{"id":"21b3ce68-0842-4e92-ab25-6c9ffc9f31c5","arxiv_id":"2608.05788","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Replacing germanium with gallium in a monolayer iron-based magnet suppresses the fourth-harmonic Fermi-sea spin-orbit torque, a change traced to the shrinking of K-point electron pockets.","lead":"First-principles simulations show that two nearly identical atomic monolayer magnets, Fe3GeTe2 and Fe3GaTe2, develop very different current-induced torques when a voltage is applied, even though their crystal structures are the same. The differences could guide engineering of atomically thin magnetic devices for memory and logic.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Rigid-rotation maps freeze εF while FGaT shifts εF by ~6 meV under in-plane m; the central FGaT fourth-harmonic claim is not benchmarked.","rationale":"The reader identified the rigid-exchange-rotation approximation as the weakest assumption, and I agree that this is where the argument is least secure. My stress-test sharpens the concern: within that approximation, the chemical potential is frozen at its reference value while the self-consistent FGaT calculation shows a ~6 meV shift of εF for in-plane magnetization. This is not a minor detail because the central comparative claim—the suppression of the fourth-harmonic TR-even torkance in FGaT—is made precisely at in-plane magnetization angles and is attributed to K-point Fermi-pocket evolution, which is exponentially sensitive to the placement of εF when pockets are small. The existing benchmark is only for FGT, where the εF shift is <0.1 meV, so it does not validate the regime in which the main FGaT claims are made. This is an internal methodological gap rather than a disagreement with external consensus, and it is directly testable. I credit the paper for its transparent error discussion, the symmetry validation of the velocity operator, and the explicit benchmarks; these make the gap identifiable and fixable. However, until the fixed-εF issue is checked, the quantitative fourth-harmonic suppression and the factor-of-30 hidden-torque claim for FGaT should be treated as provisional. The reader's CONDITIONAL verdict remains appropriate, so I do not change the verdict; my concern is more specific than the reader's general statement, hence 'partial' agreement.","tokens_in":25549,"tokens_out":7624,"duration_ms":70078,"concrete_test":"Compute FGaT torkances within the rigid-rotation scheme but at each magnetization angle determine εF self-consistently by requiring the integrated density below εF to equal the reference electron count (or use the εF from a noncollinear DFT calculation at that angle). Re-extract the TR-even harmonic coefficients of Table I and the momentum-resolved teven_θ at φ=π/4 (Fig. 5). If the FGaT m=4 coefficients remain suppressed relative to FGT (e.g., ψ^c_5,4 still <20% of the FGT value), the claim survives; if they become comparable, the central comparison is an artifact of the fixed chemical potential. Additionally, run the Appendix E yz-plane benchmark for FGaT, where the εF shift is ~6 meV, to quantify the rigid-rotation error in the regime used for the main claims.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Sec. II.D introduces the rigid-exchange-rotation scheme, which computes torkances from one reference Wannier model and, implicitly, at the reference chemical potential. Sec. III.A states that in self-consistent DFT, rotating m into the plane changes the chemical potential by <0.1 meV in FGT but by ~6 meV in FGaT. The full-sphere maps of Fig. 3 and the fourth-harmonic analysis of Table I and Fig. 5 are computed without re-adjusting εF, so for FGaT the in-plane angles are evaluated at a fixed chemical potential that is up to 6 meV away from the charge-neutral value. The only rigid-rotation benchmark (Appendix E, Fig. 9) is performed for FGT, where the shift is negligible; no FGaT benchmark is provided. The central claim that the TR-even fourth harmonic is 'strongly reduced' in FGaT rests on momentum-space compensation that is sensitive to the placement of εF relative to the K-point pockets. A 6 meV misplacement could change the size of the small K pockets and hence the harmonic coefficients (e.g., ψ^c_5,4 in Table I), or even the sign of the TR-even channel. Similarly, the hidden site-resolved FeI torques in FGaT (factor ~30) are computed at fixed εF and are not covered by any DFT validation. The concern is internal to the method: the rigid-rotation approximation conflates a change of magnetization direction with a rigid band rotation while freezing the carrier density, whereas the self-consistent system adjusts εF to maintain charge neutrality.