{"id":"b117a2b0-dbfd-48e4-add8-e7a9a0deb579","arxiv_id":"2608.08269","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Solving the exact eigenstates of a rotating Weyl semimetal shows the chiral vortical current arises from a non-thermal, ground-state-free spectrum, not from a Fermi distribution.","lead":"A theory paper derives microscopic quantum states of a rotating Weyl semimetal and computes the chiral vortical current directly from them. It shows the current is a non-equilibrium quantum effect and identifies exactly when the standard semiclassical formula applies.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central non-equilibrium claim rests on Eq. (1)'s orbital-only rotation coupling; a physical rotating frame requires -ω·J, under which the paper's quench protocol produces no CVE dynamics.","rationale":"The algebraic derivation of Eq. (13) from the chosen Hamiltonian appears internally consistent, and the paper does reproduce the known semiclassical coefficient under its stated conditions with no fitted parameters, which I credit. However, the central claim is about what physical rotation does to a Weyl semimetal, and that is fixed by the Hamiltonian input. Eq. (1) assumes a rotation coupling that acts only on L while leaving the chiral spin/pseudospin degree of freedom inert. The standard rotating-frame transformation U = exp(−iωtJ_z/ℏ) gives H' = H0 − ωJ_z, including −ω·S; the paper's own Eq. (2) shows [H0,J_z]=0, so under the physical coupling the quench dynamics vanishes and no ground-state-free Floquet spectrum is generated from an initial H0 eigenstate. The paper offers only a heuristic statement that spin+orbital rotation corresponds to observer rotation and therefore no CVE, which conflates the generator of reference-frame rotations with the physical inertial coupling. A concrete calculation with H' = H0 − ωJ_z would settle whether this concern lands. Since this Hamiltonian-choice issue is the entry point for all subsequent claims, the result must remain conditional on the outcome of that test; I therefore keep the reader's CONDITIONAL verdict rather than moving to ACCEPT or REJECT.","tokens_in":30735,"tokens_out":18234,"duration_ms":192798,"concrete_test":"Recompute the spectrum and the Sec. 4.2 quench with H' = ⊕_λ λℏv_F k·σ − ω(L_z + ℏσ_z/2) and the same isotropic initial Fermi-sea state. If H' shares eigenstates with H0, then ⟨j_z(t)⟩ from Eq. (21) is stationary and vanishes for the stated initial condition, falsifying the ground-state-free unitary-CVE claim. To rule out the alternative, derive the rotating-frame low-energy Weyl Hamiltonian from the Dirac equation with the spin connection (or from a microscopic tight-binding model on a rotating lattice) and identify whether −ω·S appears; if it does, Eq. (1) must include this term before the subsequent results can be attributed to CVE.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive assumption is Eq. (1): H = ⊕_λ λℏv_F k·σ − ω·L. The paper explicitly rejects the alternative −ω·J (Sec. V) because it commutes with H0 and would give no bulk current. But that is precisely what a rotating frame does: a Hamiltonian in a frame rotating at ω is obtained from H0 by H_rot = H0 − ω·J = H0 − ω·(L + ℏσ/2), with the spin-rotation term required by the spin connection and present in standard chiral kinetic/Dirac treatments. If −ω·S is included, [H0,J_z]=0 makes every H0 eigenstate stationary under the paper's quench protocol: switching on rotation changes nothing, so the dynamically generated f_CVE, the ground-state-free Floquet structure of Eq. (13), and the three semiclassical validity conditions cannot arise from that protocol. The paper provides no microscopic argument that a mechanically rotated Weyl semimetal realizes a rotation coupling that acts on the orbital part but not on the spin/pseudospin part, nor does it justify treating the σ degree of freedom as rotationally inert. Without this, the central claim that CVE is a non-equilibrium unitary phenomenon over a ground-state-free spectrum is predicated on a model Hamiltonian that may not be the physical rotating-frame Hamiltonian. A separate finite-R/basis-transform issue remains, but the spin-rotation term is more fundamental