pith. sign in

arxiv: 2604.17672 · v1 · submitted 2026-04-19 · ❄️ cond-mat.mes-hall

Magnetoresistance from decoherence

Xian-Peng Zhang , Yan-Qing Feng , Haiwen Liu , Yugui Yao This is my paper

Pith reviewed 2026-05-10 04:53 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords magnetoresistancequantum decoherencedensity matrixFermi seaimpurity scatteringmagnetotransport
0
0 comments X

The pith

Magnetoresistance arises from the decay of off-diagonal density-matrix elements throughout the entire Fermi sea.

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

The paper shows that magnetoresistance can originate from quantum decoherence affecting the whole Fermi sea instead of only surface scattering. Conductivity is expressed through two complex decoherence times whose real and imaginary parts control the decay of off-diagonal density-matrix components. This produces a conductivity that rises linearly with impurity density, opposite to the conventional Drude dependence. The same framework accounts for a temperature-driven sign change in magnetoresistance and a conductivity peak that resembles Kondo behavior when magnetic and exchange fields interact.

Core claim

Magnetoresistance is generated by the decoherence of off-diagonal density-matrix elements across the full Fermi sea; the resulting conductivity is parameterized by two complex decoherence times and therefore scales linearly with impurity density rather than inversely as in the Drude model.

What carries the argument

Two complex decoherence times that capture the decay rate and phase relaxation of off-diagonal density-matrix components throughout the Fermi sea.

If this is right

  • Conductivity becomes directly proportional to impurity density instead of inversely proportional.
  • External magnetic field and internal exchange field together produce a crossover from positive to negative magnetoresistance with rising temperature.
  • A nonmonotonic temperature dependence appears, with a maximum in conductivity at intermediate temperatures.
  • Quantum decoherence becomes an electrically measurable quantity in bulk transport.

Where Pith is reading between the lines

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

  • The linear scaling offers an electrical route to quantify decoherence rates in materials where conventional mobility measurements are dominated by other scattering channels.
  • The mechanism may be relevant in devices where coherence loss across many states, rather than at the Fermi surface alone, limits performance.
  • Similar density-matrix decoherence terms could be added to transport calculations in other systems exhibiting non-Drude scaling.

Load-bearing premise

Decoherence of off-diagonal density-matrix elements can be fully described by two complex times and remains the dominant process controlling magnetoresistance.

What would settle it

Measurement showing conductivity increasing linearly with added impurity density at fixed low temperature, or absence of the predicted temperature-driven crossover in magnetoresistance sign when exchange fields are present.

Figures

Figures reproduced from arXiv: 2604.17672 by Haiwen Liu, Xian-Peng Zhang, Yan-Qing Feng, Yugui Yao.

Figure 1
Figure 1. Figure 1: (a) The ni dependence of σ dia L [Eq. (12)], σ off L = σ ∥ L + σ ⊥ L [Eqs. (8) and (9)], and σL = σ dia L + σ off L where ϵF = 0 meV the Fermi energy only cuts η = − band. Panel (b) corresponds to ϵF = 0.5 meV where the Fermi energy cuts both η = − and η = + bands. Other parameters: vR/a = 1 meV, and ϵL = 0.2 meV, U/a2 = 0.8 eV, and a = 3.5 nm. largely neglecting the dynamics of quantum coherence and its d… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The ni dependence of dependence of σ ∥ L [Eqs. (8)], σ ⊥ L [Eq. (9)], and σ off L = σ ∥ L + σ ⊥ L . Panel (b) plots the corresponding ϵL dependence. Here, vR/a = 3 meV and ϵF = 0.5meV. Other parameters are the same as [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a-b) B dependence of σ off L for (a) nSJsd = +0.2 meV and (b) nSJsd = −0.2 meV. Panels (c) and (d) plot the corresponding T dependence. Here, nia 2 = 0.02%, S = 1, Tc = 100 K, and other parameters are the same as [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Microscopic theories of magnetoresistance have traditionally focused on momentum relaxation and the plasma frequency of itinerant electrons. Here, we uncover a distinct mechanism in which magnetoresistance originates from quantum decoherence throughout the whole Fermi sea, specifically the decay of the off-diagonal components of the density matrix. The resulting conductivity, parameterized by two complex decoherence times, scales linearly with impurity density-markedly contrasting the conventional Drude picture, where conductivity is governed by momentum relaxation of Ferm-surface quasiparticles and is inversely proportional to impurity density. This unconventional scaling provides a direct electrical probe of quantum decoherence, a quantity central to both fundamental studies and emerging nanoscale technologies. Furthermore, the interplay between the external magnetic field and the exchange field gives rise to rich magnetotransport phenomena, including temperature-drive crossover from positive to negative magnetoresistance and a nonmonotonic temperature dependence with a conductivity maximum reminiscent of the Kondo effect. Our results establish quantum decoherence as a key ingredient in magnetoresistance and our findings should have an unprecedented impact on advancing research and applications involving magnetoresistance.

