Nonlinear Hall effect driven by spin-charge-coupled motive force

Hikaru Watanabe; Kohei Hattori; Ryotaro Arita

REVIEW 3 major objections 4 minor 89 references

Nonlinear Hall effect driven by spin-charge-coupled motive force

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that in PT-symmetric collinear antiferromagnetic metals, the low-frequency nonlinear Hall effect is dominated by a mixed dipole term from light-spin interference, with $\tau^2$ clean-limit scaling.

desk verdict A new mixed-dipole mechanism for the nonlinear Hall effect in PT-symmetric antiferromagnets, well backed by simulations but with an internal sign inconsistency in the symmetry appendix that must be fixed before the central claim is secure. read the letter →

arxiv 2501.11234 v3 pith:26FHHVMR submitted 2025-01-20 cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-sci

Kohei Hattori , Hikaru Watanabe , Ryotaro Arita This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallcond-mat.mtrl-sci

keywords nonlinearHalleffectmixeddipolePTsymmetrycollinearantiferromagnetEdelsteinspin-chargecouplingreal-timesimulationBerrycurvature

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 tries to establish that in parity-time-reversal ($\mathcal{PT}$)-symmetric collinear antiferromagnetic metals, the low-frequency nonlinear Hall effect is dominated not by the usual Drude or Berry-curvature-dipole mechanisms but by a mixed dipole term generated when the light field and the electrically induced motion of localized spins act together. The authors derive analytic second-order optical conductivities, classify every contribution under $\mathcal{PT}$ symmetry, and confirm the classification with real-time spin-charge coupled simulations. Their central numerical and analytical result is that this mixed dipole term carries an extra factor of the relaxation time through the Edelstein-type spin susceptibility, so the clean-limit scaling is $\tau^2$ rather than the $\tau^1$ of the Berry curvature dipole. If the paper is right, a distinct, electrically controllable nonlinear Hall signal should appear in materials such as CuMnAs and Mn$_2$Au, offering a readout of Néel spin dynamics through a DC transverse current.

What carries the argument

The machine at the center of the argument is the mixed dipole $D^{\mu;\nu\lambda}_M=\int dk/(2\pi)^d\sum_{a\neq b}\partial_\lambda\,\mathrm{Im}[A^\mu_{ab}S^\nu_{ba}]f_a$, a momentum-space dipole formed from the interband Berry connection $A^\mu$ and the interband spin operator $S^\nu$; it plays the role the Berry curvature dipole plays in time-reversal-symmetric metals, except that one of the two velocity operators is replaced by spin. The argument runs on two coupled pieces: a real-time simulation that solves the von Neumann equation for itinerant electrons together with the Landau-Lifshitz-Gilbert equation for localized spins, and an analytic decomposition of the resulting photocurrent into Drude, Berry curvature dipole, mixed dipole, injection, shift, gyration, and intrinsic Fermi-surface terms in a U(2)-gauge-invariant form. The $L_y$ staggered mode is the one linearly coupled to the electric field, and its electromagnetic susceptibility $\mathrm{Re}\,\chi^{L_y}_{E_x}\propto\tau$ is what upgrades the mixed dipole's bare $\tau^1$ scaling to $\tau^2$.

What would settle it

Compute the $\mathcal{PT}$ transformation of the integrand $\partial_x\,\mathrm{Im}[A^y_{ab}S^{L_y}_{ba}]f_a$ directly from the model's Bloch states: if the transformed integrand is odd under the same sign convention used in Eq. (C71), the mixed dipole integral vanishes and the claimed $\tau^2$ enhancement collapses. A complementary experimental check: measure the low-frequency nonlinear Hall conductivity in a $\mathcal{PT}$-symmetric collinear antiferromagnet such as CuMnAs as disorder or temperature changes the relaxation time; the paper predicts $\tau^2$ scaling, distinguishable from the $\tau^1$ Berry-curvature-dipole and $\tau^0$ shift-current scalings.

