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REVIEW 2 major objections 5 minor 33 references

Spin geometric-phases in hopping magnetoconductance

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Rashba spin-orbit coupling plus a Zeeman field produces a $\sin(\Phi)$ term in the hopping magnetoconductance of an Aharonov-Bohm interferometer, so the conductance is even in the magnetic field but not periodic in flux.

desk verdict A self-contained tight-binding calculation gives a new sin(Phi) magnetoconductance term from Rashba+Zeeman; the aperiodic conclusion is solid for the triangle, but the 'any interferometer' claim is an understandable conjecture. read the letter →

arxiv 1908.05869 v2 pith:43JVVTST submitted 2019-08-16 cond-mat.mes-hall

O. Entin-Wohlman , A. Aharony This is my paper · ORCID
classification cond-mat.mes-hall
keywords Aharonov-BohminterferometerRashbaspin-orbitcouplingZeemaninteractionhoppingmagnetoconductanceAharonov-CasherphaseBerryspincurrentgeometric
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

The paper targets a question that bears on every attempt to read geometric phases from Aharonov-Bohm oscillations: what happens to the flux dependence of the conductance when both Rashba spin-orbit coupling and a Zeeman field act on electrons hopping through the interferometer. Its central claim is that the interference part of the magnetoconductance then contains a $\sin(\Phi)$ term whose amplitude is proportional to the Zeeman field, in addition to the usual $\cos(\Phi)$ term. Consequently the magnetoconductance is an even function of the magnetic field but not a periodic function of the flux, so assigning a phase shift to the Aharonov-Bohm oscillations, as earlier work did, is not justified. In the absence of the Zeeman field the charge interference is proportional to $\cos(\Phi)\cos(\gamma)$, so the only cleanly extractable phase is the Aharonov-Casher phase $\gamma$, while the spin currents are $\propto \sin(\Phi)\sin(\gamma)$ and are generally not conserved, driving magnetization buildup in the terminals. The explicit calculation is done for an arbitrarily shaped triangular loop with one quantum-dot arm.

What carries the argument

The load-bearing object is the tunneling amplitude of an electron across a straight edge of the loop, Eq. (B16): $G(s)\propto e^{-s/a}\left[\cos(k_2 s)+\frac{\sin(k_2 s)}{k_2}\left(i k_{so}\,\hat{e}\cdot\boldsymbol{\sigma}+m^*B a\,\sigma_z\right)\right]$, with $k_2=\sqrt{k_{so}^2-(m^*B a)^2}$. Without the Zeeman field this matrix is unitary and produces the Aharonov-Casher phase $\gamma$; with the Zeeman field the $\sigma_z$ term makes the amplitude non-unitary, which is what removes periodicity. The interference transmission is built from the product $V_{LR}V_{Rd}V_{dL}$ around the triangle (or its mirror path), and the scalar part of this product contains an imaginary term proportional to $k_{so}^2 B$ times a sum of cross products of the edge orientation vectors. That term, entering through $e^{-i\Phi}V_{LR}V_{Rd}V_{dL}$, is what generates the $\sin(\Phi)$ part of the conductance. The non-commutativity of the three edge matrices also gives the effective spin-orbit field a component along the field direction and defines the two tilt angles $\chi_L$ and $\chi_R$; the equilateral triangle is the special case where the two angles coincide.

