Quantum Formulation of Chiral Vortical Effect in Weyl Semi-metals

B. Q. Song

REVIEW 3 major objections 5 minor 96 references

Quantum Formulation of Chiral Vortical Effect in Weyl Semi-metals

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.08269 v1 pith:VGFPEPCG submitted 2026-08-08 cond-mat.str-el cond-mat.mes-hallhep-th

B. Q. Song This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallhep-th

keywords chiralvorticaleffectWeylsemimetalFloquetspectrumorbitalangularmomentumnon-equilibriumtransportBesselfunctionsmagnetizationcontributionchargepumping

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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 $v_F$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

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.

The 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.

Secondary 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.

Beyond 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Sec. II.1, Eq. (1); Sec. V] 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.
[Sec. II.1, Eq. (13); Sec. IV.5, Eqs. (62)–(63)] 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.
[Appendix A, Eqs. (79)–(87); Eq. (48)] 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.

minor comments (5)

[Sec. IV.2] The phrase “density density operator” should be corrected to “density operator”.
[Sec. IV.3] “Water-proof derivation” should read “watertight derivation”.
[Sec. IV.4] The sentence “For quantum, the frame transformation is specified by Eq.” is incomplete; the intended equation reference is missing.
[Fig. 2 and Sec. II.1] 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.
[Sec. IV.6, Eq. (65)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the quantum derivation is self-contained and benchmarks are external.

full rationale

The paper's central derivation is self-contained. The Hamiltonian in Eq. (1) is presented explicitly as a model choice ('We find the below (isotropic) H respects these general principles'), not as a consequence of the CVE formula or of the semiclassical current. The spectrum in Eq. (13) is obtained by solving this Hamiltonian in the Bessel-function basis, and the current in Eq. (24) is the standard quantum current operator evaluated on an initial density operator; the coefficients |a|^2 are arbitrary initial occupancies, not fitted parameters. The recovery of the semiclassical CVE formula in Eq. (31) and the magnetization ratio 2/3 in Eq. (43) are comparisons against external benchmarks (Chen-Son-Stephanov-Yee-Yin; Nanda-Hosur), and neither derivation imports those benchmark formulas as inputs. The Fermi distribution is tested, not assumed: Section 4.4 computes f(epsilon) from unitary evolution and compares it with f_F, and the three validity conditions in Sections 4.5 and 5 follow from scale analysis (omega R/v_F, mu/(hbar v_F/R), isotropy), not from fitting to the semiclassical result. Self-citations appearing in the paper (e.g., [9], [54], [55]) are background references for phase-space distributions, Floquet properties, and boundary-condition discussion; none is load-bearing for the main derivation. The choice of -omega.L instead of -omega.J is a physical modeling decision explicitly discussed in Section V; whether it is correct is a scientific question, not a circularity. No equation reduces to its own input by construction, and no parameter is fitted and then renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivation is analytic and no constants are fitted to data. The main pulled-in inputs are the band model, the rotation-coupling choice, the finite-N regularization, and the unitary-evolution protocol. R enters as a physical scale, but its microscopic origin in the infinite-plane spectrum is not specified.

free parameters (1)

system radius R
Entered through N_k⊥≈k_F R and the two-scale conditions ωR/v_F and μ/(ℏv_F/R), although the spectrum Eq. (13) is solved in an infinite plane with no boundary condition specifying R.

assumptions (5)

domain assumption Isotropic two-node Weyl semimetal dispersion H_λ0=λℏv_F σ·k.
The whole calculation starts from Eq. (1) with two isotropic Weyl nodes and a single Fermi velocity v_F.
ad hoc to paper Rotation acts only through the orbital angular momentum term −ω·L, with no spin-rotation coupling.
This choice is asserted from the principles in Sec. 2.1; a microscopic rotating-frame derivation is not given.
ad hoc to paper The plane-wave to Bessel basis transformation can be regularized with finite N_k⊥ and the continuum limit reproduces observables.
Appendix A shows the direct continuum inner product diverges; the finite-N prescription and its limit are assumed rather than proven.
domain assumption The initial state is a Fermi-filled set of H_0 eigenstates that evolves unitarily without relaxation.
Sec. 4.2 defines f_CVE from this unitary evolution, excluding interactions and thermal relaxation.
domain assumption Inter-band transitions are negligible for small ω (intra-band approximation).
Sec. 4.5 and Appendix A rely on negligible transitions between the two Weyl bands to justify the basis transformation.