Watch

Extended reading notes

Core claim

The paper's central claim is that in a $\mathcal{PT}$-symmetric collinear antiferromagnet, the leading low-frequency nonlinear Hall conductivity from spin-charge coupling is the mixed dipole term $$\$sigma^{{\mu;\nu\lambda}}$_{\mathrm{MD},L}=\frac{J/\tau}{\$omega^{2}$+1/\$tau^{2}$}\int\frac{dk}{(2\pi)^d}\sum_{a\neq b}\partial_\$\lambda$\,\mathrm{Im}[A^\mu_{ab}S^\nu_{ba}]f_a,$$ the exact analogue of the Berry curvature dipole with one Berry connection $A$ replaced by the interband spin operator $S$ in the U(2) gauge required by the Kramers degeneracy. This term arises from the interference of one photon and one spin fluctuation, and symmetry analysis allows it for linearly polarized light through the staggered $L_y$ mode. Because the light-induced spin response is itself the Edelstein-type susceptibility $\mathrm{Re}\,\chi^{L_y}_{E_x}\propto\tau$, the mixed dipole contribution to the nonlinear Hall signal scales as $\tau^2$ in the clean limit, and it is not suppressed by the $1/(\omega-\epsilon_g)$ factor that limits injection and intrinsic Fermi-surface contributions. The paper claims this term, not the Drude term or any Berry-curvature term, is what makes the nonlinear Hall effect sizable in these magnets.

Load-bearing premise

The load-bearing premise is the symmetry bookkeeping in Appendix C that decides whether the alternating y-component of the spins can pair with the light field; if the sign assigned to that component under the parity-time-reversal operation is wrong, the mixed dipole integral the paper identifies as dominant would be forced to vanish by symmetry.

Editorial extensions

If this is right

In the clean low-frequency limit the nonlinear Hall conductivity in $\mathcal{PT}$-symmetric antiferromagnetic metals should grow as $\tau^2$, steeper than the $\tau^1$ scaling of the ordinary Berry curvature dipole.
The nonlinear Hall spectrum should show a resonance at the collective spin excitation frequency (here near $\omega=0.25$) that is absent in independent-particle calculations.
The mixed dipole channel should remain significant even when the optical gap is large, because it lacks the $1/(\omega-\epsilon_g)$ suppression carried by injection and intrinsic Fermi-surface terms.
A purely transverse current $J^y$ is allowed for linearly polarized light along $x$ even though $\mathcal{PT}$ symmetry forces the Berry curvature to vanish at every wave vector.
Materials such as CuMnAs and Mn$_2$Au, where electric fields already control the Néel vector, are natural settings to look for this effect.

Reading between the lines

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

A testable extension the paper leaves implicit: varying the relaxation time through temperature or disorder and plotting the low-frequency nonlinear Hall conductivity against $\tau$ should separate the $\tau^2$ mixed-dipole channel from the $\tau^1$ Berry-dipole and $\tau^0$ shift channels.
If the mixed dipole dominates, the nonlinear Hall signal could serve as an all-electrical readout of Néel-vector orientation, detecting the same spin dynamics that electrical switching protocols already excite in CuMnAs-type devices.
The mechanism should be generic to $\mathcal{PT}$-symmetric metals with sublattice-dependent spin-orbit coupling and an optically active staggered mode, so the particular square-lattice model likely represents a broader class of antiferromagnets.
The predicted resonance at the magnon frequency suggests terahertz or pump-probe experiments could detect the spin-dynamics contribution spectroscopically, separating it from electronic interband contributions by its frequency position.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing here is the claim that in PT-symmetric collinear antiferromagnetic metals the low-frequency nonlinear Hall effect is dominated by a previously unidentified 'mixed dipole' term—an integral of ∂_λ Im[A^μ S^ν] weighted by occupation—rather than the usual Berry curvature dipole or Drude terms. The paper derives this via a systematic second-order density-matrix expansion, writes everything in U(2)-gauge-invariant form appropriate to the Kramers degeneracy, and supports it with real-time spin-charge dynamics simulations. The τ-scaling checks are genuine predictions: the spin-influenced conductivity gains an extra factor of τ from the Edelstein susceptibility, giving τ² in the clean limit, and the numerics reproduce that. That is a real result if the symmetry bookkeeping holds.

What the paper does well: the decomposition of the photocurrent into EE, col-E, and col-col channels is sensible, the analytical formulas are general enough to be reusable, and the classification tables give a practical map of allowed mechanisms. The connection to CuMnAs and Mn₂Au makes it experimentally relevant.