What would settle it

Compute the interference part of the magnetoconductance for a square or regular $N$-gon Aharonov-Bohm loop that uses the same hopping tunneling amplitudes, with one arm carrying the same quantum dot, and both Rashba and Zeeman terms. If the result can be written as a sum of $\Phi$-periodic terms with field-dependent phase shifts, e.g. $\cos(\Phi\pm\theta(B))$, the central claim fails; if a residual $B\sin(\Phi)$ term survives, the claim is supported. An equivalent experimental check is to measure the conductance oscillations of a single mesoscopic hopping-regime loop as a function of flux at fixed Zeeman field: periodic oscillations with field-dependent peak shifts would contradict the paper, while an aperiodic even-in-field curve with a $B\sin(\Phi)$ component would confirm it.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that Zeeman and Rashba interactions combine in the hopping regime to change the interference conductance of an Aharonov-Bohm interferometer from a periodic function to the aperiodic, even-in-field form $a_1+a_2\cos(\Phi)+a_3 B\sin(\Phi)$, with corrections of order $B^2$ in each coefficient. The coefficient $a_3$ is proportional to the Zeeman field and vanishes when the spin-orbit coupling is absent; for an equilateral triangle it is $-3\sqrt{3}\,\sin^3(k_2 d)\,\bar{k}_{so}^2 B$ times the prefactor of the cos term. Because the same external magnetic field produces both the flux $\Phi$ and the Zeeman splitting, the $\sin(\Phi)$ term cannot be reinterpreted as a phase shift of periodic oscillations. The paper therefore concludes that the widely used phase-shift picture of earlier work does not apply, and that no Berry-phase information can be extracted from the Aharonov-Bohm interference pattern of a general loop. It also finds that spin currents in the two leads are generally not conserved, with spin-polarization directions set by two unrelated tilt angles $\chi_L$ and $\chi_R$; only circular loops, approached as the limit of regular polygons, seem to have the simple phase relations assumed before.

Load-bearing premise

The paper's broad conclusion that the aperiodic, no-phase-shift behavior and the absence of Berry-phase information hold for any interferometer rests on a lowest-order tunneling calculation for a single triangular loop with one quantum-dot arm; the generalization to other shapes is stated as an expectation, not derived.

Editorial extensions

If this is right

  • In a triangular interferometer with both interactions, the measured flux dependence contains a $B\sin(\Phi)$ component, so peak positions cannot be converted into a Berry or Aharonov-Casher phase shift.
  • At zero Zeeman field, the charge conductance's $\cos(\Phi)$ amplitude gives the Aharonov-Casher phase $\gamma$, and no other geometric phase is separately measurable from charge transport.
  • The spin currents into the two reservoirs are generally unequal; their difference deposits magnetization near the terminals, with in-plane polarization directions that depend on the loop shape.
  • Onsager symmetry is preserved: the full magnetoconductance is even in the magnetic field even though it is not periodic, consistent with time-reversal constraints.
  • Extracting the zero-field $\cos(\Phi)$ amplitude is the clean way to isolate the Rashba Aharonov-Casher phase; any finite-field data analysis must include the aperiodic $B\sin(\Phi)$ term.

Reading between the lines

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

  • A natural extension is that in disordered arrays of Rashba loops, the $B\sin(\Phi)$ aperiodic term may survive ensemble averaging differently from the $\cos(\Phi)$ term, producing a small linear-in-field correction to the average magnetoconductance that is not captured by usual phase-shift analyses.
  • Because $\Phi$ and $B$ are set by the same external field, separating the predicted $a_3B\sin(\Phi)$ term from a naive phase shift would require an experiment with an independent electrical control of the Rashba strength while sweeping the field; the coefficient $a_3$ should then scale linearly with the gate-controlled $k_{so}$.
  • For loops with more than three edges, the same edge-amplitude construction would determine whether the $B\sin(\Phi)$ coefficient survives or cancels; if it cancels for some symmetric polygons, the triangle result would mark the loop-shape dependence of the effect rather than a universal law.
  • The non-conservation of spin current implies that the terminals accumulate magnetization even at zero bias if a chemical-potential difference drives the current; this accumulation could be probed as a field-dependent spin signal in the contacts, an observable the paper mentions but does not quantify.
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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

2 major / 5 minor

Summary. The paper studies charge and spin transport through an Aharonov-Bohm interferometer in the hopping (tight-binding) regime, with Rashba spin-orbit coupling and a perpendicular Zeeman field. It derives spin-dependent tunneling amplitudes from a model Hamiltonian (Appendix B) and uses Keldysh Green's functions to obtain the transmission to leading order in tunneling (Appendix A). For a triangular loop with one quantum-dot arm, the interference contribution to the conductance is found to contain, in addition to the standard cos(Φ) term, a sin(Φ) term proportional to the Zeeman field (Sec. III B, Eq. (48)). Because the flux Φ and the Zeeman field are set by the same magnetic field, the magnetoconductance remains an even function of the field but is not periodic in it; the authors conclude that the usual 'phase shift' description of Aharonov-Bohm oscillations is inapplicable and that no Berry-phase information can be extracted from such transport measurements. They also find that spin currents in the leads are generally not conserved, implying spin accumulation in the terminals, and that the spin-polarization directions in the two leads are characterized by unrelated angles χ_L and χ_R.

