{"id":"aa7dabc4-7a24-4613-84cc-85461b4d7c33","arxiv_id":"1908.04946","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A first-principles Fe 3d model reproduces the size and sign of the magnetoelectric polarization change in Fe2Mo3O8 and shows that two candidate charge states are magnetically similar but electrically distinguishable.","lead":"This paper builds a microscopic model for the magnetic iron electrons in the polar magnetoelectric material Fe2Mo3O8 and uses it to explain the size and sign of the electric polarization jump at the magnetic transition. A generalist may read it because it offers a concrete electronic mechanism for switching polarization by magnetism, with a testable difference between two possible charge states.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The magnetoelectric magnitude claim rests on the unvalidated four-parameter bilinear polarization ansatz of Eq. (4); a direct transferability test of the fitted Pij values is needed.","rationale":"The reader's weakest-assumption field identifies Eq. (4) and the Pij mapping as the key unverified step; I agree. This is the load-bearing point because the quantitative magnetoelectric claims do not rest on the exchange parameters, which are checked against two methods and against TN, but on the polarization parameters. A direct test of transferability would settle whether the fitted Pij values describe the polarization outside the two collinear states used in the mapping. The paper is otherwise appropriately hedged, and the qualitative conclusion—that the isotropic electronic contribution can be large and charge-state dependent—does not require the disputed quantitative agreement. The verdict CONDITIONAL with moderate confidence is therefore appropriate; my stress-test does not move it.","tokens_in":18102,"tokens_out":6385,"duration_ms":79337,"concrete_test":"Using the d7_t d5_o Hamiltonian of Sec. II B and Table I parameters, compute the Berry-phase polarization P_z in the same HF framework for at least three spin configurations not used in the Pij fit: (i) a canted state with one t-Fe spin rotated by 90°, (ii) the q=(0,0,1/2) noncollinear state predicted in Sec. II D, and (iii) a state with uniformly reduced sublattice magnetizations. Compare each HF P_z with the value predicted by Eq. (4) and Table III. If the discrepancies exceed about 30% of the experimental ΔP_z≈0.1 µC/cm², the bilinear mapping is not validated and the claimed quantitative agreement is not established. Repeat for the d6_t d6_o scenario with orbital order relaxed self-consistently in each tested configuration.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative central claim—P_z(0)=0.27 µC/cm² and ΔP_z≈−0.1 µC/cm²—is computed from Eqs. (7)–(9) using the Pij parameters of Table III. Those parameters are obtained by mapping Hartree-Fock polarizations of a limited set of collinear magnetic configurations onto Eq. (4), which assumes the polarization is a pairwise, state-independent, isotropic function of e_i·e_j. The paper gives no independent validation of this mapping. The HF solutions are self-consistent, the screened Hubbard parameters are not in the asymptotic large-U regime (U≈1.5–1.8 eV, J≈0.8 eV), and for the d6_t d6_o scenario the orbital order differs between AFM and FRM states. If the true polarization contains non-bilinear, multispin, or magnetic-state-dependent terms, the fitted Pij will absorb them and the molecular-field P_z(T) and ΔP_z are not a genuine prediction. The authors' caveat about lattice and spin-orbit/orbital contributions is relevant but does not test whether the bilinear form itself is adequate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript constructs a first-principles-derived Hubbard model for the Fe 3d electrons of the polar magnetoelectric Fe2Mo3O8, treating the Mo3 trimers as nonmagnetic spectators. The model is solved in Hartree-Fock for two charge states: the symmetry-preserving charge-disproportionated d7_t d5_o state and the constrained homogeneous d6_t d6_o state with orbital order. Exchange parameters are extracted both by mapping total energies of collinear configurations and by Green's-function perturbation theory; the two scenarios give nearly identical Jij, and RPA estimates of TN (54–55 K) are close to the experimental 60 K. The spin-dependent electric polarization is parametrized by the bilinear form Pz = (1/2) sum_ij Pij e_i·e_j (Eq. (4)), with four Pij