

  47. [49]

    R. B. Atencia, D. P. Arovas, and D. Culcer, Intrinsic torque on the orbital angular momentum in an electric field, Phys. Rev. B110, 035427 (2024)

    2024

  48. [50]

    Liu and D

    H. Liu and D. Culcer, Dominance of extrinsic scattering mechanisms in the orbital hall effect: Graphene, transi- tion metal dichalcogenides, and topological antiferromag- nets, Phys. Rev. Lett.132, 186302 (2024)

    2024

  49. [51]

    J. H. Cullen, D. P. Arovas, R. Raimondi, and D. Culcer, Quantum geometry and dipolar dynamics in the orbital magneto-electric effect (2025), arXiv:2505.02911 [cond- mat.mes-hall]

    work page arXiv 2025

  50. [52]

    Wunderlich, B

    J. Wunderlich, B. Kaestner, J. Sinova, and T. Jung- wirth, Experimental observation of the spin-hall effect in a two-dimensional spin-orbit coupled semiconductor system, Phys. Rev. Lett.94, 047204 (2005)

    2005

  51. [53]

    Onoda, N

    S. Onoda, N. Sugimoto, and N. Nagaosa, Intrinsic ver- sus extrinsic anomalous hall effect in ferromagnets, Phys. Rev. Lett.97, 126602 (2006)

    2006

  52. [54]

    S.Onoda, N.Sugimoto,andN.Nagaosa,Quantumtrans- port theory of anomalous electric, thermoelectric, and thermal hall effects in ferromagnets, Phys. Rev. B77, 165103 (2008)

    2008

  53. [55]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys.82, 1959 (2010)

    1959

  54. [56]

    Niimi and Y

    Y. Niimi and Y. Otani, Reciprocal spin hall effects in conductors with strong spin–orbit coupling: a review, Reports on Progress in Physics78, 124501 (2015)

    2015

  55. [57]

    Sekine, D

    A. Sekine, D. Culcer, and A. H. MacDonald, Quantum kinetic theory of the chiral anomaly, Phys. Rev. B96, 235134 (2017)

    2017

  56. [58]

    G. Zala, B. N. Narozhny, and I. L. Aleiner, Interac- tion corrections at intermediate temperatures: Longitu- dinal conductivity and kinetic equation, Phys. Rev. B64, 214204 (2001)

    2001

  57. [59]

    Adroguer, W

    P. Adroguer, W. E. Liu, D. Culcer, and E. M. Han- kiewicz, Conductivity corrections for topological insula- tors with spin-orbit impurities: Hikami-larkin-nagaoka formula revisited, Phys. Rev. B92, 241402 (2015)

    2015

  58. [60]

    W. E. Liu, E. M. Hankiewicz, and D. Culcer, Weak localization and antilocalization in topological materi- als with impurity spin-orbit interactions, Materials10, 10.3390/ma10070807 (2017)

    work page doi:10.3390/ma10070807 2017

  59. [61]

    H. Liu, J. H. Cullen, D. P. Arovas, and D. Culcer, Quan- tum correction to the orbital hall effect, Phys. Rev. Lett. 134, 036304 (2025)

    2025

  60. [62]

    Bhalla, K

    P. Bhalla, K. Das, A. Agarwal, and D. Culcer, Quantum kinetic theory of nonlinear optical currents: Finite fermi surface and fermi sea contributions, Phys. Rev. B107, 165131 (2023)

    2023

  61. [63]

    V.Pandey, D.Joy, D.Culcer,andP.Bhalla,Longitudinal dc conductivity in dirac nodal line semimetals: Intrinsic and extrinsic contributions, Phys. Rev. B110, 155108 (2024)

    2024

  62. [64]

    R. B. Atencia, S. Liu, K. P. Loh, and D. Culcer, Room-temperature disorder-driven nonlinear transport in topological materials (2025), arXiv:2506.13869 [cond- mat.mes-hall]

    work page arXiv 2025

  63. [65]

    Bhalla, A

    P. Bhalla, A. H. MacDonald, and D. Culcer, Resonant photovoltaic effect in doped magnetic semiconductors, Phys. Rev. Lett.124, 087402 (2020)

    2020

  64. [66]