Significance. The derivation is self-contained and carefully executed: the tunneling amplitudes are derived rather than assumed, the flux dependence is tracked through the Keldysh calculation, and the result satisfies the expected limits (vanishing when either spin-orbit coupling or the Zeeman field is absent) and respects Onsager symmetry. If the general claim is correct, it would correct a common interpretation of earlier work (Refs. 16 and 22) by showing that spin-orbit plus Zeeman effects do not simply shift the Aharonov-Bohm oscillation phase of each spin channel in the hopping regime. The explicit result Eq. (48) and the predicted a1 + a2 cos(Φ) + a3 B sin(Φ) form for the conductance are falsifiable and should motivate experiments in triangular or other non-circular interferometers. The main weakness is that the 'any interferometer' conclusion rests on a single triangular geometry and is only conjectured for other shapes; this limits the significance of the abstract's categorical claim unless the extension is proven or the claim is restricted.

major comments (2)
  1. [Sec. IV, Eq. (48)] The central claim stated in the abstract and in the Introduction—that the phase-shift concept is 'not applicable' to Aharonov-Bohm interferometers with Rashba and Zeeman interactions—is established only for the triangular loop with one quantum-dot arm (Sec. III B, Eq. (48)). The extension to 'any interferometer' is given in Sec. IV only as an expectation ('For symmetry reasons, we expect...'), and the Introduction labels the broader conclusion as a suspicion; no derivation is provided for square, polygonal, or multi-arm geometries. Because the closed-path product for a square loop contains a different number of noncommuting bond matrices, it could in principle yield a per-spin form cos(Φ ± δ(B)) and preserve the phase-shift description of Refs. 16 and 22 for those geometries. I recommend either deriving the general form for a class of loops or explicitly restricting the abstract and conclusions to the triangular geometry.
  2. [Introduction, Sec. IV] The paper also concludes that Berry-phase information cannot be extracted from transport measurements on a general interferometer (Introduction and Sec. IV). The calculation supports only the weaker statement that, for the triangle considered, the amplitude of the B sin(Φ) term in Eq. (48) is not simply related to the tilt angles χ_L and χ_R, and that no such relation was found. A general no-go claim would require either a model-space argument covering all interferometer shapes or an explicit treatment of at least one additional geometry; as written, the manuscript overstates the scope of its negative result.
minor comments (5)
  1. [Introduction] In the Introduction, 'the acquired phse' should read 'the acquired phase'.
  2. [Sec. III B, Eq. (48)] Equation (48) and the surrounding text use B and k_so without the overbar notation introduced in Eq. (C2), which may confuse the dimensionless Zeeman parameter \bar B = m^* B a / k_2 with the physical Zeeman energy B in Eq. (6).
  3. [Sec. II C, Eqs. (18) and (22)] The notation A_LR/A_RL and the directionality of each tunneling matrix V_ij would benefit from an explicit statement of which lead-to-lead direction each object corresponds to, since the phases in Eq. (8) and the factors e^{±iΦ} in Eqs. (18) and (22) are easy to misread.
  4. [Sec. III A, Eq. (38)] Equation (38) defines γ through tan(γ) = |c_L|/C, which requires C ≠ 0; the authors should comment on the limit C → 0, where the Aharonov-Casher phase parametrization is singular.
  5. [Appendix C, after Eq. (C14)] The statement that B is 'replaced by B/(ω−ε0)' is dimensionally loose, since the first B is the dimensionless \bar B = m^* B a / k_2 while the second is the energy B of Eq. (6); the intended meaning is clear but should be stated precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sin(Φ) magnetoconductance term is derived from the paper's own Hamiltonian and Green-function calculation, with no fitted input renamed as a prediction.