fitted to the Hartree-Fock polarizations. Molecular-field theory then yields Pz(T) and the AFM-to-FRM polarization jump ΔPz ≈ −0.1 µC/cm², which agrees in sign and order of magnitude with experiment. The authors conclude that the isotropic electronic polarization at fixed crystal structure can account for the magnetoelectric effect, while lattice, orbital, and spin-orbit effects are deferred to future work.","tokens_in":18276,"tokens_out":9883,"duration_ms":96520,"significance":"The paper is a valuable contribution because it offers a transparent microscopic mechanism for a material whose magnetoelectric effect has been alternatively attributed to magnetostriction or to Dzyaloshinskii-Moriya coupling. Its strengths are the downfolding to a minimal Fe-3d model, the double validation of Jij by total-energy mapping and the Green's-function method, the reproduction of TN within about 10% in RPA without fitting to experimental magnetic data, and the prediction of a distinguishing low-temperature net-magnetization signature for the two charge scenarios. Both charge scenarios give ΔPz of the correct sign and magnitude, suggesting that the qualitative mechanism is robust. The main weakness is that the Pij parameters are fitted to the same magnetic configurations whose polarization difference is the target of the prediction; an out-of-sample check is needed before the quantitative claim is fully supported. If such a check confirms the bilinear ansatz, the result would be a clear demonstration that an isotropic electronic mechanism can explain the giant magnetoelectric effect in Fe2Mo3O8.","major_comments":[{"comment":"The central quantitative comparison with experiment is not independent of the inputs used to build the model. The Pij in Table III are obtained by mapping the Hartree-Fock polarizations of a limited set of collinear magnetic configurations (including the AFM and FRM states) onto the four-parameter bilinear form of Eq. (4). Since Eqs. (7)–(9) are then evaluated with these same Pij, the resulting Pz(0) = 0.27 µC/cm² and ΔPz ≈ −0.1 µC/cm² are at least partially a reconstruction of the polarizations of those fitted configurations, rather than a genuine prediction. The paper states that Eq. (4) can be derived rigorously in the large-U limit (Refs. [34,35]), but it does not present or compare with such a derivation. I request an out-of-sample test: compute the Berry-phase polarization for at least one additional magnetic configuration not used in the mapping (for example a noncollinear state or a different collinear order) and verify Eq. (4); alternatively, derive the Pij from the large-U expansion and compare with Table III. Without this, the claim that the order of magnitude is reproduced is vulnerable to circularity.","section":"Sec. II E, Eq. (4), Table III"},{"comment":"For the d6_t d6_o scenario, the orbital order differs between the AFM and FRM states (Fig. 8), so the orbital state is not fixed when the spin order changes. For the exchange couplings the authors provide a state-dependence test through the Green's-function analysis of Sec. II D, but no analogous test is given for the polarization parameters. The large difference between the Pij sets for d7_t d5_o and d6_t d6_o in Table III shows that Pij are highly sensitive to the local electronic state. If Pij themselves depend on the magnetic state through orbital order, a single set of Pij inserted into Eqs. (7)–(9) is not justified, and the d6 panel of Fig. 11 would not be a valid molecular-field prediction. Please provide evidence that the bilinear Pij are transferable between magnetic states, or restrict the quantitative polarization analysis to the case where orbital order is absent.","section":"Sec. II C and II E, Fig. 8"},{"comment":"The paper's most favorable comparison for the zero-temperature polarization (Pz(0) = 0.27 µC/cm² versus the experimental 0.34 µC/cm²) comes from the d7_t d5_o solution, which is obtained without a double-counting correction and with screened Coulomb parameters U ≈ 1.5–1.8 eV that are not in the strongly correlated limit. The authors explicitly note that this solution \"may also be an artifact of calculations.