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arxiv: 2501.16792 · v2 · submitted 2025-01-28 · ❄️ cond-mat.str-el

Anomalous Landau levels and quantum oscillation in rotation-invariant insulators

Jianlong Fu , Chun Yu Weng , Hoi Chun Po This is my paper

Pith reviewed 2026-05-23 05:03 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords anomalous Landau levelsquantum oscillationsrotation-invariant insulatorseffective-band descriptionLandau level spectruminsulators
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The pith

An effective-band description predicts anomalous Landau levels and their induced quantum oscillations in rotation-invariant insulators.

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

The paper examines whether anomalous Landau levels, which protrude into the zero-field energy gap, can produce Fermi-surface-like quantum oscillations in insulators that lack a zero-field Fermi surface. Focusing on two-dimensional multi-band low-energy models with continuous rotation symmetry, the authors show that an effective-band description similar to semiclassical methods in metals can predict the full Landau level spectrum including anomalous levels. Numerical calculations confirm that this framework accounts for quantum oscillations driven by anomalous Landau levels in certain insulating models.

Core claim

Landau levels in certain models protrude into the zero-field energy gap as anomalous Landau levels. For two-dimensional multi-band low-energy models of electrons with continuous rotation symmetry, an effective-band description akin to the semiclassical treatment of Landau level problems in metals can be used to predict the Landau level spectrum, including possible anomalous Landau levels. This description then accounts for anomalous Landau level induced quantum oscillation for certain insulating models, demonstrated through numerical calculations.

What carries the argument

effective-band description akin to the semiclassical treatment of Landau level problems in metals, used to predict the Landau level spectrum including anomalous Landau levels

If this is right

  • The effective-band description accurately predicts the presence and positions of anomalous Landau levels in the spectrum.
  • Anomalous Landau levels can induce quantum oscillations resembling Fermi-surface behavior in insulating models without a zero-field Fermi surface.
  • Numerical calculations validate the description for certain rotation-invariant insulating models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested by comparing predicted oscillation frequencies against magnetotransport data in candidate insulating materials.
  • If the description holds, it might simplify analysis of magnetic-field responses in other gapped systems that possess continuous rotation symmetry.

Load-bearing premise

The effective-band description remains valid and predictive for the multi-band low-energy models of electrons with continuous rotation symmetry when applied to insulating cases without a zero-field Fermi surface.

What would settle it

A numerical computation of the Landau level spectrum in one of the insulating models that shows significant mismatch with the effective-band prediction would falsify the claim.

Figures

Figures reproduced from arXiv: 2501.16792 by Chun Yu Weng, Hoi Chun Po, Jianlong Fu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

Landau levels in certain models are known to protrude into the zero-field energy gap. These are known as anomalous Landau levels (ALLs). We study whether ALLs can lead to Fermi-surface like quantum oscillation in the absence of a zero-field Fermi surface. Focusing on two-dimensional multi-band low-energy models of electrons with continuous rotation symmetry, we show that an effective-band description, akin to the semiclassical treatment of Landau level problems in metals, can be used to predict the Landau level spectrum, including possible ALLs. This description then describes ALL induced quantum oscillation for certain insulating models, which we demonstrate through numerical calculations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that in two-dimensional multi-band low-energy models of electrons with continuous rotation symmetry, an effective-band description (modeled on the semiclassical treatment of Landau levels in metals) can predict the full Landau level spectrum, including anomalous Landau levels (ALLs) that enter the zero-field gap. This description is then used to account for ALL-induced quantum oscillations in certain gapped insulating models that lack a zero-field Fermi surface; the claim is supported by numerical calculations on specific models.

Significance. If the effective-band mapping remains quantitatively predictive once the zero-field Fermi surface disappears, the work would provide a practical semiclassical tool for computing Landau-level spectra and oscillations in rotation-symmetric insulators, extending beyond the conventional Onsager quantization applicable only to metals. The numerical demonstrations for selected models constitute a concrete test of the idea.

major comments (2)
  1. [Section introducing the effective-band description (likely §3 or equivalent)] The central extension of the semiclassical effective-band ansatz from metals (with a well-defined cyclotron orbit and Fermi surface) to gapped insulators is load-bearing. The manuscript must explicitly derive or justify why interband mixing and the absence of a zero-field Fermi surface do not invalidate the mapping; without this, the claim that the description 'predicts' ALLs and oscillations rests on an uncontrolled extrapolation.
  2. [Numerical calculations and figures demonstrating ALL-induced oscillations] Numerical results for the insulating models must include a direct, quantitative comparison (e.g., level positions, oscillation frequencies) between the effective-band prediction and the exact diagonalization or tight-binding spectra, with explicit checks for parameter robustness. The abstract states the description 'describes' the oscillations, but the load-bearing step is whether agreement survives when the zero-field gap is opened.
minor comments (2)
  1. [Model and effective-band sections] Clarify the precise definition of the effective band (e.g., whether it is obtained by projecting onto a single band or by some other reduction) and state any assumptions about the rotation symmetry explicitly.
  2. [Discussion or conclusions] Add a brief discussion of the range of validity, for example by contrasting the insulating cases with a metallic reference model where the standard Onsager quantization is recovered.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the manuscript accordingly to strengthen the presentation and validation of the effective-band description.