because it attacks the input Hamiltonian itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a quantum-mechanical formulation of the chiral vortical effect in Weyl semimetals. The central model is the Weyl-node Hamiltonian augmented by an orbital rotation term, H = ⊕_λ λℏv_F k·σ − ω·L (Eq. (1)). Solving this model in a Bessel-function basis yields the spectrum of Eq. (13), which is claimed to be unbounded below and therefore ground-state-free. The paper then evaluates the axial current and orbital magnetization from the exact eigenstates, reproduces the semiclassical CVE coefficient and the 2/3 magnetization contribution under stated conditions, and proposes that the semiclassical results hold only when ωR/v_F≪1, μ/(ℏv_F/R)≫1, and isotropy are simultaneously satisfied. Beyond that regime, the paper predicts void states, a non-Fermi distribution f_CVE, deviations from the μ² law, and a Fermi-velocity-independent charge pumping.","tokens_in":31057,"tokens_out":13739,"duration_ms":153017,"significance":"If the model of Eq. (1) is accepted, the paper provides a fully solvable microscopic quantum framework for the CVE, recovering known semiclassical results from an exact wavefunction calculation rather than from a postulated distribution. The explicit spectrum, the derivation of the 2/3 magnetization contribution, and the three validity conditions are concrete and testable. The prediction of v_F-independent charge pumping and the identification of the CVE as a non-equilibrium, ground-state-free phenomenon are striking claims that would substantially change the conceptual status of the effect. The manuscript is also transparent about its assumptions, and the algebra from Eq. (13) through the current and magnetization response is coherent. However, the physical status of the orbital-only rotation coupling and the finite-size regularization of the spectrum are not yet established, and these issues directly affect the central non-equilibrium claim.","major_comments":[{"comment":"The Hamiltonian couples rotation only through −ω·L and excludes the spin-rotation term. For a Dirac/Weyl fermion in a rotating frame, the standard transformation with the spin connection gives H_rot = H − ω·J = H − ω·(L + ℏσ/2); this is the Hamiltonian used in the standard chiral kinetic and vortical literature. The manuscript neither derives Eq. (1) from a lattice model nor explains why the pseudospin degree of freedom is rotationally inert. The argument in Sec. V that −ω·J_z gives j_z=0 is not a microscopic derivation: it only shows that a sudden quench of −ω·J_z leaves the H_0-diagonal density operator unchanged because [H_0,J_z]=0, but it does not analyze the rotating-frame current operator or the possibility of a rotating equilibrium state. Since the entire non-equilibrium evolution of Secs. 4.2–4.4 and the characterization of f_CVE as non-equilibrium rest on the L-only coupling, this choice needs a derivation from a microscopic model or an explicit experimental protocol that realizes orbital-only rotation.","section":"Sec. II.1, Eq. (1); Sec. V"},{"comment":"The unbounded spectrum used to conclude that the CVE state is ground-state-free is obtained in the infinite-plane limit with continuous k⊥∈[0,∞). The same paper introduces a finite radius R for the degeneracy N_k⊥≈k_F R and for the slow-rotation condition ωR/v_F≪1, but no radial boundary condition is imposed on the Bessel solutions. In a finite disk with a physical boundary, the smallest allowed radial momentum for angular momentum n scales as |n|/R, so the large-|n| energies are approximately ℏ|n|(v_F/R − sign(n)ω), which are bounded below precisely in the regime ωR/v_F≪1. Thus the claim that the spectrum is unbounded below, and the associated conclusion that f_CVE is qualitatively distinct from a Fermi distribution, is an artifact of combining the infinite-plane spectrum with the finite-size degeneracy and conditions. A controlled finite-size regularization is needed before this central claim can be assessed.","section":"Sec. II.1, Eq. (13); Sec. IV.5, Eqs. (62)–(63)"},{"comment":"The transformation between plane-wave and Bessel-function bases is not established. The scaling argument correctly shows that the continuum inner product