\" Because the two scenarios differ in Pz by a factor of about four, the numerical agreement for Pz(0) is not robust against this ambiguity. For ΔPz the two scenarios are similar, so the issue is less severe for the AFM-FRM jump, but the Pz(0) comparison in the text is affected. A sensitivity analysis (for example varying the t-o level offset or including a double-counting term) is needed to show that the order-of-magnitude conclusion does not depend on the choice of d7_t d5_o.","section":"Sec. II C, Table I"}],"minor_comments":[{"comment":"There are several typos and grammatical errors: \"scenaria\" in the abstract, \"tends to low\" in the abstract, \"chesk\" in Sec. II D, \"nagnetization\" in Sec. IV, \"spites\" in Sec. IV, \"Fr sites\" in the Table I caption, and \"antiferromagnetic d5_t d7_o\" for \"d7_t d5_o\" in the Fig. 9 caption. These should be corrected.","section":"Throughout"},{"comment":"In the paragraph before Table III, the sentence listing the four independent parameters says \"J‖, J⊥, J⊥^o, and J⊥^t\" but should refer to the polarization parameters P‖, P⊥, P⊥^o, and P⊥^t. This is confusing and should be fixed.","section":"Sec. II E"},{"comment":"The statement that the RPA estimate TN = 55 K is close to the experimental 60 K should be qualified: this agreement holds for the reduced four-parameter exchange set, whereas using the full Green's-function exchange set yields TN = 32 K for d7_t d5_o. Please make this explicit so the reader does not overestimate the accuracy of the full calculation.","section":"Sec. III"},{"comment":"Fig. 11 would be substantially more informative if the experimental Pz(T) and ΔPz data from Refs. [4,5] were overlaid on the theoretical curves; currently the experimental values appear only in the text.","section":"Fig. 11"},{"comment":"The statement in Ref. [25] that all model parameters are available upon request is a reproducibility concern; I encourage providing the parameters and the Wannier-model construction details as ancillary files with the arXiv submission.","section":"Ref. [25]"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript is a serious downfolding study well within the journal's scope, and the main concern is validation rather than novelty. The authors' reliance on their own previously developed methods (Refs. [26,30,33–35]) is appropriate in this context. A revised version that adds an out-of-sample test of Eq. (4), or a direct comparison with the large-U derivation of Pij, would address the principal objection and could make the paper acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing: the paper does something genuinely useful. It constructs a Fe 3d-only Wannier model for Fe2Mo3O8, derives site-dependent cRPA U and J, and maps both exchange and electric polarization onto isotropic spin models for two charge configurations. The result that the homogeneous d6t d6o and charge-disproportionated d7t d5o scenarios are magnetically almost identical but dielectrically distinct is a sharp, testable statement. The RPA Neel temperature (55 K) sits close to the measured 60 K, and the polarization magnitude and sign at T=0 (0.27 µC/cm² vs 0.34) and the AFM-FRM jump (-0.1 µC/cm²) are within a factor of 1.3-1.4 of experiment. That is an honest semi-quantitative success, and the authors do not oversell it: they flag the missing double-counting term, the need for lattice effects, and the neglect of spin-orbit coupling.\n\nThe soft spot is exactly where the stress-test note points. The Pij parameters in Table III are obtained by fitting Hartree-Fock polarizations of the AFM and FRM configurations to Eq. (4), which assumes a pairwise, state-independent, isotropic bilinear form. The same fitted parameters then generate Pz(T) and ΔPz through the molecular-field equations (7)-(9). That is not an external test of the ansatz; it is a consistency check. If the true polarization has significant multispin, orbital, or state-dependent contributions, the fit absorbs them and the temperature dependence is not a genuine prediction. The authors' caveat about lattice and orbital degrees of freedom is relevant, but it does not test Eq. (4) itself. The d6t d6o scenario is further complicated by the orbital-order change between AFM and FRM, so the polarization mapping is especially fragile there.\n\nI also note the noncollinear ground state predicted by the exchange model is not checked against experiment, and the competing Dzyaloshinskii-Moriya mechanism is mentioned but not computed. Those are omissions, not errors. The paper ships no code or data, only \"available upon request,\" which limits reproducibility; given the richness of the model, a parameter table or repository would help.