The soft spot is in Appendix C, and the stress-test note is right: the sign factors are inconsistent. Table I says M_z is odd under PT, but Eq. (C6) gives σ_{S M_z} = +1. The explicit PT operator quoted in the footnote, (−iσ_y K)⊗τ_z, makes S^{L_y}=σ^y τ_z odd under PT, which would zero out the mixed dipole in Eq. (C71) and kill the main claim. This looks like a typo—the physical PT operation should include sublattice exchange, and with τ_x in the inversion part L_y is even—but as written it is an internal contradiction that a referee cannot ignore. The authors need to correct the sign table and the PT operator and verify that the L_y mixed dipole is indeed allowed. The numerical evidence is consistent with the claim, so I do not think the conclusion is wrong; the paper just has a load-bearing appendix error.

My recommendation: send it to review. A good referee will catch the sign issue, and the authors can fix it. The paper deserves a serious referee because the mechanism is new and the analytical framework is broadly useful. I would not cite it in its current form until the sign problem is resolved.

Referee Report

3 major / 4 minor

Summary. This paper analyzes the nonlinear Hall response of a two-dimensional PT-symmetric collinear antiferromagnetic metal in a model of itinerant electrons coupled to classical localized spins. The authors perform real-time simulations of the coupled von Neumann and Landau-Lifshitz-Gilbert equations, decompose the second-order photocurrent into field-only, field-spin interference, and spin-spin contributions, and derive analytic formulas in Appendices A and B. Their central claim is that the dominant low-frequency nonlinear Hall signal is a 'mixed dipole' term, sigma^{y;x nu}_{MD,L} = (J/tau)/(omega^2 + 1/tau^2) times the integral of d_lambda Im[A^y_ab S^nu_ba] f_a, with nu = Ly, enhanced by the Edelstein-type susceptibility Re chi^{Ly}_{Ex} proportional to tau, giving an overall tau^2 scaling in the clean limit. They propose this mechanism as a distinct nonlinear Hall channel relevant to electrically switchable antiferromagnets such as CuMnAs and Mn2Au.

Significance. The paper has genuine strengths: the perturbative derivation in Appendix A is detailed, the tau-scaling of the numerically decomposed contributions is checked explicitly, and the Edelstein-enhanced tau^2 scaling is a falsifiable prediction. If the symmetry classification were correct, the proposed mixed-dipole mechanism would extend nonlinear Hall physics beyond Berry-curvature and Drude mechanisms and would be of interest to the antiferromagnetic spintronics community. However, the PT bookkeeping that decides whether the mixed dipole is allowed is internally inconsistent, so the central claim is not currently established.

major comments (3)

[Appendix C, Eq. (C71)] The central claim requires the Ly mixed dipole integral over dk sum_{a neq b} d_x Im[A^y_ab S^{Ly}_ba] f_a to be nonzero. Eq. (C71) states that this quantity is proportional to (1 + sigma_{S nu})/(4i) times the antisymmetrized sum, so a nonzero result requires sigma_{S Ly} = +1. For the spin operator S^{Ly} = sigma_y tau_z defined in Eq. (A6), the explicit operator in footnote 1, PT = (-i sigma_y K) tensor tau_z, gives PT S^{Ly} (PT)^{-1} = -S^{Ly}, i.e. sigma_{S Ly} = -1. Inserted into Eq. (C71), this makes the Ly mixed dipole vanish and forbids the term identified as dominant in Sec. IV C. The paper must correct the sign convention or show explicitly that a different definition of S^{Ly} is used in the mixed-dipole formula.
[Table I and Eq. (C6)] The sign assignments are mutually inconsistent. Table I lists Lx even and Mz odd under PT, while Eq. (C6) states sigma_{S Lx} = -1 and sigma_{S Mz} = +1; direct application of the footnote-1 operator also gives PT(sigma_x tau_z)(PT)^{-1} = -sigma_x tau_z and PT(sigma_z tau_0)(PT)^{-1} = -sigma_z tau_0. Because Eq. (C6) does not list sigma_{S Ly}, the value needed in Eq. (C71) is left ambiguous, and Table I and Eq. (C6) imply opposite Ly/Mz classifications. This ambiguity is load-bearing: Tables III and IV, and the Conclusion, depend on which convention is adopted.
[Sec. IV C, Eq. (82)] The tau^2 scaling of sigma^{inter}_{col-E} is derived as sigmaMD,L Re chi^{Ly}_{Ex} proportional to tau times tau. If the PT constraint in Eq. (C71) forbids sigmaMD,L for nu = Ly, this argument collapses even though Fig. 8 shows tau^2 numerically. The paper should verify the Ly mixed dipole directly, for instance by computing the momentum integral integral dk sum_{a neq b} d_x Im[A^y_ab S^{Ly}_ba] f_a under a fixed, consistent PT convention and by comparing the result with the numerical sigma^{inter}_{col-E}. This check is necessary to distinguish the mixed-dipole interpretation from alternative mechanisms contained in sigma_{SE}.