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents first-principles Kubo–Bastin linear-response calculations of the spin–orbit torkance in monolayer Fe3GeTe2 and Fe3GaTe2, using symmetry-adapted Wannier interpolation of fully relativistic DFT. The authors decompose the torkance by time-reversal parity (Fermi-sea/Fermi-surface), by atomic sublattice, and by momentum, and they compare the two compounds as a controlled hole-doping pair. Their central qualitative claims are that the time-reversal-odd (Fermi-surface) torkances are similar in the two materials, that the time-reversal-even (Fermi-sea) torkance has a pronounced fourth-harmonic component in FGT that is strongly suppressed in FGaT, and that site-resolved hidden torques on the FeI sublattices can exceed symmetry-allowed components by nearly a factor of 30 in FGaT. The full angular dependence is computed with a rigid-exchange-rotation scheme in which only the time-reversal-odd part of the Hamiltonian is rotated and all evaluations use a single reference chemical potential.","tokens_in":25842,"tokens_out":3826,"duration_ms":38067,"significance":"If the quantitative claims hold, the paper offers a valuable microscopic picture of current-induced torques in two-dimensional ferromagnets and identifies a concrete materials-design route through sublattice engineering of hidden torques. The work has notable strengths: the Kubo–Bastin formalism and Wannier interpolation are standard but carefully implemented; the treatment of the Wannier velocity, including the Berry-connection term, is explicit and is supported by symmetry checks (Appendix E); convergence with respect to the k mesh and broadening is documented; and the calculation is essentially parameter-free apart from the phenomenological broadening Γ. The momentum-resolved decomposition in Fig. 5 is a useful diagnostic. However, the central FGaT-specific claims rest on the rigid-exchange-rotation approximation at a fixed chemical potential, and the only quantitative benchmark of that approximation is performed for FGT, where the chemical-potential shift under magnetization rotation is negligible. The 6 meV chemical-potential shift reported for FGaT in Sec. III A is therefore a load-bearing unresolved issue for the fourth-harmonic and hidden-torque conclusions.","major_comments":[{"comment":"The rigid-exchange-rotation scheme evaluates all magnetization directions from one reference Wannier model and therefore at a fixed chemical potential. Section III A states that rotating the magnetization into the plane changes the chemical potential by <0.1 meV in FGT but by about 6 meV in FGaT. The full-sphere maps, the harmonic coefficients in Table I (notably the TR-even m=4 coefficient ψ^c_5,4), and the momentum-space analysis of Fig. 5 are computed without re-adjusting εF to preserve charge neutrality for in-plane magnetization. Since the suppression of the fourth harmonic in FGaT is attributed to the small K-point Fermi pockets, an uncontrolled 6 meV misplacement of εF can alter the pocket sizes and the compensation pattern, potentially changing the harmonic coefficients or even their signs. The benchmark in Appendix E is performed only for FGT, where the shift is negligible, so it does not validate the FGaT results. Please provide a FGaT benchmark that either includes the self-consistent εF shift or quantifies the sensitivity of Table I and Fig. 5 to a 6 meV shift in εF.","section":"Sec. II D, Sec. III A, Table I, Fig. 5"},{"comment":"Even apart from the chemical-potential issue, the rigid-exchange-rotation approximation itself shows up to about 17% deviation in the TR-even component t^even_zx near m parallel to ±y (Fig. 9). The central claim of a strongly reduced fourth-harmonic TR-even torque in FGaT concerns precisely in-plane and near-in-plane magnetization angles, yet no error propagation from this benchmark is provided for the harmonic coefficients or for the site-resolved hidden torques. The authors should state how the 17% deviation of an individual tensor component translates into uncertainty in the coefficients of Table I and in the factor-of-30 hidden-torque enhancement, ideally by evaluating the harmonic decomposition on both the DFT and rigid-rotation data shown in Fig. 9.","section":"Appendix E, Sec. III B 2, Table I"},{"comment":"The claim that site-resolved FeI hidden torques can exceed symmetry-allowed components by nearly a factor of 30 in FGaT is computed with the same