\n\nIs this a fatal problem? No. The central mechanism—electronic polarization from the magnetic-state-dependent electronic structure at fixed crystal structure—is physically plausible and consistent with the sign and rough magnitude of the effect. The distinction between the two charge scenarios is qualitative and robust. But the quantitative numbers for Pz(T) should be read as model-dependent estimates, not first-principles predictions.\n\nRecommendation: send to peer review. A competent referee can push for a transferability test of the Pij mapping (e.g., compute polarization for a magnetic configuration not used in the fit) and for clarification of the uniqueness of the four-parameter fit. That would turn a suggestive paper into a solid one.","headline":"Solid downfolding study with a real semi-quantitative match, but the polarization numbers hinge on an unvalidated four-parameter bilinear fit; still worth refereeing.","tokens_in":18836,"tokens_out":2857,"would_cite":true,"duration_ms":26637,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["75.85.+t","71.10.Fd"],"model":"deepseek-v4-flash","headline":"The giant magnetoelectric signal of Fe2Mo3O8 is reproduced by the spin-dependent electronic polarization alone, with the crystal structure held fixed.","keywords":["magnetoelectric effect","Fe2Mo3O8","charge disproportionation","orbital ordering","spin Hamiltonian","electric polarization","exchange interactions","interacting-electron model"],"falsifier":"Measure the low-temperature net magnetization of the honeycomb layers: the charge-disproportionated $\\mathrm d^7_t\\mathrm d^5_o$ scenario predicts a finite net moment that persists as $T\\to 0$, while the homogeneous $\\mathrm d^6_t\\mathrm d^6_o$ scenario predicts it to vanish at $T=0$ and appear only at elevated temperature. The shape of $P_z(T)$—nearly monotonic for $\\mathrm d^7_t\\mathrm d^5_o$, peaked near half the magnetic ordering temperature for $\\mathrm d^6_t\\mathrm d^6_o$—also distinguishes the two.","tokens_in":17818,"feed_emoji":"🧲","tokens_out":17022,"duration_ms":159089,"temperature":0.7,"pith_summary":"Fe2Mo3O8 (kamiokite) displays a giant jump in electric polarization when its magnetic order changes, and this paper asks whether the effect can come from the electrons alone rather than from magnetostriction or spin-orbit coupling. The authors construct a minimal first-principles model of the iron 3d electrons, solve it self-consistently, and map it onto an isotropic spin model. They compare two charge arrangements—homogeneous Fe2+ on both tetrahedral and octahedral sites, and charge-disproportionated Fe1+ (tetrahedral) / Fe3+ (octahedral)—and find nearly identical magnetic interactions but strongly different spin-dependent electric polarizations. The calculated polarization and its jump match the order of magnitude and sign of experiment, supporting an electronic, spin-order-driven origin of the magnetoelectric effect. The paper cautions that the charge-disproportionated solution could be an artifact of neglecting a double-counting correction, and notes that a quantitative description of the temperature dependence will probably require lattice degrees of freedom.","feed_headline":"Spin order alone can explain Fe2Mo3O8's giant polarization jump","feed_subtitle":"A minimal model of iron's 3d electrons matches the measured size and sign of the polarization change.","key_machinery":"The load-bearing object is a minimal interacting-electron model for the Fe 3d electrons, built from localized basis functions and solved in a mean-field self-consistent approximation. From its solutions the paper extracts interatomic exchange couplings $J_{ij}$ and, separately, polarization parameters $P_{ij}$ by assuming that the magnetic part of the polarization has the same bilinear form as the spin Hamiltonian: $P_z=\\frac{1}{2}\\sum_{ij}P_{ij}\\mathbf e_i\\cdot\\mathbf e_j$, with only four independent parameters $P_\\parallel$, $P_\\perp$, $P^o_\\perp$, and $P^t_\\perp$. These parameters are fixed by mapping the polarization computed for the antiferromagnetic and ferrimagnetic configurations onto that form, in exact analogy to the mapping of total energies onto the spin model. The four-parameter polarization form is what carries the argument: exchange couplings come out almost identical for the two charge scenarios, while the polarization parameters differ strongly, and that difference is traced to the different bond