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Cite this review

Pith. "Pith review of Quantum Formulation of Chiral Vortical Effect in Weyl Semi-metals." pith.science (2026). https://pith.science/paper/VGFPEPCG

@misc{pith2026260808269,
  author       = {Pith},
  title        = {Pith review of: Quantum Formulation of Chiral Vortical Effect in Weyl Semi-metals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VGFPEPCG}},
  note         = {Machine review of arXiv:2608.08269}
}

abstract

The chiral vortical effect (CVE) is the generation of an axial current in a rotating Weyl fermion; its description is presently based on semiclassical frameworks. In this work, we develop a fully quantum formulation for CVE, solving the exact evolution of microscopic spinful wavefunctions, which enables a bottom-up quantitative test of semi-classical theories and postulated distributions $f_{\text{CVE}}$ in different reference frames. Notably, it shows that $f_{\text{CVE}}$ is over a ground-state-free Floquet spectrum, qualitatively distinct from a thermal equilibrium distribution (i.e., fermi form $f_F$), underscoring CVE as a non-equilibrium phenomenon, distinguished from other chiral transports. The $f_F$ only approximately holds when three conditions are simultaneously fulfilled: (1) slow rotation $\omega R/v_F\ll 1$, (2) high chemical potential $\mu/(\hbar v_F R)\gg 1$, (3) isotropic symmetry, where $R$ is the size, $v_F$ is fermi velocity. In these conditions, the theory recovers established semiclassical results, including the current-response coefficients and the magnetization contribution; otherwise, it uncovers quantum phenomena such as ``void states", deviation from the semiclassical formula $j_{\text{CVE}} \sim \mu^2$, a $v_F$-independent charge pumping. The theory is based on semimetals, providing more experimentally accessible detection than fundamental Weyl particles.

Figures

Figures reproduced from arXiv: 2608.08269 by the authors.

**Figure 2.** FIG. 2. (color online): Spectra for (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. (color online): Causality under classical and quantum [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. (color online): For [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. (color online): (a) An initial distribution and (b) its [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: FIG. 6. (color online): The time-dependence of occupancy of [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. (color online) [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. (color online): Distributions over [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. (color online): (a) Quantum or classical regimes tuned [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10. (color online): (a) The shift [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

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Reference graph

Works this paper leans on

96 extracted references · 75 canonical work pages

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Thej CVE was expressed in terms of a spinless (or spin-averaged) distributionf(p,x) in the phase space{p,x}

Chiral current under rotation A key result from kinetic theory is that, for isotropic Weyl fermion, the CVE current isj CVE = ω (2π)2 R ∞ 0 2f·ϵdϵ, givenv F =ℏ= 1 [1, 21, 23–25] – the basic connotation of CVE. Thej CVE was expressed in terms of a spinless (or spin-averaged) distributionf(p,x) in the phase space{p,x}. With the microscopic quantum theory, w...
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(13)) varies with the rotating speedω 0 of the reference frame

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correction

Observables in Quantum formulation The quantum theory is more than adding “correction” terms. It changes the paradigm (Fig. 3) by reshaping the causality. It will also supply the valid conditions of semi-classical arguments, such as 2/3 of CVEj z being contributed by∇×M[21, 24, 65], which concerns two observables: the currentjand the magnetizationM. Parad...
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bottom-up

Magnetization contribution. Another semi-classical result to deduce is that a mag- netic contribution∇ x×Mamounts to 2/3 of the CVE current [24]. To be concrete, Eq. (20) suggests the spatial curl of M(x) of Weyl fermion will contribute in addition to the Liouville current; they together form the axial current jCVE, and∇ x×Mis termedmagnetization contribu...
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void states

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ripple-shaped

Quantum evolution. After clarifying the Hilbert space, we solve the quan- tum evolution and examine such questions: From an initial eigenstate ofH 0, what a state does it evolve into? What a microscopic wavefunction is responsible for CVE? An initial state is characterized by a density density operator that is specified by coefficientsaassociated with H0’...