is divergent and that any formally convergent definition gives c_n,k⊥=0, meaning the plane-wave states are not in the Hilbert space spanned by the Bessel states in the continuum. The finite-N_k⊥ Fourier prescription of Eq. (48) is then introduced with N_k⊥>1, but no proof is given that the limit N_k⊥→∞ (together with R→∞) exists, nor that the observables computed from it, such as the current formula in Eq. (24) and the diagonal matrix element in Eq. (59), are independent of the regularization. Because this transformation is used to connect the exact H-eigenbasis calculation to the plane-wave-labeled semiclassical current, this is a load-bearing gap.","section":"Appendix A, Eqs. (79)–(87); Eq. (48)"}],"minor_comments":[{"comment":"The phrase “density density operator” should be corrected to “density operator”.","section":"Sec. IV.2"},{"comment":"“Water-proof derivation” should read “watertight derivation”.","section":"Sec. IV.3"},{"comment":"The sentence “For quantum, the frame transformation is specified by Eq.” is incomplete; the intended equation reference is missing.","section":"Sec. IV.4"},{"comment":"The term “Floquet features” is potentially misleading because the Hamiltonian in Eq. (1) is time-independent; the n-index ladder resembles a Floquet spectrum, but the terminology should be clarified to avoid implying an explicit time-periodic Hamiltonian.","section":"Fig. 2 and Sec. II.1"},{"comment":"The v_F-independent pumping result should state more explicitly the assumption that μ (or the carrier density) is held fixed while v_F is varied; otherwise the prefactor in Eq. (65) appears to depend on v_F through the density of states.","section":"Sec. IV.6, Eq. (65)"}],"recommendation":"major_revision","confidential_remarks":"The paper is ambitious and the algebra is mostly coherent, but the two load-bearing issues — the physical justification of the orbital-only rotation coupling and the finite-size regularization of the unbounded-spectrum claim — determine whether the central non-equilibrium interpretation survives. If the authors can derive Eq. (1) from a microscopic lattice or synthetic-gauge model and re-derive the spectrum on a finite disk, the paper could become a strong contribution. As it stands, the central claim is defensible only under a specific model assumption whose physical status is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a genuine attempt at a wavefunction-level theory of the CVE. It constructs exact Bessel eigenstates for a rotating Weyl node, derives f_CVE over a ground-state-free Floquet spectrum, and recovers the standard j_CVE ~ μ^2 coefficient and the 2/3 magnetization contribution without fitting. That is real work, and the semiclassical recovery is a benchmark, not a circular step. The three-condition validity analysis (slow rotation, high chemical potential, isotropy) is also sharp and useful.\n\nThe soft spot is the Hamiltonian, and it is load-bearing. Eq. (1) couples rotation only to the orbital angular momentum: H = H0 - ω·L. The paper explicitly rejects -ω·J because it commutes with H0 and would give no bulk current. But the standard rotating-frame Hamiltonian for a spin-1/2 particle is H0 - ω·J, with the spin-rotation term required by the spin connection. Since [H0, J_z] = 0, switching on ω·J under the paper's quench protocol changes nothing: the initial eigenstates remain eigenstates, no dynamics, no CVE. The paper has no microscopic argument for why a mechanically rotated Weyl semimetal should realize orbital-only coupling. Without that, the central claim that CVE is a non-equilibrium unitary phenomenon over a ground-state-free spectrum is an artifact of the model, not a property of the physical rotating frame. This is the first thing a referee must push on. The authors could perhaps justify the orbital-only coupling as a model for a specific spin-orbit-locked configuration, but the paper does not supply that justification.\n\nSecondary issues: the plane-wave-to-Bessel basis transformation in Appendix A diverges in the continuum and is replaced by a finite-N_k⊥ Fourier ansatz without a proof that a well-defined limit exists. The paper admits this, and it is technically honest, but unsupported. The unbounded spectrum also sits uneasily with the finite-R validity conditions, although the paper handles this explicitly via Eqs. (62)-(63); the logic needs tightening but is not incoherent.