polarity of Fe2+–Fe2+ versus Fe1+–Fe3+ pairs.","core_discovery":"The central claim is that the giant magnetoelectric response of Fe2Mo3O8 can be understood without invoking atomic displacements or antisymmetric spin-orbit exchange: the electric polarization acquires a spin-dependent part $P_z=\\frac{1}{2}\\sum_{ij}P_{ij}\\mathbf e_i\\cdot\\mathbf e_j$ because the electronic structure of the Fe 3d shell changes with the magnetic pattern, even with the crystal structure fixed. The polarization parameters $P_\\parallel$, $P_\\perp$, $P^o_\\perp$, and $P^t_\\perp$ are obtained by mapping self-consistent mean-field polarizations of the antiferromagnetic and ferrimagnetic states onto this bilinear form. In the homogeneous $\\mathrm d^6_t\\mathrm d^6_o$ scenario the in-plane $P_\\parallel$ is small and the octahedral interlayer term dominates; in the charge-disproportionated $\\mathrm d^7_t\\mathrm d^5_o$ scenario all four parameters are large, with $P_\\parallel$ strongly enhanced because the large Fe1+–Fe3+ ionic charge difference compensates for the small vertical separation within the bond. The two scenarios are nearly indistinguishable magnetically, yet they give different $P_z(T)$. Molecular-field evaluation for $\\mathrm d^7_t\\mathrm d^5_o$ yields $P_z(0)\\approx 0.27\\,\\mu\\mathrm C/\\mathrm{cm}^2$, close to the measured $\\approx 0.34\\,\\mu\\mathrm C/\\mathrm{cm}^2$, and a jump $\\Delta P_z\\approx -0.1\\,\\mu\\mathrm C/\\mathrm{cm}^2$ near the magnetic ordering temperature, matching the experimental magnitude and sign.","pith_inferences":["A natural extension is to apply the same four-parameter polarization construction to other polar $Me_1Me_2Mo_3O_8$ compounds; the Fe2Mo3O8 result suggests that a large difference in formal charge between tetrahedral and octahedral cations is a generic amplifier for spin-driven electric polarization.","If lattice relaxation were included, it could enhance or partially cancel the electronic polarization found here; comparing the fixed-lattice prediction with measurements under hydrostatic pressure could isolate the electronic contribution.","The paper's preliminary discussion of spin-orbit coupling implies that in the homogeneous $\\mathrm d^6_t\\mathrm d^6_o$ scenario a canted magnetic state—antiferromagnetic $c$-axis components, ferrimagnetic $ab$-plane components—may allow continuous tuning of the polarization by an in-plane magnetic field, a consequence the paper does not quantify.","The paper itself notes in Sec. II C that the charge-disproportionated solution could be an artifact of neglecting a double-counting correction; if that artifact is real, the homogeneous $\\mathrm d^6_t\\mathrm d^6_o$ scenario remains the operative one, and the predicted peak shape of $P_z(T)$ would identify it experimentally."],"forward_implications":["If the electronic mechanism is right, the giant polarization jump does not require magnetostriction; a magnetic field that switches antiferromagnetic order to ferrimagnetic order should switch the polarization through the spin-dependent electronic term.","Because the two charge scenarios give almost identical exchange couplings, magnetic measurements alone cannot easily tell them apart; the temperature dependence of the electric polarization is the discriminating observable.","In the charge-disproportionated scenario, a finite net magnetization appears already at low temperature, making the antiferromagnetic-to-ferrimagnetic transition field-driven, with the model estimating critical fields of order tens of tesla.","The sign pattern—positive $P_z$ and negative $\\Delta P_z$—comes out consistent with experiment in both scenarios, so sign compatibility alone is not sufficient to select between them.","The model predicts a noncollinear magnetic ground state with wave vector near $\\mathbf q=(0,0,1/2)$, a testable neutron-scattering signature."],"supporting_citations":[{"why":"Experimental report of the giant polarization jump and the antiferromagnetic-to-ferrimagnetic transition that the calculations target.","marker":"[4]"},{"why":"Independent experimental data on the field-induced switching and polarization change used as a second baseline.","marker":"[5]"},{"why":"Source of the experimental hexagonal crystal structure that fixes all atomic positions in the calculations.","marker":"[9]"},{"why":"Earlier proposal of an antisymmetric spin-orbit exchange mechanism, the competing explanation the paper