minor comments (4)

[Appendix C, Eqs. (C72)-(C77)] Several terms in Eqs. (C72)-(C77) are written with S^x and A^x where the index nu of the spin operator is intended; this makes the PT constraints hard to follow.
[Appendix B and Appendix C] The heading 'spinfull' in Appendix B should read 'spinful', and 'less torelant' before Ref. [73] should be 'less tolerant'.
[Fig. 1] The notation in Fig. 1(a), D^{mu;nu x}_M, differs from the text definition D^{mu;nu lambda}_M; please unify the notation.
[Table II] The Field column uses 'S E' for the mixed dipole; this shorthand should be defined in the caption or in the text.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity in the central derivation; the mixed dipole formula is derived analytically, not fitted. Minor methodological self-citations do not carry the argument.

full rationale

The central claim is self-contained. The mixed dipole expressions of Eqs. (46)-(47) and Eqs. (80)-(81) are derived in Appendices A and B from the perturbative von Neumann equation using independent definitions of the interband Berry connection A and interband spin operator S; they are not fitted to the numerical NHE spectra. The tau^2 scaling is obtained by combining the analytic tau^1 of sigma_MD,L with the simulated Edelstein susceptibility Re chi^{Ly}_{Ex} proportional to tau^1 (Eq. 78), and the result is then checked against the real-time simulation (Fig. 8 inset). This is a consistency argument rather than a fitted parameter renamed as a prediction. The decomposition into J0, Jcol-E, and Jcol-col in Eqs. (57)-(66) is a response-theory identity, not a circular definition. Self-citations to Refs. [29,30] supply the von Neumann+LLG numerical scheme, which is re-implemented here, and Ref. [58] supplies the U(2)-gauge formalism, an independent mathematical framework rather than a uniqueness theorem; neither reduces the physical claim to authorial assertion. One internal inconsistency exists: Appendix C Eq. (C6) assigns sigma_{S Mz}=1 while Table I and the explicit PT operator imply Mz is odd under PT. This appears to be a typographical sign error and affects Mz, not the Ly channel used for the mixed dipole, so it is a correctness issue rather than a circularity. The concern that PT forbids the Ly mixed dipole is not supported by the explicit operator, which leaves S^{Ly} invariant. Overall, the paper's derivation is self-contained; score 2 reflects only minor reliance on the authors' prior methodology.

Assumptions & free parameters 9 free parameters · 8 assumptions · 0 invented entities

The central claim rests on a specific tight-binding model, classical spin dynamics, and a phenomenological relaxation scheme. The free parameters are model inputs and numerical settings, not fitted to external data. The main structural assumptions are the classical fixed-length spin approximation, restriction to uniform k = 0 spin dynamics, the relaxation time approximation, and the U(2)-gauge treatment of Kramers-degenerate bands. No new physical entities are introduced; the mixed dipole is a derived momentum-space integral.

free parameters (9)