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 claims that magnetoresistance originates from quantum decoherence throughout the whole Fermi sea via decay of off-diagonal density-matrix elements, rather than conventional momentum relaxation at the Fermi surface. Conductivity is parameterized by two complex decoherence times and scales linearly with impurity density (contrasting Drude inverse scaling). The model predicts temperature-driven crossover from positive to negative magnetoresistance and nonmonotonic temperature dependence with a conductivity maximum, arising from interplay between external magnetic field and exchange field.

Significance. If the central derivation holds, the result would establish quantum decoherence as a distinct, measurable contributor to magnetotransport and provide an electrical probe of decoherence times in mesoscopic systems. The linear impurity-density scaling and predicted crossovers could explain anomalous behaviors beyond standard Fermi-surface pictures, with potential relevance to nanoscale technologies. The approach is novel in shifting focus from surface to bulk decoherence but requires explicit microscopic grounding to be convincing.

major comments (2)
  1. [conductivity derivation and results sections] The conductivity parameterization by two complex decoherence times (introduced to capture off-diagonal decay across the Fermi sea) is the load-bearing step for the linear scaling with impurity density. The manuscript must explicitly derive or benchmark these times from a microscopic model (e.g., via impurity scattering Hamiltonian) and demonstrate that they remain independent of impurity density while conventional momentum-relaxation channels do not dominate; otherwise the claimed contrast to Drude scaling is not secured.
  2. [magnetotransport calculation and comparison to Drude] Standard Kubo or Boltzmann derivations localize conductivity contributions to the Fermi surface and yield inverse impurity scaling via momentum relaxation. The paper needs to show, with explicit equations, how the bulk decoherence terms produce the opposite linear dependence without violating current conservation or being overwhelmed by other relaxation processes (as flagged in the weakest assumption).
minor comments (2)
  1. [introduction and model section] Clarify the notation for the two complex decoherence times early in the text and specify their physical interpretation (real and imaginary parts) to aid readability.
  2. [discussion] Add a brief discussion or reference to how the model reduces to known limits (e.g., zero magnetic field or high temperature) to strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The concerns about the microscopic foundation of the decoherence times and the explicit contrast to standard transport calculations are well taken. We address each major comment below and have revised the manuscript to incorporate additional derivations and clarifications.

read point-by-point responses

  1. Referee: [conductivity derivation and results sections] The conductivity parameterization by two complex decoherence times (introduced to capture off-diagonal decay across the Fermi sea) is the load-bearing step for the linear scaling with impurity density. The manuscript must explicitly derive or benchmark these times from a microscopic model (e.g., via impurity scattering Hamiltonian) and demonstrate that they remain independent of impurity density while conventional momentum-relaxation channels do not dominate; otherwise the claimed contrast to Drude scaling is not secured.

    Authors: We agree that an explicit microscopic derivation is required to substantiate the parameterization. In the revised manuscript we have added a dedicated subsection deriving the two complex decoherence times directly from the impurity scattering Hamiltonian via the time evolution of the off-diagonal density-matrix elements under the Born approximation. The resulting decoherence rates are proportional to the impurity density, yet the times themselves are defined independently of the conventional Fermi-surface momentum-relaxation time. We include a benchmark calculation for a model impurity potential that confirms the bulk decoherence channel is not subsumed by momentum relaxation, thereby preserving the linear conductivity scaling with impurity density. revision: yes

  2. Referee: [magnetotransport calculation and comparison to Drude] Standard Kubo or Boltzmann derivations localize conductivity contributions to the Fermi surface and yield inverse impurity scaling via momentum relaxation. The paper needs to show, with explicit equations, how the bulk decoherence terms produce the opposite linear dependence without violating current conservation or being overwhelmed by other relaxation processes (as flagged in the weakest assumption).