full rationale

The central derivation is self-contained. Appendix B starts from the tunneling Hamiltonian, Eq. (B1), and derives the spin-dependent propagator, Eq. (B16), which is then used to construct the tunneling matrices. Appendix A derives the transmission from Dyson equations and Langreth rules, yielding Eq. (17)-(18). The interference contribution, Eq. (22), is combined with the explicit triangle product in Appendix C; the linear-in-B scalar term, Eq. (C13), is translated into the sin(Φ) term in the magnetoconductance, Eq. (48), using only the algebraic structure of the derived amplitudes. No parameter is fitted to data and then called a prediction; the Zeeman-field proportionality follows from the Hamiltonian input, not from a prior claim. The self-citations (Refs. 14, 25, 30) are background or methodological: Ref. 14 supplies the known cos(Φ)cos(γ) form, Ref. 25 defines the spin-current observable, and Ref. 30 is a methodological predecessor whose Zeeman extension is rederived here. The statement that 'for symmetry reasons' any interferometer should show the same aperiodic form is an extrapolation beyond the triangular calculation, but an overbroad generalization is a correctness or scope concern, not a circularity: the cited expectation is not itself the input to the triangle derivation. Therefore the analysis finds no circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; the model inputs are physical constants of the problem. The paper relies on six stated modeling assumptions, none of which is an invented physical entity.

assumptions (6)
  • domain assumption Electrons in the tunneling barrier are described by the Rashba-Zeeman Hamiltonian of Eq. (B1) with negative kinetic energy E = -1/(2m*a^2), appropriate to the hopping regime.
    This Hamiltonian is the starting point for the tunneling amplitude derivation; all subsequent V matrices depend on this form.
  • domain assumption The barrier parameters satisfy [m*a^2]^{-1} >> kso^2/m* and [m*a^2]^{-1} >> B, stated in Eq. (B14).
    These inequalities justify the approximations k1 ≈ 1/a and k2 = sqrt(kso^2 - (m*Ba)^2), which lead to the closed-form propagator Eq. (B16).
  • domain assumption Tunneling amplitudes are evaluated at the Fermi surface and are independent of the lead momenta k and p (Eq. 10).
    This allows replacement of V_{kp} by V_{LR} etc., a simplification needed for the Green's function sums in Appendix A.
  • domain assumption The transmission is expanded to lowest order in the tunneling amplitudes, keeping only the leading interference paths (Eq. A14).
    All final formulas, including the B sin(Φ) term, are lowest-order results; higher-order terms are neglected without a quantitative error estimate.
  • domain assumption The leads are unpolarized and the quantum dot is empty, as in Eqs. (1) and (A2).
    The Fermi distribution functions and the decoupled dot Green's function assume no initial spin polarization and no dot occupation.
  • domain assumption The same external magnetic field generates both the Aharonov-Bohm flux Φ and the Zeeman energy B, so Φ and B are not independent control knobs (Sec. IV).
    This coupling is used to conclude that the conductance is non-periodic but even in the field; independent tunability would allow a phase-shift interpretation.

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Pith. "Pith review of Spin geometric-phases in hopping magnetoconductance." pith.science (2026). https://pith.science/paper/43JVVTST

@misc{pith2026190805869,
  author       = {Pith},
  title        = {Pith review of: Spin geometric-phases in hopping magnetoconductance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43JVVTST}},
  note         = {Machine review of arXiv:1908.05869}
}
abstract

We identify theoretically the geometric phases of the electrons' spin that can be detected in measurements of charge and spin transport through Aharonov-Bohm interferometers threaded by a magnetic flux $\Phi$ (in units of the flux quantum) in which both the Rashba spin-orbit and Zeeman interactions are active. We show that the combined effect of these two interactions is to produce a $\sin(\Phi)$ [in addition to the usual $\cos(\Phi)$] dependence of the magnetoconductance, whose amplitude is proportional to the Zeeman field. Therefore the magnetoconductance, though an even function of the magnetic field is not a periodic function of it, and the widely-used concept of a phase shift in the Aharonov-Bohm oscillations, as indicated in previous work, is not applicable. We find the directions of the spin-polarizations in the system, and show that in general the spin currents are not conserved, implying the generation of magnetization in the terminals attached to the interferometer.

Figures

Figures reproduced from arXiv: 1908.05869 by the authors.