fixed-εF rigid-rotation approximation and is not covered by any direct benchmark. Because this claim is presented as a promising engineering route, the authors should either validate it with the FGaT benchmark requested above or soften the conclusion and provide a sensitivity estimate with respect to εF and to the rigid-rotation error.","section":"Sec. III B 3, Fig. 4"}],"minor_comments":[{"comment":"Figure 3 appears twice in the manuscript with different color scales (the first set shows FGT TR-even up to 2.5×10^-2 and the second up to 4.2×10^-2, with different FGaT ranges as well). This looks like a duplicated figure from an earlier draft; please retain only the intended version with consistent scales and captions.","section":"Fig. 3"},{"comment":"The sentence fragment \"lead to hidden torques s\" appears to contain a typo and should read \"lead to hidden torques.\"","section":"Sec. III B 3, after Eq. (11)"},{"comment":"The phrase \"We note that, due to D3h symmetry, magnetization-independent torque begins at ℓ = 2\" is potentially confusing: in D3h a magnetization-independent torque is symmetry-forbidden, and the ℓ = 2 sector is the leading allowed magnetization-dependent contribution. Please rephrase to avoid implying a nonzero rigid-field torque.","section":"Sec. III B 2, around Eq. (10)"},{"comment":"The description of the parameters a, b, and c as \"real parameters material-dependent coefficients\" is grammatically incomplete and should be rephrased, for example as \"real, material-dependent coefficients.\"","section":"Eq. (12) and surrounding text"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the core methodology is sound. The main obstacle is the missing FGaT-specific validation of the rigid-exchange-rotation scheme in the regime where the self-consistent chemical-potential shift is 6 meV; this directly affects the headline fourth-harmonic and hidden-torque claims. I do not see grounds for rejection if the authors can supply the requested benchmark or a convincing sensitivity analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The controlled Ge-vs-Ga comparison is the real selling point: same D3h symmetry, same Fe 3d character, one valence electron difference, and the paper shows that the TR-odd torkances are similar while the TR-even fourth-harmonic content is strongly suppressed in FGaT. The atom- and momentum-resolved decompositions, including the hidden FeI torques that cancel in the total, are new and properly tied to symmetry. The methodology is careful: Kubo-Bastin with symmetry-adapted Wannier functions, full velocity with Berry connection, explicit checks of symmetry-forbidden zeros, and an honest appendix on the rigid-exchange-rotation approximation.\n\nThe stress-test concern holds up. The rigid-rotation scheme keeps the chemical potential at the reference value (m along z), while self-consistent DFT says rotating m into the plane shifts εF by about 6 meV in FGaT (and <0.1 meV in FGT). The FGT benchmark in Appendix E shows up to ~17% deviation in t_even_zx even with negligible εF shift; for FGaT, where the Fermi surface is much smaller, the error could be larger, and the central fourth-harmonic suppression claim plus the factor-30 hidden torques are computed at this fixed εF with no FGaT validation. The authors state the approximation honestly, but they don't quantify how much the 6 meV shift changes the K-pocket contributions or the harmonic coefficients. That's a real gap, not a manufactured one. It doesn't kill the paper—the qualitative picture is probably right—but it needs to be addressed before the numbers are used for quantitative engineering.\n\nMinor point: Fig. 5's momentum-resolved panels are suggestive but not quantified beyond the harmonic coefficients; a direct correlation between pocket size and fourth-harmonic amplitude would strengthen the causal claim.\n\nOverall: worthwhile paper, honest about approximations, with a methodological gap that a referee should probe. Send it to review; ask for an FGaT benchmark of the rigid-rotation scheme and a check of the fixed-εF assumption's effect on the key coefficients. I'd cite it for the comparison and the hidden-torque analysis.","headline":"Careful SOT comparison of two 2D magnets, but the main quantitative claim rests on a rigid-rotation approximation that freezes the chemical potential, which shifts by ~6 meV in FGaT and is never benchmarked for that material.","tokens_in":26414,"tokens_out":2755,"would_cite":true,"duration_ms":27147,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In Fe3GaTe2, swapping