{"id":"999a1415-a408-415c-b122-efd8302f3820","arxiv_id":"2607.13559","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Intraband and interband magnetoelectric couplings in magnetic Weyl semimetals produce emergent inductance that is enhanced when the chemical potential lies in the Weyl dispersion.","lead":"This paper derives formulas for two distinct magnetoelectric contributions in magnetic Weyl semimetals and uses them to predict an emergent inductance from magnetization dynamics, enhanced near the Weyl points. A generalist reader might care because it offers a mechanism-based route toward inductive circuit elements built from topological materials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Γ-point reduction to the two-band model (Eqs. 16–20) lacks quantitative validation; uncomputed higher-order and remote-band terms could shift the interband ME tensor C_I and thus the predicted inductance.","rationale":"The mapping from Eq. (15) to Eq. (5) is the linchpin of the quantitative claims. Without a direct numerical or analytical verification that the Γ-point truncation preserves the interband ME response, the inductance curves in Figs. 2–3 are not proven to represent the lattice model. The reader's weakest assumption identifies exactly this issue. My proposed test—comparing the full lattice Kubo result with the effective-model formula—would settle it. If the test passes, the quantitative claims are solid; if it fails, the paper's application to PrAlGe would need revision, though the general mechanism might survive. Since the reader already issued CONDITIONAL, my concern does not change the verdict.","tokens_in":12987,"tokens_out":24293,"duration_ms":234566,"concrete_test":"Numerically compute the interband ME tensor C_I for the full lattice model Eq. (15) with the Fig. 3 parameters (λ=0.3t0, M_w=0.7t0, δt=0.1t0, m_z=3.5t0, α=0.3) using the Kubo formula Eq. (6) at μ=0, with a dense BZ grid. Compare each component to the effective-model prediction from Eq. (7) with v,m,η,P,M from Eqs. (16)–(20) and I from Eq. (9). If any component differs by more than ~20%, the Γ-point expansion is quantitatively inadequate. A corollary check: scan the BZ for eigenvectors at the band-touching energy to confirm only two Weyl points exist.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's quantitative predictions (Figs. 2–3, Eqs. (7),(11)) rely on computing the interband ME tensor C_I from the effective two-band model Eq. (5), whose parameters (v,m,η,P,M) are obtained by a leading-order Γ-point expansion of the lattice Hamiltonian Eq. (15) (Appendix B, Eqs. (B1)–(B7)). The central assumption, unstated, is that this truncated expansion captures the full Brillouin-zone integral in Eq. (6). This is not self-evident: the integral for C_I (Eq. A7) runs over all momenta, and although it is UV convergent, its value can depend on the band structure far from the Weyl nodes. The Γ-point expansion neglects (i) higher-order k terms (cubic, quartic) that modify the dispersion and Berry curvature away from the nodes, and (ii) all transitions between the Weyl bands and the other two bands of the 4-band lattice model. It also assumes no additional Weyl points or band crossings exist elsewhere in the BZ; the paper does not report a full-BZ search. If any of these neglected contributions are significant, the computed C_I and hence the inductance (Fig. 3c) would differ from the actual lattice-model response. This directly affects the central claim of Weyl-region-enhanced emergent inductance and the comparison with the companion experiment (Ref. 38).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of emergent inductance in magnetic Weyl semimetals, driven by the magnetoelectric (ME) response of Weyl fermions to magnetization dynamics. The authors derive analytical expressions for the ME tensor, decomposing it into an intraband Fermi-surface contribution proportional to the relaxation time and the square of the chemical potential (Eq. (4)), and an interband contribution proportional to the Weyl-node separation/Fermi-arc length (Eq. (7), with the integral evaluated in Eq. (9)). These ME tensors are then used to construct an emergent inductance formula (Eq. (11)), which is applied to a model of polar Weyl ferromagnets (Eq. (15)) via a low-energy expansion around the Γ point (Appendix B). The paper claims that the inductance is enhanced when the chemical potential lies in the Weyl regime, reflecting the topological nature of the Weyl dispersion, and connects the results to the recent experimental observation of toroidal induction in PrAlGe (Ref. [38]).","tokens_in":13325,"tokens_out":5112,"duration_ms":54913,"significance":"If the central claims are correct, the paper provides a useful analytical framework for emergent inductance in magnetic Weyl semimetals, complementing numerical and