read point-by-point responses

  1. Referee: [Section introducing the effective-band description (likely §3 or equivalent)] The central extension of the semiclassical effective-band ansatz from metals (with a well-defined cyclotron orbit and Fermi surface) to gapped insulators is load-bearing. The manuscript must explicitly derive or justify why interband mixing and the absence of a zero-field Fermi surface do not invalidate the mapping; without this, the claim that the description 'predicts' ALLs and oscillations rests on an uncontrolled extrapolation.

    Authors: We agree that an explicit justification for the extension is necessary to avoid any appearance of uncontrolled extrapolation. The manuscript introduces the effective-band ansatz via rotation symmetry but does not derive its validity in the gapped case in full detail. In the revised version we will add a dedicated derivation subsection showing that, for Hamiltonians with continuous rotation symmetry, the semiclassical quantization condition follows from the Berry phase and orbital magnetic moment contributions even when the zero-field Fermi surface is absent; interband mixing is controlled by the magnetic length scale and does not invalidate the leading-order mapping to anomalous Landau levels. revision: yes

  2. Referee: [Numerical calculations and figures demonstrating ALL-induced oscillations] Numerical results for the insulating models must include a direct, quantitative comparison (e.g., level positions, oscillation frequencies) between the effective-band prediction and the exact diagonalization or tight-binding spectra, with explicit checks for parameter robustness. The abstract states the description 'describes' the oscillations, but the load-bearing step is whether agreement survives when the zero-field gap is opened.

    Authors: We concur that direct quantitative comparisons and robustness checks are required to substantiate the claim once the gap is opened. The present numerical results demonstrate qualitative consistency but do not provide side-by-side quantitative metrics. In the revision we will augment the relevant figures and add a table comparing effective-band predicted Landau-level energies and oscillation frequencies against exact diagonalization results for multiple gap sizes and parameter sets, explicitly confirming that quantitative agreement persists in the insulating regime. revision: yes

Circularity Check

0 steps flagged

No circularity: effective-band description presented as independent predictor, verified numerically

full rationale

The paper introduces an effective-band description modeled on semiclassical Landau-level methods for metals and applies it to predict the spectrum (including ALLs) in gapped, rotation-symmetric insulators. This prediction is then checked against direct numerical diagonalization for specific models. No quoted step reduces the claimed prediction to a fitted parameter, self-citation, or definitional identity; the numerical verification supplies an external benchmark. The extension from metallic to insulating regimes is an assumption whose validity is tested rather than presupposed by construction. Hence the derivation chain remains self-contained against external checks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields insufficient detail to enumerate free parameters, axioms, or invented entities; no explicit fitting or new postulates are mentioned.

pith-pipeline@v0.9.0 · 5628 in / 1046 out tokens · 49589 ms · 2026-05-23T05:03:27.450450+00:00 · methodology

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Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    For parabolic bands the parametersβ >0 has the dimension of mass-inverse

    +β(k 2 −k 2 0) which crosses energy zero at a ring of momentumk= k0. For parabolic bands the parametersβ >0 has the dimension of mass-inverse. The approach can be readily generalized to higher number of bands, in general forn- band model in canonical basis, the off-diagonal elements of the Hamiltonian readsH ij =γ ij(k−)Qii−Qjj . III. LANDAU LEVELS IN THE...

    work page

  2. [2]

    + 2BQii. (6) The translation (6) suggests that the operator form of ˜k2 acting on the ansatz states|n+q i⟩in number-basis calculation produces an eigenvalue identical to polyno- mialk 2 ξ + 2BQii evaluated atξ=n+|ζ|withk ξ defined in (5). The diagonal elements of the effective Hamilto- nian can be readily obtained. Generally, thei-th diag- onal entry of t...

    work page

  3. [3]

    hybridization

    in the matrix element (7). This pro- cedure leads to the following results for off-diagonal ele- ments. Given thatQ ii ≥ Q jj ≥ζ, we define two non- negative numbersr ij =Q ii−Qjj ands ij =Q jj −ζ(=q j). For the caser ij ≥1,Q ii ̸=Q jj,g ij(k) must contain nonzero order ofk ±, then the off-diagonal elements of the effective band are given by (HE)ij =γ ij ...

    work page

  4. [4]