Nearest-neighbor hopping t1 = 1.0
Sets the energy unit and defines the minimal PT-symmetric antiferromagnet model from Ref [47].
Next-nearest-neighbor hopping t2 = 0.08
Chosen to give the desired band dispersion in the tight-binding model.
Sublattice-dependent antisymmetric spin-orbit coupling lambda = 0.8
Chosen to enhance magnetoelectric coupling, following refs [40,41,48].
Exchange coupling J = 1.0
Sets the electron-localized-spin coupling strength and stabilizes the collinear antiferromagnetic order.
Easy-axis anisotropy Kx = 0.05
Positive value stabilizes spins along x and sets the spin resonance frequency near omega = 0.25.
Chemical potential mu = -0.6
Places the Fermi level in the metallic regime and controls which Fermi-surface terms contribute.
Relaxation time tau = 25.0
Phenomenological electron relaxation time; the tau scaling of the response is the paper's main diagnostic.
Gilbert damping alpha_G = 0.05
Phenomenological damping for the localized spin dynamics.
Pulse and grid parameters = E0 = 1e-5, t0 = 0.2, sigma = 0.03, N = 1000
Numerical settings for the real-time response calculation; E0 is chosen weak to stay in the linear/nonlinear perturbative regime.

assumptions (8)

domain assumption Localized spins are classical vectors of fixed length |S_alpha| = 1.
Invoked in Eq. (12) and Section II A; neglects quantum spin fluctuations that could alter the mixed dipole magnitude.
domain assumption Spin dynamics is uniform across unit cells, so only k = 0 magnons are considered.
Section II A states uniform spin dynamics consistent with the dipole approximation; finite-wavevector magnons are excluded.
domain assumption Relaxation time approximation with a single tau in the von Neumann equation, plus Gilbert damping alpha_G in the LLG equation.
Eqs. (15) and (16) in Section II B; ignores vertex corrections and frequency-dependent scattering.
domain assumption The equilibrium state is the zero-temperature Fermi-Dirac occupation Theta(mu - epsilon_kn).
Eq. (19) in Section II B; ignores thermal occupation and disorder broadening.
domain assumption Electronic bands are exactly doubly degenerate and the response is U(2)-gauge invariant.
Requires exact PT symmetry and no level splitting; used in Appendix B to replace the U(1) Berry connection with a U(2) connection.
domain assumption Light-matter coupling is treated in the length gauge with well-localized Wannier functions.
Eq. (14) and discussion in Section II A; neglects spatial dispersion of the light-matter coupling.
ad hoc to paper The photocurrent decomposition replays spin trajectories from the full simulation with the light field removed.
Section IV B; this is an analysis decomposition, not a self-consistent calculation, and its error is unquantified.
ad hoc to paper The PT transformation signs in Appendix C are the ones that classify the Ly mixed dipole as allowed.
Eqs. (C6) and (C71). As written, C71 with sigma_{S Lx} = -1 and the factor (1 + sigma_{S nu})/(4i) appears to forbid the Ly mixed dipole, so a consistent sign convention is load-bearing.

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Cite this review

Pith. "Pith review of Nonlinear Hall effect driven by spin-charge-coupled motive force." pith.science (2026). https://pith.science/paper/26FHHVMR

@misc{pith2026250111234,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Hall effect driven by spin-charge-coupled motive force},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/26FHHVMR}},
  note         = {Machine review of arXiv:2501.11234}
}

abstract

Parity-time-reversal symmetric ($\mathcal{PT}$-symmetric) magnets have garnered much attention due to their spin-charge coupled dynamics enriched by the parity-symmetry breaking. By real-time simulations, we study how localized spin dynamics can affect the nonlinear Hall effect in $\mathcal{PT}$-symmetric magnets. To identify the leading-order term, we derive analytical expressions for the second-order optical response and classify the contributions by considering their transformation properties under $\mathcal{PT}$ symmetry. Notably, our results reveal that the sizable contribution is attributed to the mixed dipole effect, which is analogous to the Berry curvature dipole term.

Figures

Figures reproduced from arXiv: 2501.11234 by the authors.

**Figure 2.** FIG. 2. (a) Square lattice model with collinear antiferro [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Collective mode of collinear antiferromagnetic mo [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. (a), (b) Linear electromagnetic susceptibility of the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. (a) Photocurrent spectra for [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Diagrams of three different processes of the pho [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. (a) Relaxation time dependence of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. Relaxation time dependence of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

89 extracted references · 58 canonical work pages

[1]

To facilitate the analysis, the position operator in Eq

Light field induced photocurrent Here, we derive the photocurrent induced by the light field based on ρEE . To facilitate the analysis, the position operator in Eq. (A9) is decomposed into intra-band ( ri) and inter-band ( re) component as (ri)ab = δab(i∇k + ξaa), (re)ab = (1 − δab)ξab. (A22) Based on this decomposition, we can classify ρEE (ω) into the f...
[2]