gallium for germanium shrinks the K-point pockets, silencing the Fermi-sea torque's fourth harmonic, barely moving the Fermi-surface torque, and exposing hidden sublattice torques up to ~30 times the allowed size.","keywords":["spin-orbit torque","torkance","monolayer Fe3GeTe2","monolayer Fe3GaTe2","time-reversal-even torque","hidden torques","Kubo-Bastin linear response","Wannier interpolation"],"falsifier":"Recompute the torkance at full self-consistent noncollinear DFT, without the rigid rotation, for both monolayers at the key in-plane directions, $\\hat{m}\\parallel\\hat{x}$ where Fe3GeTe2's TR-even peak and fourth harmonic sit, and $\\hat{m}\\parallel\\hat{y}$ where the benchmark already deviates by about 17%, and check whether the FGT/FGaT contrast in the TR-even peak or the fourth-harmonic coefficient survives. The experimental counterpart is to electrostatically hole-dope monolayer Fe3GeTe2 and measure the angular torque map: if the fourth-harmonic component does not shrink as holes are added, the K-pocket suppression mechanism is wrong.","tokens_in":25323,"feed_emoji":"🧲","tokens_out":27261,"duration_ms":226186,"temperature":0.7,"pith_summary":"This paper tries to establish which microscopic electronic states carry the spin-orbit torque in two nearly identical monolayer ferromagnets, Fe3GeTe2 and Fe3GaTe2. Because gallium has one fewer valence electron than germanium, the gallium compound is effectively a hole-doped twin of the germanium one, and the paper shows that this single change reshapes the torque landscape: the Fermi-surface (time-reversal-odd) torques stay similar, while the Fermi-sea (time-reversal-even) torque loses a pronounced fourth-harmonic angular component that is strong in Fe3GeTe2, an effect traced to the shrinkage of the $K$-point Fermi pockets. The paper also establishes that large hidden torques act on the two iron sublattices, reaching up to about thirty times the symmetry-allowed component in Fe3GaTe2, although they cancel exactly in the pristine monolayer. The finding matters because it identifies band filling, not crystal symmetry, as the control parameter for torque engineering in these van der Waals magnets, and it suggests that breaking the mirror symmetry between sublattices could activate otherwise hidden torques.","feed_headline":"One missing electron per cell rewrites a magnet's spin torque","feed_subtitle":"Hole doping shrinks K-point pockets, silencing the Fermi-sea fourth harmonic and unmasking ~30x hidden torques.","key_machinery":"The argument runs on the torkance tensor, the torque per unit applied electric field, computed by Kubo-Bastin linear response and split by time-reversal parity into a Fermi-sea (TR-even) channel $t^{even}_{ab}$ and a Fermi-surface (TR-odd) channel $t^{odd}_{ab}$. The electronic structure is represented by symmetry-adapted, fully spin-orbit-coupled Wannier functions built from relativistic density-functional theory; they carry the Hamiltonian, the velocity operator (including its Berry-connection contribution, which enforces symmetry-protected zeros of the response), and the spin operator used to build the exchange-torque operator. To sample the whole magnetization sphere the paper uses a rigid-exchange-rotation scheme: for each direction $\\hat{m}$ only the TR-odd (exchange) part of the Hamiltonian is rotated by an SU(2) spinor rotation $U_R$, with the TR-even part fixed at the reference $\\hat{m}\\parallel\\hat{z}$, so that a single reference calculation serves every magnetization direction. The angular maps are decomposed in vector spherical harmonics, the tangent families $\\Psi_{\\ell m}=\\nabla_s Y_{\\ell m}$ and $\\Phi_{\\ell m}=\\hat{m}\\times\\nabla_s Y_{\\ell m}$, and the fourth-harmonic content of the TR-even channel lives in the $m=4$ pair of harmonics; sublattice projectors resolve the same response onto individual atomic sites, which is what exposes the hidden torques.","core_discovery":"The central claim is that in two monolayers with the same point group ($D_{3h}$) and the same dominant Fe $3d$ bands, the substitution Ge$\\to$Ga, which is equivalent to removing one valence electron per formula unit, changes the torque through band filling rather than symmetry. The density of states at $\\varepsilon_F$ drops by roughly a factor of three and its spin polarization reverses sign, and individual band features shift by 30-130 meV relative to the Fermi level even though the two work functions differ by only about 28 meV. These shifts shrink the minority-spin pockets at $K$ and $K'$. As a result the time-reversal-odd (Fermi-surface) torkances of the two compounds are similar, whereas the time-reversal-even (Fermi-sea) torkance differs markedly: Fe3GeTe2 shows a strong fourth-harmonic angular component, concentrated at in-plane magnetization along $\\hat{x}$, which is strongly reduced in Fe3GaTe2 because the positive and negative momentum-space contributions compensate more completely. A second discovery is site-resolved hidden torque: the two FeI sublattices, related by the horizontal mirror $\\sigma_h$, carry components in directions the global symmetry forbids, and in Fe3GaTe2 these hidden components can exceed the symmetry-allowed torque by nearly a factor of 30; they cancel exactly in the ideal monolayer, but breaking the sublattice symmetry would activate them.","pith_inferences":["A natural test the paper leaves implicit: electrostatically hole-doping monolayer Fe3GeTe2 should reproduce Fe3GaTe2's torque signature, namely smaller $K$-point pockets, a reduced Fermi-level density of states, and a weakened fourth-harmonic TR-even torque, without any chemical substitution.","The rigid-rotation benchmark error (up to about 17% in one TR-even component) is not propagated into the quantitative claims; a fully self-consistent check at the in-plane peak directions ($\\hat{m}\\parallel\\hat{x}$ and $\\hat{m}\\parallel\\hat{y}$) would pin down whether the factor-of-two TR-even contrast between the compounds is quantitatively robust.","Because the momentum-resolved maps show the TR-even integrand as alternating positive and negative densities concentrated near the $K$ pockets and at band anticrossings, strain or moire potentials that move those pocket energies should act as a continuous, non-chemical dial on the fourth-harmonic torque.","One internal inconsistency to note when reading: the abstract and main text state that the Fermi-level density-of-states polarization reverses from majority in Fe3GeTe2 to minority in Fe3GaTe2, but the tabulated values in Appendix F give the opposite (minority-like $P\\approx -0.41$ in Fe3GeTe2, majority-like $P\\approx +0.28$ in Fe3GaTe2); the torque conclusions do not depend on this direction, but"],"forward_implications":["The Fermi-surface (TR-odd) torque is largely transferable between Fe3GeTe2 and Fe3GaTe2, while the Fermi-sea (TR-even) torque changes sharply; the even channel is therefore the sensitive indicator of band filling in this family.","The band-feature shifts between the compounds (30-130 meV) lie within the reach of electrostatic or ionic gating, so the fourth-harmonic torque should be tunable in a single material by shifting the chemical potential toward or away from the $K$-point pockets.","Since the two FeI sublattices carry opposite hidden torques that cancel only while the horizontal mirror $\\sigma_h$ is intact, any local or global breaking of that mirror, by a defect, adsorbate, substrate, or van der Waals partner, should activate torques that in Fe3GaTe2 reach up to about 30 times the symmetry-allowed size.","The symmetry phenomenology predicts that in the ideal monolayer the current-induced spin accumulation is strictly in-plane and linear in the magnetization, forbidding the conventional dampinglike torque; removing the mirror unlocks out-of-plane spin accumulation and dampinglike torques, a route the paper proposes for deterministic switching of perpendicular ferromagnets.","Because the fourth-harmonic content controls the field-free switching trajectory, as established for monolayer Fe3GeTe2 by the earlier work this paper builds on, the suppression of that harmonic in Fe3GaTe2 implies different current directions and thresholds for field-free switching in the two compounds."],"supporting_citations":[{"why":"The earlier first-principles study of monolayer FGT that established the angular-dependent torkance maps and the vector-spherical-harmonics expansion; the present paper recovers its higher-harmonic content and reassigns it to the FeI sublattice.","marker":"[32]"},{"why":"Supplies the Kubo-Bastin linear-response formulas whose analytic energy integrals define the Fermi-sea (TR-even) and Fermi-surface (TR-odd) torkance channels used throughout.","marker":"[33]"},{"why":"Establishes the D3h symmetry argument that the leading current-induced spin polarization must be linear in the in-plane magnetization, the basis of the paper's self-torque phenomenology.","marker":"[26]"},{"why":"Provides the symmetry-adapted Wannier-function construction that pins projectors and centers to the site symmetries, letting the interpolated model inherit the fully relativistic spin-orbit