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Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Thej CVE was expressed in terms of a spinless (or spin-averaged) distributionf(p,x) in the phase space{p,x}

Chiral current under rotation A key result from kinetic theory is that, for isotropic Weyl fermion, the CVE current isj CVE = ω (2π)2 R ∞ 0 2f·ϵdϵ, givenv F =ℏ= 1 [1, 21, 23–25] – the basic connotation of CVE. Thej CVE was expressed in terms of a spinless (or spin-averaged) distributionf(p,x) in the phase space{p,x}. With the microscopic quantum theory, w...

[2] [2]

(13)) varies with the rotating speedω 0 of the reference frame

T ransforming reference frames under Quantum protocol It is a misconception to think thatωinϵ(Eq. (13)) varies with the rotating speedω 0 of the reference frame. It is also mistaken that the measured energy or distribu- tion necessarily vary with reference framesω 0, – a classi- cal impression that is no longer true for quantum. Here, we discuss the quant...

[3] [3]

Since [H, J z] = 0, theHis in- variant, except for a phase flexibilityδin off-diagonals of H(Eq

by H→e −iJzδ/ℏHe iJzδ/ℏ, (15) whereδ= (ω ′ 0 −ω 0)t. Since [H, J z] = 0, theHis in- variant, except for a phase flexibilityδin off-diagonals of H(Eq. (12)) due toK +|n⟩=c +|n+ 1⟩→K +einδ|n⟩= c+ei(n+1)δ|n+ 1⟩. That is, rotation shiftsc +→eiδc+ (c−→e−iδc−), because|n⟩and|n+ 1⟩will mismatch the phase under rotation. Indeed, the eigenvalue (Eq. (13)) is indep...

[4] [4]

correction

Observables in Quantum formulation The quantum theory is more than adding “correction” terms. It changes the paradigm (Fig. 3) by reshaping the causality. It will also supply the valid conditions of semi-classical arguments, such as 2/3 of CVEj z being contributed by∇×M[21, 24, 65], which concerns two observables: the currentjand the magnetizationM. Parad...

[5] [5]

Physical phenomena typically involve scales in energy, length, etc

Quantum regimes for CVE: Two scales uncovered. Physical phenomena typically involve scales in energy, length, etc. that dictate classical-quantum crossover. Quantum formulation (defined in Sec. 2) underscores two scales, which might have been overlooked by semi- classical treatments. The first concerns the ratio of two velocities: the rotat- ing speed at ...

[6] [6]

bottom-up

Magnetization contribution. Another semi-classical result to deduce is that a mag- netic contribution∇ x×Mamounts to 2/3 of the CVE current [24]. To be concrete, Eq. (20) suggests the spatial curl of M(x) of Weyl fermion will contribute in addition to the Liouville current; they together form the axial current jCVE, and∇ x×Mis termedmagnetization contribu...

[7] [7]

void states

Uncover “void states” due to quantization. Quantization might qualitatively alter a classical re- sult, such as breaking classical symmetry, namelyquan- tum anomaly[6, 26, 28]. Here, quantization distinguishes k⊥ = 0 fromk ⊥ >0, unlike the uniform classicalk-space. The structure beyond semi-classical pictures will affect dynamics and transports. The casek...

[8] [8]

ripple-shaped

Quantum evolution. After clarifying the Hilbert space, we solve the quan- tum evolution and examine such questions: From an initial eigenstate ofH 0, what a state does it evolve into? What a microscopic wavefunction is responsible for CVE? An initial state is characterized by a density density operator that is specified by coefficientsaassociated with H0’...