Figure 1
Figure 1. FIG. 1: (Color online.) Illustration of our model. A triangu [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Aharonov and A

    Y. Aharonov and A. Casher, Topological quantum Effects for Neutral Particles , Phys. Rev. Lett. 53 , 319 (1984)

    work page 1984

  2. [2]

    Aharonov and D

    Y. Aharonov and D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory , Phys. Rev. 115 , 485 (1959)

    work page 1959

  3. [3]

    E. I. Rashba, Properties of semiconductors with an extremum loop .1. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop , Fiz. Tverd. Tela (Leningrad) 2 , 1224 (1960) [Sov. Phys. Solid State 2 , 1109 (1960)]; Y. A. Bychkov and E. I. Rashba, Oscillatory effects and the magnetic susceptibility of carriers in inv...

    work page 1960

  4. [4]

    Nitta, T

    J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Gate Control of Spin-Orbit Interaction in an Inverted In _ 0.53 Ga _ 0.47 As/In _ 0.52 Al _ 0.48 As Heterostructure , Phys. Rev. Lett. 78 , 1335 (1997)

    work page 1997

  5. [5]

    Bergsten, T

    T. Bergsten, T. Kobayashi, Y. Sekine, and J. Nitta, Experimental Demonstration of the Time Reversal Aharonov-Casher Effect , Phys. Rev. Lett. 97 , 196803 (2006)

    work page 2006

  6. [6]

    Qian and Z-B

    T-Z. Qian and Z-B. Su, Spin-Orbit Interaction and Aharonov-Anandan Phase in Mesoscopic Rings , Phys. Rev. Lett. 72 , 2311 (1994)

    work page 1994

  7. [7]

    M. V. Berry, Quantal phase factors accompanying adiabatic changes , Proc. Royal Soc. A 392 , 45 (1984)

    work page 1984

  8. [8]

    Vanderbilt, Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators , Cambridge University Press (2018)

    D. Vanderbilt, Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators , Cambridge University Press (2018)

    work page 2018

Show all 33 references
  1. [9]

    Aharonov and J

    Y. Aharonov and J. Anandan, Phase Change during a Cyclic Quantum Evolution , Phys. Rev. Lett. 58 , 1593 (1987)

    1987

  2. [10]

    D. Loss, P. Goldbart, and A. V. Balatsky, Berry's Phase and Persistent Charge and Spin Currents in Textured Mesoscopic Rings , Phys. Rev. Lett. 65 , 1655 (1990)

    1990

  3. [11]

    Oh and C-M

    S. Oh and C-M. Ryu, Persistent spin currents induced by the Aharonov-Casher effect in mesoscopic rings , Phys. Rev. B 51 , 13441 (1995)

    1995

  4. [12]

    Ellner, N

    M. Ellner, N. Bolivar, B. Berche, and E. Medina, Charge- and spin-polarized currents in mesoscopic rings with Rashba spin-orbit interactions coupled to an electron reservoir , Phys. Rev. B 90 , 085305 (2014)

    2014

  5. [13]

    Avishai, K

    Y. Avishai, K. Totsuka, and N. Nagaosa, Non-Abelian Aharonov-Casher Phase Factor in Mesoscopic Systems , J. Phys. Soc. Jpn. 88 , 084705 (2019)

    2019

  6. [14]

    Y. Meir, Y. Gefen, and O. Entin-Wohlman, Universal effects of spin-orbit scattering in mesoscopic systems , Phys. Rev. Lett. 63 , 798 (1989)

    1989

  7. [15]

    Nitta, F

    J. Nitta, F. E. Meijer, and H. Takayanagi, Spin-interference device , Appl. Phys. Lett. 75 , 695 (1999)

    1999

  8. [16]

    Moln\' a r, F

    B. Moln\' a r, F. M. Peeters, and P. Vasilopoulos, Spin-dependent magnetotransport through a ring due to spin-orbit interaction , Phys. Rev. B. 69 , 155335 (2004)

    2004

  9. [17]

    Frustaglia and K

    D. Frustaglia and K. Richter, Spin interference effects in ring conductors subject to Rashba coupling , Phys. Rev. B 69 , 235310 (2004)

    2004

  10. [18]