    Without off-diagonal terms, the two di- agonal terms cross at|k|=k 0 with energy zero

    andH (±1) 22 =−α ′(k4 −k 4 0)− β′(k2 −k 2 0)). Without off-diagonal terms, the two di- agonal terms cross at|k|=k 0 with energy zero. In an external magnetic fieldB, the diagonal piece of the effective Hamiltonian (10) changes to (H (±1 E ))11 = α[(k2 ±B) 2 −k 4 0] +β(k 2 ±B−k 2

    work page

  5. [5]

    The quantization rule isξ= 1 2 ,1 + 1 2 ,2 + 1 2 ,· · ·, so the selected momenta for LLs arek ξ = √ 2Bnforn= 1,2,3,· · ·

    and (H (±1) E )22 = −α′[(k2 ∓B) 2 −k 4 0]−β ′(k2 ∓B−k 2 0). The quantization rule isξ= 1 2 ,1 + 1 2 ,2 + 1 2 ,· · ·, so the selected momenta for LLs arek ξ = √ 2Bnforn= 1,2,3,· · ·. Beside these, H(+1) has one extra ansatz state that is not captured in the sequence, (|0⟩,0) T , with energyE ex =α(B 2 −k 4

    work page

  6. [6]

    Similarly,H (−1) has one extra state (0,|0⟩) T , with energyE ex =−α ′(B2 −k 4 0)−β ′(B−k 2 0)

    + β(B−k 2 0). Similarly,H (−1) has one extra state (0,|0⟩) T , with energyE ex =−α ′(B2 −k 4 0)−β ′(B−k 2 0). In Fig. 2, we plot the effective bands as a function ofB, which illustrates their evolution. As shown in the Fig. 2 (a), the movement of the effective bandH (−1) E corresponds to a simple orbital Zeeman shift, the LLs carried into the original ban...

    work page

  7. [7]

    +β 1(2B)( 1 3)γ 1 q k2 − B 3 γ2k0 γ1 q k2 − B 3 β(k2 −k 2

    work page

  8. [8]

    spread of LLs

    +β(2B)(− 2 3)γ 3 q k2 − B 3 γ2k0 γ3 q k2 − B 3 −β2(k2 −k 2 0)−β 2(2B)( 1 3)   (12) There is one extra ansatz state (|0⟩,0,|0⟩) T , correspond- ingly there are two extra LLs 49. As shown in Fig. 2 (b), the effects ofBfield goes beyond a simple orbital Zeeman term with modification of off-diagonal terms at the same time. Consequently the effective band...

    work page

  9. [9]

    fermi surface

    +β(k 2 −k 2 0),−α ′(k4 −k 4 0)−β ′(k2 −k 2 0)]), and com- pare the QO for the case with finiteα(quartic band) and vanishingα(parabolic band). A finiteαlowers the critical fieldB c at which QO begins. This leads to a larger field range for QO before the quantum limit, ren- dering the QO more pronounced. The result is shown in Fig. 4 (d), from which one can...

    work page

  10. [10]

    aperiodic quantum oscillation

    is the area of the effective-band fermi surface49. This equation shows that the frequency of the ALL QO is not determined by the fermi surface area of the effective-band; rather, there is a modification. Whenεhas no explicit dependence onB, 9 (a) (b) (c) (d) FIG. 4:QO from effective band picture. (a) The LLs that are going to cross the chemical potential ...

    work page doi:10.1080/14786440908521019 1984

  11. [11]

    The general three-band Hamiltonian reads Hk = ˆD(k) + u(k) ˆU+ +v(k) ˆV+ +i(k) ˆI+ + h.c

    ˆY= diag( 2p 3 + q 3 ,− p 3 + q 3 ,− p 3 − 2q 3 ) with two integers (p, q) characterizing the relative angular momenta. The general three-band Hamiltonian reads Hk = ˆD(k) + u(k) ˆU+ +v(k) ˆV+ +i(k) ˆI+ + h.c. ,(S3) in which ˆD(k) =h 0 ˆI+h 1 ˆY+h 3 ˆI3 is the diagonal piece. For infinitesimal rotation, the adjoint action (2) and the Lie algebra ofSU(3) (...

    work page

  12. [12]

    2 −3 −2 −1 0 1 2 3 4 5 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2 E/(βk2 0) 2B/k 2 0 FIG

    and (|1⟩,0) T with energyβ(3B−k 2 0). 2 −3 −2 −1 0 1 2 3 4 5 0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2 E/(βk2 0) 2B/k 2 0 FIG. S1: Effective band evolution under magnetic field for four-bandH (+2) ⊕ H(−2) with β′ β = 0.5 and γ β = 0.1. The left-most band is the original band forB= 0. The chemical potential is set at µ βk 2 0 = 0. From the two-band mod...

    work page

  13. [13]