    Authors: We have expanded the magnetotransport section with the requested explicit equations. The conductivity is obtained by integrating the current operator matrix elements over the entire Fermi sea, where the off-diagonal decay terms contribute a factor proportional to the decoherence rate; because this rate scales linearly with impurity density, the conductivity does likewise. Charge conservation is maintained because the diagonal elements of the density matrix remain unaffected. We have added a paragraph discussing the weakest assumption, clarifying that our model isolates the decoherence mechanism and that other relaxation channels are subdominant in the parameter regime where bulk decoherence dominates; a direct comparison with the Drude result is now shown in a new figure. revision: yes

Circularity Check

1 steps flagged

Linear impurity-density scaling of conductivity follows from parameterization by two complex decoherence times

specific steps
  1. fitted input called prediction [Abstract]
    "The resulting conductivity, parameterized by two complex decoherence times, scales linearly with impurity density-markedly contrasting the conventional Drude picture, where conductivity is governed by momentum relaxation of Ferm-surface quasiparticles and is inversely proportional to impurity density."

    Conductivity is defined via the two complex decoherence times; the linear scaling with impurity density is then stated as the resulting behavior. Because the times themselves are introduced as parameters without a first-principles derivation of their density dependence shown in the text, the scaling is forced by the parameterization choice rather than independently obtained.

full rationale

The abstract presents the conductivity as directly parameterized by two complex decoherence times whose impurity-density dependence is not shown to arise from an independent microscopic calculation. This makes the claimed linear scaling a direct consequence of the chosen parameterization rather than an emergent prediction. No other load-bearing steps are visible in the provided text, and the conventional Drude contrast is stated but not reduced to a self-citation chain or redefinition. The central claim therefore contains partial circularity at the level of the modeling assumption, but remains partially independent because the paper asserts a distinct whole-Fermi-sea mechanism.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Central claim depends on introducing two complex decoherence times as parameters and assuming density-matrix decoherence applies uniformly across the Fermi sea to produce the linear impurity scaling.

free parameters (1)
  • two complex decoherence times
    Conductivity is parameterized by them to obtain the linear impurity-density scaling and temperature crossovers; no derivation or external calibration given in abstract.
axioms (1)
  • domain assumption Quantum density matrix formalism with off-diagonal decay describes transport in the Fermi sea under magnetic and exchange fields
    Invoked to link decoherence directly to conductivity without detailing how it reduces to the stated linear scaling.

pith-pipeline@v0.9.0 · 5487 in / 1357 out tokens · 44905 ms · 2026-05-10T04:53:23.573199+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Theory of quantum decoherence in macroscopic topological insulators

    cond-mat.mes-hall 2026-04 unverdicted novelty 7.0

    Decoherence in macroscopic topological insulators produces quadratic corrections with impurity density, a stronger second-order skew-scattering channel for the extrinsic spin Hall effect, and a quadratic scaling law b...

  2. Metalization of topological insulators

    cond-mat.mes-hall 2026-04 unverdicted novelty 6.0

    Impurity-scattering-induced coherence decay produces finite longitudinal conductivity in Berry-curvature-dominated topological insulators without Fermi-level carriers, with linear impurity scaling and 1/T temperature ...

Reference graph

Works this paper leans on

51 extracted references · 4 canonical work pages · cited by 2 Pith papers

  1. [1]

    Thomson, XIX

    W. Thomson, XIX. on the electro-dynamic qualities of metals: Effects of magnetization on the electric conduc- tivity of Nickel and of Iron, Proc. R. Soc. Lond. , 546 (1857)

  2. [2]

    N. F. Mott, The electrical conductivity of transition met- als, P. Roy. Soc. A-Math. Phys.153, 699 (1936)

    1936

  3. [3]

    N. F. Mott, The resistance and thermoelectric properties of the transition metals, P. Roy. Soc. A-Math. Phy.156, 368 (1936)

    1936

  4. [4]

    N. F. Mott, Electrons in transition metals, Adv. Phys. 13, 325 (1964)

    1964

  5. [5]

    D. A. Goodings, Electrical resistivity of ferromagnetic metals at low temperatures, Phys. Rev.132, 542 (1963)

    1963

  6. [6]

    Yosida, Anomalous electrical resistivity and magne- toresistance due to ans−dinteraction in Cu-Mn alloys, Phys

    K. Yosida, Anomalous electrical resistivity and magne- toresistance due to ans−dinteraction in Cu-Mn alloys, Phys. Rev.107, 396 (1957)

    1957

  7. [7]

    Raquet, M

    B. Raquet, M. Viret, E. Sondergard, O. Cespedes, and R. Mamy, Electron-magnon scattering and magnetic re- sistivity in3dferromagnets, Phys. Rev. B66, 024433 (2002)

    2002

  8. [8]

    A. P. Mihai, J. P. Attané, A. Marty, P. Warin, and Y. Samson, Electron-magnon diffusion and magnetiza- tion reversal detection inFePtthin films, Phys. Rev. B 77, 060401 (2008)

    2008

  9. [9]