(A33) 18 By replacing η with 1/τ phenomenologically, we can obtain the Drude term formula as σµ;νλ D (0; −Ω, Ω) = 1 Ω2 + (1/τ )2 Z dk (2π)d X a ∂µ∂ν∂λϵkaf (ϵka)

F ermi surface effect I: Drude term Firstly, we focus on σµ;νλ EE, (ii) as σµ;νλ EE, (ii)(0; −Ω, Ω) = 1 2 Z dk (2π)d X a −vµ aad0 aadΩ aa∂ν∂λf (ϵka) + [(ν, −Ω) ↔ (λ, Ω)] , (A31) = − 1 2iη Z dk (2π)d X a vµ aa 1 Ω + iη + 1 −Ω + iη ∂ν∂λf (ϵka), (A32) = 1 Ω2 + η2 Z dk (2π)d X a vµ aa∂ν∂λf (ϵka). (A33) 18 By replacing η with 1/τ phenomenologically, we can obt...
[3]

F ermi surface effect II: Berry curvature dipole Secondly, we derive the Berry curvature dipole term from σµ;νλ EE, (ei). σµ;νλ EE, (ei)(0; −Ω, Ω) = 1 2(Ω + iη) Z dk (2π)d X a̸=b ξµ abξν ba∂λfba + [(ν, −Ω) ↔ (λ, Ω)] , = 1 2(Ω + iη) Z dk (2π)d X a̸=b (ξµ baξν ab − ξµ abξν ba)∂λfa + [(ν, −Ω) ↔ (λ, Ω)] , = − iΩ + η Ω2 + η2 Z dk (2π)d X a̸=b Im[ξµ abξν ba]∂λf...
[4]

(A38) Here, ∆ µ ac = J µ aa − J µ cc = ∂µϵka − ∂µϵkb is the velocity difference matrix

Interband effect I: Injection current Here, we focus on σµ;νλ EE, (ee) with the diagonal component of the current operator in the band basis denoted as σµ;νλ EE, (ee;d); σµ;νλ EE, (ee;d)(ω, ω1, ω2) = 1 2 Z dk (2π)d X a̸=c J µ aadaa(ω) dca(ω − ω1)ξν acξλ cafac − dac(ω − ω1)ξν caξλ acfcb + [(ν, ω1) ↔ (λ, ω2)] = 1 2 1 ω + iη Z dk (2π)d X a̸=c (J µ aa − J µ c...
[5]

The photocurrent responses that we derive in this section come from the off-diagonal part of the current operator J µ in the output vertex

Interband effect II: Shift current and intrinsic F ermi surface effect Here, we derive the shift current term and intrinsic Fermi surface term based from σµ;νλ EE, (ee) and σµ;νλ EE, (ie). The photocurrent responses that we derive in this section come from the off-diagonal part of the current operator J µ in the output vertex. First, we focus on σµ;νλ EE,...
[6]

Spin dynamics induced photocurrent Here, we derived the photocurrent formula related to spin dynamics. Using the SPDM ρSS, we can express pho- tocurrent response to the spin field as J µ SS = Z dk (2π)d X abc J µ abρ(2) SS,ba(ω) (A61) =: Z dω1dω2 (2π)2 σµ;νλ SS (ω, ω1, ω2)∆Sν(ω1)∆Sλ(ω2)2πδ(ω − ω1 − ω2) (A62) 22 In the previous subsection, we derived the p...
[7]

Interference of light field and spin dynamics Following the previous subsection, we consider the photocurrent response coming from the interference of the light field and spin dynamics. Using ρES and ρSE, we can write photocurrent formula as J µ ES(ω) = Z dk (2π)d X abc J µ ab(ρ(2) ES,ba(ω) + ρ(2) SE,ba(ω)) (A71) =: Z dω1dω2 (2π)2 h σµ;νλ MD (ω, ω1, ω2) +...
[8]

In the previous section, we can derive the formulas for the photocurrent responses in spinless systems

Light field induced photocurrent in PT -symmetric spinful systems Here, we show the formulas for the photocurrent responses from the light field in PT -symmetric systems. In the previous section, we can derive the formulas for the photocurrent responses in spinless systems. In the same manner, we can get the formulas for the photocurrent responses in PT -...