\n\nBeyond the main derivation, the paper offers several genuinely interesting and falsifiable predictions: void states at k⊥=0, the quantitative failure of the Fermi distribution at low T in this model, and a v_F-independent charge pumping. Those are worth engaging with.\n\nWho is this for: specialists in chiral transport and topological semimetals. It deserves a serious referee, but it is not close to acceptance as-is. Send it out with the Hamiltonian question front and center. If the authors can justify -ω·L physically, this could be an important contribution. If not, the formalism still stands as a well-posed model calculation, but not as a theory of the CVE.","headline":"A real wavefunction-level CVE calculation with a clean semiclassical limit, but the model Hamiltonian only couples rotation to orbital L; the standard rotating frame uses J, which would give no dynamics, so the central non-equilibrium claim is currently unsupported.","tokens_in":31528,"tokens_out":5518,"would_cite":false,"duration_ms":51357,"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":"The chiral vortical effect is a non-equilibrium phenomenon supported on a ground-state-free Floquet spectrum, according to a fully quantum derivation for Weyl semimetals.","keywords":["chiral vortical effect","Weyl semimetal","Floquet spectrum","orbital angular momentum","non-equilibrium transport","Bessel functions","magnetization contribution","charge pumping"],"falsifier":"Numerically diagonalize Eq. (1) on a finite cylinder of radius $R$ with a physical boundary and check whether the spectrum remains unbounded below; if a lowest-energy state appears, the central claim that CVE lives on a ground-state-free spectrum fails for finite samples.","tokens_in":30525,"feed_emoji":"🔄","tokens_out":14446,"duration_ms":119163,"temperature":0.7,"pith_summary":"The paper sets out a fully quantum account of the chiral vortical effect (CVE) in Weyl semimetals: rather than postulating a phase-space distribution $f(p,x)$, it solves the evolution of spinful wavefunctions under the rotating Hamiltonian $H=\\bigoplus_\\lambda \\lambda\\hbar v_F\\,\\mathbf{k}\\cdot\\boldsymbol{\\sigma}-\\boldsymbol{\\omega}\\cdot\\mathbf{L}$. Its central result is the exact spectrum $\\epsilon_{\\lambda,s,n}=s\\hbar\\sqrt{(v_F k_\\perp)^2+(\\lambda v_F k_z+\\omega/2)^2}-(n+\\tfrac{1}{2})\\hbar\\omega$ with $n\\in\\mathbb{Z}$, an infinite ladder unbounded in both directions, so the CVE distribution $f_{\\mathrm{CVE}}$ is supported on a ground-state-free Floquet spectrum and is not a Fermi function. The paper argues that the established semiclassical results, including $j_{\\mathrm{CVE}}\\sim\\mu^2$ and the $2/3$ magnetization contribution, are recovered only when three conditions hold at once: $\\omega R/v_F\\ll 1$, $\\mu/(\\hbar v_F/R)\\gg 1$, and in-plane isotropy. If this is right, the CVE is a non-equilibrium unitary phenomenon, not a thermal response, and each revolution pumps an amount of charge that is independent of $v_F$. A sympathetic reader would care because the result turns a previously semiclassical transport effect into a microscopic, quantum-mechanically testable one.","feed_headline":"Chiral vortical effect lives on a spectrum with no ground state","feed_subtitle":"Quantum Weyl-semimetal calculation recovers CVE current only for slow rotation, large chemical potential, isotropy.","key_machinery":"The load-bearing object is the rotating Weyl Hamiltonian $H=\\bigoplus_\\lambda \\lambda\\hbar v_F\\,\\mathbf{k}\\cdot\\boldsymbol{\\sigma}-\\boldsymbol{\\omega}\\cdot\\mathbf{L}$, with $\\boldsymbol{\\omega}$ along $z$. It is solvable because $H$, $J_z$, $k_z$, and $k_x^2+k_y^2$ share eigenstates built from cylindrical Bessel functions: the two spinor components are $J_n(k_\\perp r)e^{in\\phi}$ and $J_{n+1}(k_\\perp r)e^{i(n+1)\\phi}$ times a plane wave in $z$, and the operators $k_x\\pm ik_y$ ladder between neighboring $n$. The integer $n$ does the conceptual work: it labels a fermion's quantized distance from the rotation axis and generates the