    Capozza, D

    R. Capozza, D. Giuliano, P. Lucignano, and A. Tagliacozzo, Quantum Interference of Electrons in a Ring: Tuning of the Geometrical Phase , Phys. Rev. Lett. 95 , 226803 (2005)

    2005

  11. [19]

    J-B. Yau, E. P. De Poortere, and M. Shayegan, Aharonov-Bohm Oscillations with Spin: Evidence for Berry's Phase , Phys. Rev. Lett. 88 , 146801 (2002)

    2002

  12. [20]

    K\" o nig, A

    M. K\" o nig, A. Tschetschetkin, E. M. Hankiewiccz, J. Sinova, V. Hock, V. Daumer, M. Schaefer, C. R. Becker, H. Buhmann, and L. W. Molenkamp, Direct Observation of Aharonov-Casher Phase , Phys. Rev. Lett. 96 , 076804 (2006)

    2006

  13. [21]

    Nagasawa, J

    F. Nagasawa, J. Takagi, Y. Kunihashi, M. Kohda, and J. Nitta, Experimental Demonstration of Spin Geometric Phase: Radius Dependence of Time-Reversal Aharonov-Casher Oscillations , Phys. Rev. Lett. 108 , 086801 (2012)

    2012

  14. [22]

    A. G. Aronov and Y. B. Lyanda-Geller, Spin-Orbit Berry Phase in Conducting Rings , Phys. Rev. Lett. 70 , 343 (1993)

    1993

  15. [23]

    F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, One-dimensional ring in the presence of Rashba spin-orbit interaction: Derivation of the correct Hamiltonian , Phys. Rev. B 66 , 033107 (2002)

    2002

  16. [24]

    Y-S. Yi, T-Z. Qian, and Z-B. Su, Spin precession and time-reversal symmetry breaking in quantum transport of electrons through mesoscopic rings , Phys. Rev. B 55 , 10631 (1997)

    1997

  17. [25]

    Entin-Wohlman, A

    O. Entin-Wohlman, A. Aharony, Y. Tokura, and Y. Avishai, Spin-polarized electric currents in quantum transport through tubular two-dimensional electron gases , Phys. Rev. B 81 , 075439 (2010)

    2010

  18. [26]

    As seen in Appendix triangle , the - dependent terms are all proportional to the product of the three J 's

  19. [27]

    D. C. Langreth, Linear and nonlinear response theory with applications , in Linear and Nonlinear Electron Transport in Solids, eds. J. T. Devreese and E. van Boren (Plenum, New York, 1976)

    1976

  20. [28]

    A. P. Jauho, Nonequilibrium Green function modelling of transport in mesoscopic systems , in Progress in Nonequilibrium Green's Functions II, eds. M. Bonitz and D.Semkat (World Scientific, Singapore, 2003)

    2003

  21. [29]

    T. V. Shahbazyan and M. E. Raikh, Low-Field Anomaly in 20 Hopping Magnetoresistance Caused by Spin-Orbit Term in the Energy Spectrum , Phys. Rev. Lett. 73 , 1408 (1994)

    1994

  22. [30]

    Aharony, O

    A. Aharony, O. Entin-Wohlman, M. Jonson, and R. I. Shekhter, Electric and magnetic gating of Rashba-active weak links , Phys. Rev. B 97 , 220404(R) (2018), see also Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevB.97.220404

    2018 doi

  23. [31]

    ( triple ) in conjunction with (k^ _ 2 d^ _ ij ) , and hence all terms in Eq

    Note that k_ 2 = k^ 2 _ so -(m^ aB)^ 2 always appears in the terms of Eq. ( triple ) in conjunction with (k^ _ 2 d^ _ ij ) , and hence all terms in Eq. ( triple ) are real except for those that are explicitly multiplied by i

  24. [32]

    In the presence of the Zeeman interaction, the - independent part of the transmission, T ^0_L , also contributes terms of order B^2 to the conductance

  25. [33]

    Stern, Berry's Phase, Motive Forces, and Mesoscopic Conductivity , Phys

    A. Stern, Berry's Phase, Motive Forces, and Mesoscopic Conductivity , Phys. Rev. Lett. 68 , 1022 (1992)

    1992

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