    Zhang, W

    X.-P. Zhang, W. Feng, R.-W. Zhang, X. Fan, X. Wang, and Y. Yao, Theory of anisotropic magnetoresistance in altermagnets and its applications, Phys. Rev. Lett.135, 266706 (2025)

    2025

  10. [10]

    Zhang, Extrinsic spin-valley Hall effect and spin-relaxation anisotropy in magnetized and strained graphene, Phys

    X.-P. Zhang, Extrinsic spin-valley Hall effect and spin-relaxation anisotropy in magnetized and strained graphene, Phys. Rev. B106, 115437 (2022)

    2022

  11. [11]

    Zhang, Y

    X.-P. Zhang, Y. Yao, K. Y. Wang, and P. Yan, Micro- scopic theory of spin-orbit torque and spin memory loss from interfacial spin-orbit coupling, Phys. Rev. B108, 125309 (2023)

    2023

  12. [12]

    Zhang, F

    X.-P. Zhang, F. S. Bergeret, and V. N. Golovach, The- ory of spin Hall magnetoresistance from a microscopic perspective, Nano Lett.19, 6330 (2019)

    2019

  13. [13]

    Dieny, I

    B. Dieny, I. L. Prejbeanu, K. Garello, P. Gambardella, P. Freitas, R. Lehndorff, W. Raberg, U. Ebels, S. O. Demokritov, J. Akerman, A. Deac, P. Pirro, C. Adel- mann, A. Anane, A. V. Chumak, A. Hirohata, S. Man- gin, S. O. Valenzuela, M. C. Onbaşlı, M. d’Aquino, G. Prenat, G. Finocchio, L. Lopez-Diaz, R. Chantrell, O. Chubykalo-Fesenko, and P. Bortolotti, ...

    2020

  14. [14]

    Drude, Zur elektronentheorie der metalle, Annalen der Physik312, 687 (1902)

    P. Drude, Zur elektronentheorie der metalle, Annalen der Physik312, 687 (1902)

    1902

  15. [15]

    Landau, Diamagnetismus der metalle, Zeitschrift für Physik64, 629 (1930)

    L. Landau, Diamagnetismus der metalle, Zeitschrift für Physik64, 629 (1930)

    1930

  16. [16]

    Bastin, C

    A. Bastin, C. Lewiner, O. Betbeder-Matibet, and P. Nozieres, Quantum oscillations of the Hall effect of a fermion gas with random impurity scattering, J. Phys. Chem. Solids.32, 1811 (1971)

    1971

  17. [17]

    Abrikosov, Galvanomagnetic phenomena in metals in the quantum limit, Sov

    A. Abrikosov, Galvanomagnetic phenomena in metals in the quantum limit, Sov. Phys. JETP29, 746 (1969)

    1969

  18. [18]

    A. A. Abrikosov, Quantum magnetoresistance, Phys. Rev. B58, 2788 (1998)

    1998

  19. [19]

    Abrikosov, Quantum linear magnetoresistance; solu- tion of an old mystery, J

    A. Abrikosov, Quantum linear magnetoresistance; solu- tion of an old mystery, J. Phys. A: Math. Gen.36, 9119 (2003)

    2003

  20. [20]

    R. Xu, A. Husmann, T. Rosenbaum, M.-L. Saboungi, J. Enderby, and P. Littlewood, Large magnetoresistance in non-magnetic silver chalcogenides, Nature390, 57 (1997)

    1997

  21. [21]

    A. A. Abrikosov, Quantum magnetoresistance of layered semimetals, Phys. Rev. B60, 4231 (1999)

    1999

  22. [22]

    Aifer, F

    A.L.Friedman, J.L.Tedesco, P.M.Campbell, J.C.Cul- bertson, E. Aifer, F. K. Perkins, R. L. Myers-Ward, J. K. Hite, C. R. Eddy Jr, G. G. Jernigan,et al., Quantum lin- ear magnetoresistance in multilayer epitaxial graphene, Nano Lett.10, 3962 (2010)

    2010

  23. [23]

    Z.-C. Wang, L. Chen, S.-S. Li, J.-S. Ying, F. Tang, G.-Y. Gao, Y. Fang, W. Zhao, D. Cortie, X. Wang,et al., Gi- ant linear magnetoresistance in half-metallic Sr2CrMoO6 thin films, npj Quantum Mater.6, 53 (2021)

    2021

  24. [24]

    Khouri, U

    T. Khouri, U. Zeitler, C. Reichl, W. Wegscheider, N. Hussey, S. Wiedmann, and J. Maan, Linear magne- toresistance in a quasifree two-dimensional electron gas in an ultrahigh mobility GaAs quantum well, Phys. Rev. Lett.117, 256601 (2016)

    2016

  25. [25]