Show all 89 references

[9]

In the previous section, we derived the formulas for the photocurrent responses from the light field in PT -symmetric spinful systems

Spin dynamics induced photocurrent in PT -symmetric spinful systems Here, we show the formulas for the photocurrent responses from the spin dynamics in PT -symmetric systems. In the previous section, we derived the formulas for the photocurrent responses from the light field i...
[10]

Interference of light and spin dynamics in PT -symmetric spinful systems Here, we show the formulas for the photocurrent responses stemming from the interference of the light and spin dynamics in PT -symmetric spinful systems. First, the mixed dipole term can be expressed in t...
[11]

(C27) Owing to Im[ Ax abAx ba] = 0, the photocurrent conductivity σy;xx EE, Inj;E, σy;xx EE, IFSI;E and σy;xx EE, IFSI I;E vanish

Light field induced photocurrent As drawn in the previous subsection, the photocurrent along the y direction induced by the light field along the x direction is expressed as J y EE = Z dΩ 2π σy;xx EE (0; −Ω, Ω)Ex(−Ω)Ex(Ω), (C16) 28 where σy;xx can be classified into following ...
[12]

Spin dynamics induced photocurrent Photocurrent induced solely by localized spin dynamics can be described as J y SS = Z dΩ 2π σy;νλ SS (0; −Ω, Ω)∆Sν(−Ω)∆Sλ(Ω) = Z dΩ 2π σy;νλ SS (0; −Ω, Ω)[∆Sν(Ω)]∗∆Sλ(Ω). (C37) Here, σSS can be classified into the following eight components σ...
[13]

J y ES(ω) = Z dk (2π)d X abc J y ab(ρ(2) ES,ba(ω) + ρ(2) SE,ba(ω)) =: Z dω1dω2 (2π)2 [σy;νx MD (ω, ω1, ω2) + ˜σy;νx SE (ω, ω1, ω2)] ∆Sν(ω1)Ex(ω2)2πδ(ω − ω1 − ω2)

Interference of light field and spin dynamics Photocurrent arising from the interference of light field and spin dynamics can be expressed as follows. J y ES(ω) = Z dk (2π)d X abc J y ab(ρ(2) ES,ba(ω) + ρ(2) SE,ba(ω)) =: Z dω1dω2 (2π)2 [σy;νx MD (ω, ω1, ω2) + ˜σy;νx SE (ω, ω1,...
[14]

Injection current First, we discuss the relaxation time dependence of the injection current. We can express the injection current in general as σµ;νλ Inj = πτ Z dk (2π)d X a̸=b ∆µ abX ν abX λ bafabδ(Ω − ϵba), (D1) 33 where the matrix Xab is defined as the operator, such as the...
[15]

We can discuss the behavior of the shift current in the same manner as the injection current case

Shift current Second, we focus on the relaxation time dependence of the shift current. We can discuss the behavior of the shift current in the same manner as the injection current case. In general, the shift current term can be expressed as σµ;νλ shift = − iπ 2 Z dk (2π)d X a̸...
[16]

In general, we can express the formula for the intrinsic Fermi surface term as σµ;νλ IFS = − 1 2 Z dk (2π)d X a̸=b X ν abX λ ba∂µfabP 1 Ω − ϵba

Intrinsic F ermi surface term Finally, we discuss the relaxation time dependence of the intrinsic Fermi surface term. In general, we can express the formula for the intrinsic Fermi surface term as σµ;νλ IFS = − 1 2 Z dk (2π)d X a̸=b X ν abX λ ba∂µfabP 1 Ω − ϵba . (D10) 34 Cons...
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J. E. Moore and J. Orenstein, Confinement-induced berry phase and helicity-dependent photocurrents, Physical Review Letters 105, 026805 (2010)

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

To facilitate the analysis, the position operator in Eq

Light field induced photocurrent Here, we derive the photocurrent induced by the light field based on ρEE . To facilitate the analysis, the position operator in Eq. (A9) is decomposed into intra-band ( ri) and inter-band ( re) component as (ri)ab = δab(i∇k + ξaa), (re)ab = (1 − δab)ξab. (A22) Based on this decomposition, we can classify ρEE (ω) into the f...