argues against.","marker":"[15]"},{"why":"Provides the method for building the localized-orbital effective model and for self-consistent mean-field solutions and exchange parameters.","marker":"[26]"},{"why":"Supplies the constrained random-phase-approximation scheme used to obtain the screened Coulomb interaction parameters that favor charge disproportionation.","marker":"[27]"},{"why":"Gives the Green's-function perturbation formula used to compute interatomic exchange interactions.","marker":"[30]"},{"why":"Establishes the geometric-phase theory of electric polarization that underlies the polarization calculation.","marker":"[32]"},{"why":"Applies the geometric-phase theory to the effective model so that polarization can be evaluated for the magnetic configurations.","marker":"[33]"},{"why":"Derives the isotropic and anisotropic spin-dependent polarization expressions used in Eq. (4) and its extensions.","marker":"[35]"}],"fun_headline_variants":["Spin order alone explains Fe2Mo3O8's giant polarization jump","Minimal spin model captures Fe2Mo3O8's magnetoelectric jump","No atomic motion needed: spin sets Fe2Mo3O8 polarization","Charge disproportionate model reproduces Fe2Mo3O8's jump"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole quantitative result depends on the assumption that the spin-dependent polarization is described by four pairwise spin-alignment coefficients fitted from two magnetic configurations, with lattice, orbital, and spin-orbit contributions either absent or fixed; if any of those contributes substantially, the predicted magnitudes and temperature dependence change.","fun_headline_variants_meta":{"raw":{"variants":["Spin order alone explains Fe2Mo3O8's giant polarization jump","Minimal spin model captures Fe2Mo3O8's magnetoelectric jump","No atomic motion needed: spin sets Fe2Mo3O8 polarization","Charge disproportionate model reproduces Fe2Mo3O8's jump"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000607,"raw_usage":{"total_tokens":2992,"prompt_tokens":1269,"completion_tokens":1723,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":885,"completion_tokens_details":{"reasoning_tokens":1644}},"tokens_in":885,"tokens_out":1723,"duration_ms":13242,"temperature":1.0,"reasoning_tokens":1644,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:28:34.871918+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the low-temperature net magnetization of the honeycomb layers: the charge-disproportionated $\\mathrm d^7_t\\mathrm d^5_o$ scenario predicts a finite net moment that persists as $T\\to 0$, while the homogeneous $\\mathrm d^6_t\\mathrm d^6_o$ scenario predicts it to vanish at $T=0$ and appear only at elevated temperature. The shape of $P_z(T)$—nearly monotonic for $\\mathrm d^7_t\\mathrm d^5_o$, peaked near half the magnetic ordering temperature for $\\mathrm d^6_t\\mathrm d^6_o$—also distinguishes the two.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Experimental report of the giant polarization jump and the antiferromagnetic-to-ferrimagnetic transition that the calculations target."},{"cited_title":"Kurumaji, S","cited_arxiv_id":null,"evidence_quote":"Independent experimental data on the field-induced switching and polarization change used as a second baseline."},{"cited_title":"Le Page and P","cited_arxiv_id":null,"evidence_quote":"Source of the experimental hexagonal crystal structure that fixes all atomic positions in the calculations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier proposal of an antisymmetric spin-orbit exchange mechanism, the competing explanation the paper argues against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the method for building the localized-orbital effective model and for self-consistent mean-field solutions and exchange parameters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Green's-function perturbation formula used to compute interatomic exchange interactions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the geometric-phase theory of electric polarization that underlies the polarization calculation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Applies the geometric-phase theory to the effective model so that polarization can be evaluated for the magnetic configurations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the isotropic and anisotropic spin-dependent polarization expressions used in Eq. (4) and its extensions."}],"review_version":1}