infinite Floquet fold structure that removes the ground state. The same Bessel basis produces the void states at $k_\\perp=0$, where both components vanish except for $n=0,-1$, which is what makes the $k_\\perp=0$ sector current-free.","core_discovery":"The paper's claim is that rotation enters the Weyl Hamiltonian through orbital angular momentum only, $\\hat V=-\\boldsymbol{\\omega}\\cdot\\mathbf{L}$, and that the resulting spectrum is an infinite Floquet ladder $\\epsilon_{\\lambda,s,n}=s\\hbar\\sqrt{(v_F k_\\perp)^2+(\\lambda v_F k_z+\\omega/2)^2}-(n+\\tfrac{1}{2})\\hbar\\omega$, $n\\in\\mathbb{Z}$. Because the ladder has no bottom, the distribution $f_{\\mathrm{CVE}}$ formed by unitary evolution from an initial state is not a Fermi-Dirac distribution even at $T=0$; it is a non-equilibrium, dynamically attainable distribution over metastable states. In the isotropic high-$\\mu$, slow-rotation limit the quantum calculation reproduces the semiclassical formulas $j_{\\mathrm{CVE}}=\\omega/(2\\pi v_F)^2\\int 2f\\,\\epsilon\\,d\\epsilon$ and $\\nabla\\times\\mathbf{M}=\\tfrac{2}{3}\\mathbf{j}_{\\mathrm{CVE}}$, with the magnetization split into localized and itinerant parts. Beyond that regime the theory predicts deviations: void states at $k_\\perp=0$ make that sector carry no current, the $\\mu^2$ law fails, and charge pumped per revolution is independent of Fermi velocity.","pith_inferences":["A finite cylinder with a physical boundary could regularize the unbounded spectrum; if it does, the ground-state-free character of the CVE would be an idealization of the infinite-plane model rather than a property of real samples.","The paper's symmetry argument suggests directly testing CVE without mechanical rotation by rotating the spin sector, which would create the same axial current; this is a concrete route the paper sketches but does not develop.","Because anisotropic initial states produce time-dependent distributions, the CVE response coefficient should depend on how the rotating state is prepared; a measurement of $j_z$ under different preparation protocols would distinguish this quantum prediction from the semiclassical one.","If the $v_F$-independent per-revolution pumping is observed across materials with different Fermi velocities, CVE could serve as a calibrated probe of chiral carrier density $\\tilde n$ that does not require knowing $v_F$."],"forward_implications":["Under the three simultaneous conditions ($\\omega R/v_F\\ll 1$, $\\mu/(\\hbar v_F/R)\\gg 1$, isotropy), the known semiclassical CVE coefficients follow from a strictly quantum calculation: the axial current is proportional to $\\mu^2$ and $\\nabla\\times\\mathbf{M}$ accounts for exactly $2/3$ of it.","Outside that regime the current deviates from the semiclassical $\\mu^2$ law, most sharply when $\\mu/(\\hbar v_F/R)\\sim 1$, because the $k_\\perp=0$ void states do not contribute to $j_z$.","The CVE distribution is not a Fermi function even at zero temperature; it is a non-equilibrium state built from dynamically reachable metastable bands, so CVE is better classed with unitary evolution than with thermal relaxation.","Rotating the system by $2\\pi$ pumps charge $\\Delta Q=A(9\\pi \\tilde n^2/16)^{1/3}$ per Weyl node, a quantity independent of $v_F$; flatter bands therefore do not suppress this pumping the way they suppress electric-field transport.","A rotation that couples to total angular momentum ($-\\omega J_z$) produces no bulk current, so the physically active rotation is the relative motion between orbital and spin degrees of freedom."],"supporting_citations":[{"why":"Introduces the chiral vortical effect as a parity-violating axial current in rotating fermions; the phenomenon this paper re-derives from a quantum model.","marker":"[1]"},{"why":"Gives the earlier finite-temperature rotating-system treatment whose equilibrium distribution the quantum formulation tests and departs from.","marker":"[2]"},{"why":"Supplies the semiclassical $j_{\\mathrm{CVE}}\\propto\\mu^2$ formula and the Lorentz-covariance context that the quantum calculation recovers in the classical regime.","marker":"[21]"},{"why":"Source of the Lorentz-invariant chiral kinetic theory, the frame-transformation