coupling.","marker":"[49]"},{"why":"The Wannier-interpolation engine that evaluates the dense-Brillouin-zone response quantities, including the gauge-covariant Berry-connection velocity.","marker":"[50]"},{"why":"Introduces the concept of current-induced staggered relativistic fields on symmetry-related sites, the template for the hidden sublattice torques analyzed here.","marker":"[29]"},{"why":"Frames the intrinsic staggered spin-orbit torque in a monolayer system, the direct precursor of the hidden FeI torques in the present compounds.","marker":"[31]"}],"fun_headline_variants":["Hole doping silences a magnet's fourth-harmonic torque","Ge→Ga swap shrinks K pockets, unmasking 30× hidden torques","Fermi-sea torque vanishes when one electron is removed per cell","One missing electron per cell rewrites spin–orbit torque","Fe3GeTe2 vs Fe3GaTe2: One electron changes spin torque"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every angular torque map rests on the rigid-exchange-rotation approximation: the magnetization direction enters the Hamiltonian only by rigidly rotating its exchange (time-reversal-odd) part, with the time-reversal-even part frozen at its out-of-plane reference value; the paper's own benchmark shows this can deviate by about 17% for one TR-even component at in-plane magnetization, so the quantitative peak values of the maps inherit that uncertainty.","fun_headline_variants_meta":{"raw":{"variants":["Hole doping silences a magnet's fourth-harmonic torque","Ge→Ga swap shrinks K pockets, unmasking 30× hidden torques","Fermi-sea torque vanishes when one electron is removed per cell","One missing electron per cell rewrites spin–orbit torque","Fe3GeTe2 vs Fe3GaTe2: One electron changes spin torque"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00045,"raw_usage":{"total_tokens":2387,"prompt_tokens":1186,"completion_tokens":1201,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":802,"completion_tokens_details":{"reasoning_tokens":1115}},"tokens_in":802,"tokens_out":1201,"duration_ms":10116,"temperature":1.0,"reasoning_tokens":1115,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:27:34.565855+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the torkance at full self-consistent noncollinear DFT, without the rigid rotation, for both monolayers at the key in-plane directions, $\\hat{m}\\parallel\\hat{x}$ where Fe3GeTe2's TR-even peak and fourth harmonic sit, and $\\hat{m}\\parallel\\hat{y}$ where the benchmark already deviates by about 17%, and check whether the FGT/FGaT contrast in the TR-even peak or the fourth-harmonic coefficient survives. The experimental counterpart is to electrostatically hole-dope monolayer Fe3GeTe2 and measure the angular torque map: if the fourth-harmonic component does not shrink as holes are added, the K-pocket suppression mechanism is wrong.","supporting_citations":[{"cited_title":"Roemer, C","cited_arxiv_id":null,"evidence_quote":"The earlier first-principles study of monolayer FGT that established the angular-dependent torkance maps and the vector-spherical-harmonics expansion; the present paper recovers its higher-harmonic content and reassigns it to the FeI sublattice."},{"cited_title":"Zhang, F","cited_arxiv_id":null,"evidence_quote":"Supplies the Kubo-Bastin linear-response formulas whose analytic energy integrals define the Fermi-sea (TR-even) and Fermi-surface (TR-odd) torkance channels used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the D3h symmetry argument that the leading current-induced spin polarization must be linear in the in-plane magnetization, the basis of the paper's self-torque phenomenology."},{"cited_title":"Giannozzi, S","cited_arxiv_id":null,"evidence_quote":"Provides the symmetry-adapted Wannier-function construction that pins projectors and centers to the site symmetries, letting the interpolated model inherit the fully relativistic spin-orbit coupling."},{"cited_title":"Giannozzi, O","cited_arxiv_id":null,"evidence_quote":"The Wannier-interpolation engine that evaluates the dense-Brillouin-zone response quantities, including the gauge-covariant Berry-connection velocity."},{"cited_title":"MacNeill, G","cited_arxiv_id":null,"evidence_quote":"Introduces the concept of current-induced staggered relativistic fields on symmetry-related sites, the template for the hidden sublattice torques analyzed here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Frames the intrinsic staggered spin-orbit torque in a monolayer system, the direct precursor of the hidden FeI torques in the present compounds."}],"review_version":1}