    Novak, S

    M. Novak, S. Sasaki, K. Segawa, and Y. Ando, Large linear magnetoresistance in the dirac semimetal TlBiSSe, Phys. Rev. B91, 041203 (2015)

    2015

  26. [26]

    Xiao, M.-C

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

    1959

  27. [27]

    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

  28. [28]

    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

  29. [29]

    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 6 96, 035106 (2017)

    2017

  30. [31]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys.89, 041003 (2017)

    2017

  31. [32]

    W. H. Zurek, Decoherence, einselection, and the quan- tum origins of the classical, Rev. Mod. Phys.75, 715 (2003)

    2003

  32. [33]

    X. Z. Zhang, Y. Q. Feng, H. Liu, W. X. Feng, and Y. G. Yao, Theory of quantum decoherence and its application to anomalous Hall effect, arXiv preprint arXiv:2603.29690 (2026)

    work page arXiv 2026

  33. [34]

    Sekine, D

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

    2017

  34. [35]

    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, Physi- cal Review Research4, 013001 (2022)

    2022

  35. [36]

    C. M. Varma, Colloquium: Linear in temperature re- sistivity and associated mysteries including high tem- perature superconductivity, Rev. Mod. Phys.92, 031001 (2020)

    2020

  36. [37]

    R. A. Cooper, Y. Wang, B. Vignolle, O. Lipscombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike, M. Nohara, H. Takagi,et al., Anomalous criticality in the electrical resistivity of la2–x sr x cuo4, Science323, 603 (2009)

    2009

  37. [38]

    K. Jin, N. Butch, K. Kirshenbaum, J. Paglione, and R. Greene, Link between spin fluctuations and electron pairing in copper oxide superconductors, Nature476, 73 (2011)

    2011

  38. [39]

    Licciardello, J

    S. Licciardello, J. Buhot, J. Lu, J. Ayres, S. Kasahara, Y. Matsuda, T. Shibauchi, and N. Hussey, Electrical re- sistivity across a nematic quantum critical point, Nature 567, 213 (2019)

    2019

  39. [40]

    J. A. N. Bruin, H. Sakai, R. S. Perry, and A. Mackenzie, Similarity of scattering rates in metals showing t-linear resistivity, Science339, 804 (2013)

    2013

  40. [41]

    Legros, S

    A. Legros, S. Benhabib, W. Tabis, F. Laliberté, M. Dion, M. Lizaire, B. Vignolle, D. Vignolles, H. Raffy, Z. Li, et al., Universal t-linear resistivity and planckian dissi- pation in overdoped cuprates, Nat. Phys.15, 142 (2019)

    2019

  41. [42]

    Zhang, Y

    X.-P. Zhang, Y. Feng, W. Feng, and Y. Yao, Metalization of topological insulators, (Unfinished)

  42. [43]

    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

  43. [44]

    Onoda, N

    S. Onoda, N. Sugimoto, and N. Nagaosa, Quantum transport theory of anomalous electric, thermoelectric, and thermal Hall effects in ferromagnets, arXiv preprint arXiv:0712.0210 (2007)

    work page arXiv 2007

  44. [45]

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

    2009

  45. [46]

    Smit, The spontaneous Hall effect in ferromagnetics I, Physica21, 877 (1955)

    J. Smit, The spontaneous Hall effect in ferromagnetics I, Physica21, 877 (1955)

    1955

  46. [47]

    Smit, The spontaneous Hall effect in ferromagnetics II, Physica24, 39 (1958)

    J. Smit, The spontaneous Hall effect in ferromagnetics II, Physica24, 39 (1958)

    1958

  47. [48]

    Berger, Side-jump mechanism for the Hall effect of ferromagnets, Phys

    L. Berger, Side-jump mechanism for the Hall effect of ferromagnets, Phys. Rev. B2, 4559 (1970)

    1970

  48. [49]

    Karplus and J

    R. Karplus and J. Luttinger, Hall effect in ferromagnet- ics, Phys. Rev.95, 1154 (1954)

    1954

  49. [50]

    Jungwirth, Q

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

    2002

  50. [51]

    Zhang, X

    X.-P. Zhang, X. R. Wang, and Y. G. Yao, Quantum spin decoherence theory of magnetoresistance in meso- scopic ferromagnets and its applications, arXiv preprint arXiv:2406.13932 (2024)

    work page arXiv 2024

  51. [52]

    X. Z. Zhang, X. R. Wang, and Y. G. Yao, Open quan- tum theory of magnetoresistance in mesoscopic magnetic materials, arXiv preprint arXiv:2512.24586 (2025)

    work page arXiv 2025