experimental work. The decomposition into intraband and interband ME contributions, with the latter tied to Weyl-node separation, is a conceptually clean result that could guide future experiments. The paper includes detailed appendices (A and B) with explicit derivations of the interband integral and the parameter mapping for the lattice model, which is a strength. The connection to the companion experiment on polar Weyl ferromagnets is timely. However, the quantitative predictions rest on a low-energy two-band truncation whose validity is not quantitatively established, and the derivation of the inductance formula is only sketched. These gaps are significant for a paper claiming both a general theory and specific enhancement predictions.","major_comments":[{"comment":"The mapping from the lattice model Eq. (15) to the effective two-band model Eq. (5) is performed by a leading-order expansion around the Γ point. The interband ME tensor C_I in Eq. (6)/(A7) is an integral over all momenta; although UV convergent, its value can receive significant contributions from regions away from Γ, including higher-order k terms and transitions to the other two bands of the 4-band model. The paper asserts in §V that the enhancement is 'robust and does not depend on details of the microscopic Hamiltonian,' but no numerical validation is provided (no full-BZ calculation of C_I for Eq. (15), no search for additional Weyl points). This assumption is load-bearing for the quantitative predictions in Fig. 3(c) and for the claimed explanation of the experiment Ref. [38]. I request a numerical check of C_I for the full lattice model or a controlled estimate of the neglected t","section":"§IV, Appendix B (Eqs. (16)–(20), (B1)–(B7))"},{"comment":"The derivation of the spin-dynamics-mediated conductivity Σ(ω) and the resulting inductance formula is not presented. The expression for Σ(ω), with factors v and v_cell, and the step from the ME response to the impedance Z require a detailed derivation. In particular, the treatment of the magnon Green's function (-ωσ_y - H_m)^{-1}, the role of the removed z-row in C̃_O and C̃_I, and the signs in the combination (-C̃_O + C̃_I) vs (C̃_O + C̃_I) are not justified. Since the central claim of emergent inductance rests on Eq. (11), please provide the derivation in an appendix and clarify the assumptions (e.g., symmetry of H_m, linear response regime).","section":"§III, Eq. (11)"},{"comment":"The analytical expression for C_I in Eq. (7) is explicitly derived for the chemical potential at the Weyl points (µ=0). However, Fig. 1(b) and the discussion in §IV present the chemical-potential dependence of the interband ME contribution, which is central to the claimed Weyl-region enhancement. No finite-µ formula or numerical method is given. Please provide the expression used for finite µ (e.g., the generalization of Eq. (6) with f0(ξ-µ)) or describe the numerical integration procedure, so that Fig. 1(b) and Fig. 3(c) are reproducible.","section":"§II.B, Fig. 1(b)"}],"minor_comments":[{"comment":"There is a missing phrase: 'Dividing the Brillouin zone into regions with and without Fermi arcs along the k_z direction (the arc length being ),' — the arc-length expression is absent. Please fill in the missing term or remove the parenthetical.","section":"§II.B, text near Eq. (9)"},{"comment":"Typos: 'consideter' (should be 'consider'), 'obatin' ('obtain'), 'proprortional' ('proportional'), 'semimentals' ('semimetals'). In §II.B, 'τ in Eq. (5) include spin degrees of freedom' is garbled; likely 'the Pauli matrices in Eq. (5)'.","section":"Throughout"},{"comment":"The indices on C_O and C_I are inconsistent (e.g., C^x_Ox vs C^x_Iy in the text). Please define the index convention clearly (first index = magnetization component, second = electric field direction, or vice versa) and use it consistently.","section":"Eqs. (7)–(11)"},{"comment":"Parameters such as τ=10ma^2/ℏ and M diagonal elements m^{-1}a^{-2}ℏ^{-1} need explicit definitions of the units and of the mass parameter m in the effective model (Eq. (5)) to avoid confusion with the lattice mass M_w.","section":"Figure captions"},{"comment":"The definition of impedance via the inverse of (σ_DC + Σ(ω)) may deserve a brief justification in terms of the sample geometry and the sign convention, especially since the inductance is extracted from the imaginary part of ∂_ω Z.","section":"§III, Eq. (10)"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly positioned as a theory companion to the experimental arXiv:2607.12322 by the same groups. The central mechanism is plausible and the analytical derivations in the appendices are a real strength. However, the missing validation of the Γ-point reduction is a substantive issue for the quantitative claims; a full-BZ numerical check for Eq. (15) would substantially increase