relation for $f$, and the $2/3$ magnetization contribution that the paper derives bottom-up.","marker":"[24]"},{"why":"Foundational chiral kinetic theory that defines the distribution-based current framework being compared with quantum evolution.","marker":"[25]"},{"why":"Vortical effects in chiral band structures; provides the band-structure setting and Fermi-step estimate for $j_{\\mathrm{CVE}}$ that the paper refines.","marker":"[65]"},{"why":"Gives the localized/itinerant decomposition of orbital magnetization that the paper uses to compute $\\nabla\\times\\mathbf{M}$ and obtain the $2/3$ ratio.","marker":"[75]"},{"why":"Provides the Weyl-node Berry curvature $\\boldsymbol{\\Omega}_{\\lambda,s,k}=-\\lambda s\\,\\hat{\\mathbf{k}}/2k^2$ used in the magnetization and Berry-curvature derivations.","marker":"[77]"}],"fun_headline_variants":["Quantum CVE lives on a Floquet ladder with no ground state","Weyl semimetal rotation: CVE is a non-equilibrium spectrum","No ground state means no Fermi sea for chiral vortical effect","Quantum rotation shows chiral vortical effect is far from equilibrium","CVE deviates from μ² law outside slow-rotation limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire construction rests on the model $H=\\bigoplus_\\lambda\\lambda\\hbar v_F\\,\\mathbf{k}\\cdot\\boldsymbol{\\sigma}-\\boldsymbol{\\omega}\\cdot\\mathbf{L}$, in which rotation acts only on orbital angular momentum with no spin-rotation coupling and no radial boundary; if a finite sample edge regularizes the spectrum or spin also feels rotation, the ground-state-free character and the CVE distribution would change.","fun_headline_variants_meta":{"raw":{"variants":["Quantum CVE lives on a Floquet ladder with no ground state","Weyl semimetal rotation: CVE is a non-equilibrium spectrum","No ground state means no Fermi sea for chiral vortical effect","Quantum rotation shows chiral vortical effect is far from equilibrium","CVE deviates from μ² law outside slow-rotation limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000462,"raw_usage":{"total_tokens":2384,"prompt_tokens":1091,"completion_tokens":1293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":1201}},"tokens_in":707,"tokens_out":1293,"duration_ms":11444,"temperature":1.0,"reasoning_tokens":1201,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:12:43.661948+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically diagonalize Eq. (1) on a finite cylinder of radius $R$ with a physical boundary and check whether the spectrum remains unbounded below; if a lowest-energy state appears, the central claim that CVE lives on a ground-state-free spectrum fails for finite samples.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the semiclassical $j_{\\mathrm{CVE}}\\propto\\mu^2$ formula and the Lorentz-covariance context that the quantum calculation recovers in the classical regime."},{"cited_title":"P˘ atuleanu, A","cited_arxiv_id":null,"evidence_quote":"Source of the Lorentz-invariant chiral kinetic theory, the frame-transformation relation for $f$, and the $2/3$ magnetization contribution that the paper derives bottom-up."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Foundational chiral kinetic theory that defines the distribution-based current framework being compared with quantum evolution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Vortical effects in chiral band structures; provides the band-structure setting and Fermi-step estimate for $j_{\\mathrm{CVE}}$ that the paper refines."},{"cited_title":"Auletta, Foundations and Interpretation of Quantum Mechanics (World Scientific, Singapore, 2000)","cited_arxiv_id":null,"evidence_quote":"Gives the localized/itinerant decomposition of orbital magnetization that the paper uses to compute $\\nabla\\times\\mathbf{M}$ and obtain the $2/3$ ratio."},{"cited_title":"Geometric origin of supercurrents in Berry phase: Formula for computing currents from wavefunctions with correlation and particle number variation","cited_arxiv_id":"2502.16258","evidence_quote":"Provides the Weyl-node Berry curvature $\\boldsymbol{\\Omega}_{\\lambda,s,k}=-\\lambda s\\,\\hat{\\mathbf{k}}/2k^2$ used in the magnetization and Berry-curvature derivations."}],"review_version":1}