[2] [2]

(A33) 18 By replacing η with 1/τ phenomenologically, we can obtain the Drude term formula as σµ;νλ D (0; −Ω, Ω) = 1 Ω2 + (1/τ )2 Z dk (2π)d X a ∂µ∂ν∂λϵkaf (ϵka)

F ermi surface effect I: Drude term Firstly, we focus on σµ;νλ EE, (ii) as σµ;νλ EE, (ii)(0; −Ω, Ω) = 1 2 Z dk (2π)d X a −vµ aad0 aadΩ aa∂ν∂λf (ϵka) + [(ν, −Ω) ↔ (λ, Ω)] , (A31) = − 1 2iη Z dk (2π)d X a vµ aa 1 Ω + iη + 1 −Ω + iη ∂ν∂λf (ϵka), (A32) = 1 Ω2 + η2 Z dk (2π)d X a vµ aa∂ν∂λf (ϵka). (A33) 18 By replacing η with 1/τ phenomenologically, we can obt...

[3] [3]

F ermi surface effect II: Berry curvature dipole Secondly, we derive the Berry curvature dipole term from σµ;νλ EE, (ei). σµ;νλ EE, (ei)(0; −Ω, Ω) = 1 2(Ω + iη) Z dk (2π)d X a̸=b ξµ abξν ba∂λfba + [(ν, −Ω) ↔ (λ, Ω)] , = 1 2(Ω + iη) Z dk (2π)d X a̸=b (ξµ baξν ab − ξµ abξν ba)∂λfa + [(ν, −Ω) ↔ (λ, Ω)] , = − iΩ + η Ω2 + η2 Z dk (2π)d X a̸=b Im[ξµ abξν ba]∂λf...

[4] [4]

(A38) Here, ∆ µ ac = J µ aa − J µ cc = ∂µϵka − ∂µϵkb is the velocity difference matrix

Interband effect I: Injection current Here, we focus on σµ;νλ EE, (ee) with the diagonal component of the current operator in the band basis denoted as σµ;νλ EE, (ee;d); σµ;νλ EE, (ee;d)(ω, ω1, ω2) = 1 2 Z dk (2π)d X a̸=c J µ aadaa(ω) dca(ω − ω1)ξν acξλ cafac − dac(ω − ω1)ξν caξλ acfcb + [(ν, ω1) ↔ (λ, ω2)] = 1 2 1 ω + iη Z dk (2π)d X a̸=c (J µ aa − J µ c...

[5] [5]

The photocurrent responses that we derive in this section come from the off-diagonal part of the current operator J µ in the output vertex

Interband effect II: Shift current and intrinsic F ermi surface effect Here, we derive the shift current term and intrinsic Fermi surface term based from σµ;νλ EE, (ee) and σµ;νλ EE, (ie). The photocurrent responses that we derive in this section come from the off-diagonal part of the current operator J µ in the output vertex. First, we focus on σµ;νλ EE,...

[6] [6]

Spin dynamics induced photocurrent Here, we derived the photocurrent formula related to spin dynamics. Using the SPDM ρSS, we can express pho- tocurrent response to the spin field as J µ SS = Z dk (2π)d X abc J µ abρ(2) SS,ba(ω) (A61) =: Z dω1dω2 (2π)2 σµ;νλ SS (ω, ω1, ω2)∆Sν(ω1)∆Sλ(ω2)2πδ(ω − ω1 − ω2) (A62) 22 In the previous subsection, we derived the p...

[7] [7]

Interference of light field and spin dynamics Following the previous subsection, we consider the photocurrent response coming from the interference of the light field and spin dynamics. Using ρES and ρSE, we can write photocurrent formula as J µ ES(ω) = Z dk (2π)d X abc J µ ab(ρ(2) ES,ba(ω) + ρ(2) SE,ba(ω)) (A71) =: Z dω1dω2 (2π)2 h σµ;νλ MD (ω, ω1, ω2) +...

[8] [8]

In the previous section, we can derive the formulas for the photocurrent responses in spinless systems

Light field induced photocurrent in PT -symmetric spinful systems Here, we show the formulas for the photocurrent responses from the light field in PT -symmetric systems. In the previous section, we can derive the formulas for the photocurrent responses in spinless systems. In the same manner, we can get the formulas for the photocurrent responses in PT -...