    Bhalla, K

    P. Bhalla, K. Das, D. Culcer, and A. Agarwal, Resonant second-harmonic generation as a probe of quantum ge- ometry, Phys. Rev. Lett.129, 227401 (2022)

    2022

  65. [67]

    J. H. Cullen and D. Culcer, Spin-hall effect due to the bulk states of topological insulators: Extrinsic contribu- tion to the proper spin current, Phys. Rev. B108, 245418 (2023)

    2023

  66. [68]

    K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agar- wal, Intrinsic nonlinear conductivities induced by the quantum metric, Phys. Rev. B108, L201405 (2023). 19

    2023

  67. [69]

    K. Das, K. Ghorai, D. Culcer, and A. Agarwal, Nonlinear valley hall effect, Phys. Rev. Lett.132, 096302 (2024)

    2024

  68. [70]

    Culcer, The anomalous hall effect (2022), book chapter for Elsevier Encyclopedia of Condensed Matter Physics, arXiv:2204.02434 [cond-mat.mes-hall]

    D. Culcer, The anomalous hall effect (2022), book chapter for Elsevier Encyclopedia of Condensed Matter Physics, arXiv:2204.02434 [cond-mat.mes-hall]

    work page arXiv 2022

  69. [71]

    P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys.57, 287 (1985)

    1985

  70. [72]

    Rammer and H

    J. Rammer and H. Smith, Quantum field-theoretical methods in transport theory of metals, Rev. Mod. Phys. 58, 323 (1986)

    1986

  71. [73]

    A. A. Kovalev, K. Výborný, and J. Sinova, Hybrid skew scattering regime of the anomalous hall effect in rashba systems: Unifying keldysh, boltzmann, and kubo for- malisms, Phys. Rev. B78, 041305 (2008)

    2008

  72. [74]

    Kamenev and A

    A. Kamenev and A. Levchenko, Keldysh technique and non-linearσ-model: basic principles and ap- plications, Advances in Physics58, 197 (2009), https://doi.org/10.1080/00018730902850504

    work page doi:10.1080/00018730902850504 2009

  73. [75]

    A. A. Kovalev, Y. Tserkovnyak, K. Výborný, and J. Sinova, Transport theory for disordered multiple-band systems: Anomalous hall effect and anisotropic magne- toresistance, Phys. Rev. B79, 195129 (2009)

    2009

  74. [76]

    Gorini, R

    C. Gorini, R. Raimondi, and P. Schwab, Onsager rela- tions in a two-dimensional electron gas with spin-orbit coupling, Phys. Rev. Lett.109, 246604 (2012)

    2012

  75. [77]

    C.Xiao, Z.Z.Du,andQ.Niu,Theoryofnonlinearhallef- fects: Modifiedsemiclassicsfromquantumkinetics,Phys. Rev. B100, 165422 (2019)

    2019

  76. [78]

    Nandy and I

    S. Nandy and I. Sodemann, Symmetry and quantum ki- netics of the nonlinear hall effect, Phys. Rev. B100, 195117 (2019)

    2019

  77. [79]

    R. B. Atencia, A. Agarwal, and D. Culcer, Orbital an- gular momentum of bloch electrons: equilibrium for- mulation, magneto-electric phenomena, and the orbital hall effect, Advances in Physics: X9, 2371972 (2024), https://doi.org/10.1080/23746149.2024.2371972

    work page doi:10.1080/23746149.2024.2371972 2024

  78. [80]

    K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agar- wal, Intrinsic nonlinear conductivities induced by the quantum metric, Phys. Rev. B108, L201405 (2023)

    2023

  79. [81]

    Aversa and J

    C. Aversa and J. E. Sipe, Nonlinear optical susceptibili- ties of semiconductors: Results with a length-gauge anal- ysis, Phys. Rev. B52, 14636 (1995)

    1995

  80. [82]

    Morimoto and N

    T. Morimoto and N. Nagaosa, Topologi- cal nature of nonlinear optical effects in solids, Science Advances2, e1501524 (2016), https://www.science.org/doi/pdf/10.1126/sciadv.1501524

    work page doi:10.1126/sciadv.1501524 2016

Showing first 80 references.