confidence. The sketchy derivation of Eq. (11) is also a barrier to assessing the generality of the claimed inductance formula. The manuscript would be improved by narrowing the 'general theory' claim or by adding the missing derivations and numerical checks. The scope of the journal (cond-mat.mes-hall) is appropriate, provided the technical gaps are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper delivers a clean analytic decomposition of the magnetoelectric response in a two-Weyl model into a dissipative intraband term and a nondissipative interband term, then builds an emergent-inductance formula from the two. The interband tensor C_I in Eqs. (7)–(9) is new, and the resulting longitudinal/Hall inductance splitting is a useful construction. The appendices are substantive—they show real derivations, not just hand-waving. The authors also reproduce the known Boltzmann intraband result, which is a good sanity check.\n\nThe central mechanism—Weyl semimetals act as emergent inductors, with interband processes dominating near the Weyl points—is credible. The qualitative comparison to the PrAlGe experiment is reasonable, and the prediction of enhanced inductance when the chemical potential sits in the Weyl window is falsifiable.\n\nThe soft spots are real but not fatal. The biggest one is the Γ-point reduction in Appendix B. The effective parameters in Eqs. (16)–(20) come from a leading-order expansion of the lattice Hamiltonian, and the paper never checks whether that truncated model reproduces the full Brillouin-zone integral for C_I. The stress-test concern is on point: the integral is UV convergent but can still depend on band structure away from the nodes, and there is no search for other Weyl points or remote-band transitions. That means the quantitative curves in Figs. 2–3 should be read as schematic, not as measured predictions for PrAlGe. The magnon Hamiltonian H_m is likewise a toy parameter, so the absolute inductance scale is not predictive. These are addressable gaps: the authors could compute the lattice-model C_I numerically or add a justification for why the low-energy model suffices.\n\nThe circularity worry about the companion experiment is minor. The theory does not fit parameters to the data, and the mechanism stands on its own.\n\nWho is this for? People working on emergent inductance, magnetoelectric responses, or topological semimetals. I'd send it to a serious referee with a request to tighten the quantitative claims. The paper is worth engaging with.","headline":"A genuinely useful analytic theory of emergent inductance in Weyl semimetals, with a new closed-form interband ME tensor; the mechanism is credible, but the quantitative mapping from the lattice model is under-validated.","tokens_in":13843,"tokens_out":3259,"would_cite":true,"duration_ms":35635,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Magnetic Weyl semimetals act as emergent inductors, with a magnetoelectric response that splits into dissipative and topological parts.","keywords":["Weyl semimetal","emergent inductance","magnetoelectric response","interband contribution","toroidal moment","polar Weyl ferromagnet","magnetization dynamics","Fermi arc"],"falsifier":"Measure the complex impedance of a polar Weyl ferromagnet as a function of chemical potential (via gating or doping) and check whether the inductance peaks inside the predicted Weyl window |μ| < v√(2mη) and whether the Hall inductance scales with τ while the interband-dominated longitudinal inductance is τ-independent; the predicted 1/v scaling could be tested by comparing materials with different spin-orbit coupling strengths.","tokens_in":12811,"feed_emoji":"⚡","tokens_out":3821,"duration_ms":39088,"temperature":0.7,"pith_summary":"The paper argues that magnetic Weyl semimetals generally behave as emergent inductors: when magnetization dynamics couple to Weyl fermions, the magnetoelectric response converts time-varying spin order into voltage, producing an inductance. The magnetoelectric tensor is shown to split into a dissipative intraband piece proportional to the relaxation time and Fermi-surface area, and a nondissipative interband piece proportional to the separation between Weyl nodes, i.e., the Fermi-arc length. Applying this to a minimal model of a polar Weyl ferromagnet, the authors reproduce the recently observed 'emergent toroidal induction' and find that the inductance is sharply enhanced when the chemical potential lies in the Weyl-dispersion energy window. If correct, the result establishes a general mechanism for inductive responses driven by topological band structure rather than by conventional Faraday geometry.","feed_headline":"Weyl semimetals get a topological inductance","feed_subtitle":"Magnetization dynamics split the response into dissipative and Fermi-arc parts, and inductance peaks inside the Weyl energy window.","key_machinery":"The load-bearing object is the effective two-band Weyl Hamiltonian H=(v k_x, v k_y, |k|²/2m − η) P·τ + S^T M τ, which hosts two Weyl points separated along k_z. The magnetoelectric tensors C_O and C_I are obtained by Boltzmann and Kubo approaches; the interband tensor involves the integral I in Eq. (9) whose first term is Fermi-arc controlled. The impedance formula Z = (l/A)(σ_DC + Σ(ω))^{-1} then yields the inductance L via the magnon propagation matrix H_m, the ME tensors, and the DC conductivity.","core_discovery":"The central discovery is a general formula for the emergent inductance of a magnetic Weyl semimetal, built from two magnetoelectric tensors: C_O from intraband transport (Boltzmann) and C_I from interband transitions (Kubo). C_I is expressed analytically in closed form: C_I = (e|P|/16π²) M P Q I, where I is a momentum integral that reduces to a term proportional to the Weyl-node separation 2√(2mη)/v plus a correction from the region without Fermi arcs. The paper proves C_O ∝ τ μ² and C_I ∝ 1/v, so in the Weyl window the interband term dominates and gives positive longitudinal and Hall inductance. For the polar Weyl ferromagnet model, the Hall inductance direction coincides with the toroidal","pith_inferences":["The 1/v dependence of the interband ME tensor suggests that materials with weaker spin-orbit coupling (smaller v) could show larger inductance peaks, provided Weyl nodes still exist — an extension the paper does not test.","The clean separation into C_O and C_I offers an experimental way to disentangle dissipative and geometric contributions by measuring inductance versus temperature or scattering time.","The same mechanism could appear in other topological semimetals, such as Weyl fermions at phase boundaries between three-dimensional topological and trivial insulators, as the paper hints but does not develop.","A practical corollary is that magnetization reversal could switch the sign or direction of the toroidal inductance, suggesting a memory or tunable-inductor application that the paper leaves implicit."],"forward_implications":["Magnetic Weyl semimetals generally exhibit emergent inductance whenever the Weyl fermion subspace contains spin degrees of freedom.","The interband ME contribution is nondissipative and grows with Weyl-node separation / Fermi-arc length, so inductance can survive even without Fermi-surface dissipation.","In polar Weyl ferromagnets, the Hall inductance appears along the toroidal moment direction, matching emergent toroidal induction.","Chemical potential tuned into the Weyl regime enhances both the ME tensor and the inductance; this enhancement is robust because it comes from the low-energy band structure.","Positive longitudinal inductance from the interband effect can arise even when longitudinal conductivity is suppressed, potentially enabling high quality factors."],"fun_headline_variants":["Inductance emerges from Weyl node separation","Magnetization drives new inductance in Weyl semimetals","Topological inductance peaks at Weyl nodes","Weyl semimetals get emergent induction from magnetization","Interband motion sparks inductance in Weyl semimetals"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The mapping from the microscopic four-band model of a polar Weyl ferromagnet to the effective two-band model relies on a single Γ-point expansion with parameter identifications; if higher-order terms or neglected orbital/spin mixing alter the interband magnetoelectric tensor, the quantitative inductance prediction would shift, even though the qualitative Weyl-region enhancement could survive.","fun_headline_variants_meta":{"raw":{"variants":["Inductance emerges from Weyl node separation","Magnetization drives new inductance in Weyl semimetals","Topological inductance peaks at Weyl nodes","Weyl semimetals get emergent induction from magnetization","Interband motion sparks inductance in Weyl semimetals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000394,"raw_usage":{"total_tokens":1910,"prompt_tokens":758,"completion_tokens":1152,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":1075}},"tokens_in":502,"tokens_out":1152,"duration_ms":11368,"temperature":1.0,"reasoning_tokens":1075,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T04:47:39.253913+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the complex impedance of a polar Weyl ferromagnet as a function of chemical potential (via gating or doping) and check whether the inductance peaks inside the predicted Weyl window |μ| < v√(2mη) and whether the Hall inductance scales with τ while the interband-dominated longitudinal inductance is τ-independent; the predicted 1/v scaling could be tested by comparing materials with different spin-orbit coupling strengths.","supporting_citations":[],"review_version":1}