    + 2Bβ γ √ k4 −B 2 0 0 γ √ k4 −B 2 −β′(k2 −k 2 0)−2Bβ ′ 0 0 0 0β(k 2 −k 2 0)−2Bβ γ √ k4 −B 2 0 0γ √ k4 −B 2 −β′(k2 −k 2

    work page

  14. [14]

    S1, from which we can see its effective band-gap closes as magnetic fieldBgets large

    + 2Bβ′   (S7) Its effective-band is shown in Fig. S1, from which we can see its effective band-gap closes as magnetic fieldBgets large. Three-band modelH (1,−1) k For three-band modelH (1,−1) k (11), the magnetic Hamiltonian is given by H(1,−1) =   β1[2B(a†a+ 1

    work page

  15. [15]

    On the other hand, the effective band is given by Eq

    +k 2 1]γ 1 √ 2Ba† γ2k0 γ1 √ 2Ba β[2B(a †a+ 1 2)−k 2 0]γ 3 √ 2Ba γ2k0 γ3 √ 2Ba† −β2[2B(a†a+ 1 2)−k 2 0]   .(S8) The ansatz solution is given by (|n⟩,|n−1⟩,|n⟩) T , withn= 1,2,· · ·. On the other hand, the effective band is given by Eq. (12), it is governed by the following dimensionless quantities: β1 β , β2 β , k k0 , k1 k0 , 2B k2 0 , γ1 βk0 , γ2 βk0 ,...

    work page

  16. [16]

    γ1 βk0 q ( k k0 )2 − 2B k2 0 1 6 γ2 βk0 γ1 βk0 q ( k k0 )2 − 2B k2 0 1 6 ( k k0 )2 −1 + 2B k2 0 (− 2

    work page

  17. [17]

    γ3 βk0 q ( k k0 )2 − 2B k2 0 1 6 γ2 βk0 γ3 βk0 q ( k k0 )2 − 2B k2 0 1 6 −( β2 β )[( k k0 )2 −1]−( β2 β )( 2B k2 0 ) 1 3   .(S9) It has one extra ansatz solution (|0⟩,0,|0⟩) T , correspondingly there are two LLs whose energies are eigenvalues of the two-by-two matrix Hex = β1(B+k 2 1)γ 2k0 γ2k0 −β2(B−k 2 0) .(S10) 3 0 10 20 30 40 50 60 70 2 4 6 8 10 ...

    work page

  18. [18]

    spread of LLs

    +β 2(B−k 2 0)]2 + 4γ2 2 k2 0 .(S11) In Fig. S2 we plot the QO ofµDOS for the three-band modelH (1,−1) with parameters used in Fig. 2 for the bigger-gap case and a small-gap case with Fermi-surface-like behavior. Three-band modelH (−1,−1) k Following the discussion of the Lieb lattice model with SOC, we consider the effective model given by Eq. (16) in det...

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  19. [19]

    Inside magnetic field, approximately the new crossing point for these two bands is determined by β(k2 −k 2

    and−β ′(k2 −k 2 0), with corresponding eigenvalue of ˆQ beingQ ii andQ jj. Inside magnetic field, approximately the new crossing point for these two bands is determined by β(k2 −k 2

    work page

  20. [20]

    reference frame

    +Q ii(2β)B=−β ′(k2 −k 2 0)− Q jj(2β′)B.(S17) 5 The new crossing energy for the two bands is given by Ec = 2ββ ′(Qii − Qjj) β+β ′ B.(S18) If one of these bands is flat, then the energy of the band crossing remains zero. Therefore flat-band case generally does not havein-gapALLs. But for two-band models with one flat band, ALLs can appear in the half-infini...

    work page

  21. [21]

    We set the chemical potential to be at energy zero

    dB dE .(S20) Therefore we have, in the effective-band reference frame, the change of thenth LL with respect to magnetic field d ˜En dB = 2π(|ζ|+n+ 1 2) D .(S21) As a rough estimate, neglecting the changing shape of the effective bands, we have the change ofnth LL with respective of the magnetic field is given by dEn dB = d ˜En dB + dEc dB = 2π(|ζ|+n+ 1 2)...

    work page

  22. [22]

    + ∂ε ∂B ,(S24) and dε dξ = ∂ε ∂k2 ∂k2 ∂ξ = 2B ∂ε ∂k2 .(S25) The frequency of the oscillation is determined by f= dξ d( 1 B ) =B 2 dε dB dε dξ =B(ξ+ 1

    work page

  23. [23]

    + 1 2 B ∂ε ∂B ∂ε ∂k 2 .(S26) The cross section of the effective band has areaS E =πk 2 ξ, so the frequency is also written as f= SE 2π + 1 2 B ∂ε ∂B ∂ε ∂k 2 .(S27) 6 THE MODEL WITHOUT ROTATIONAL INVARIANCE Here, we discuss a simple two-band model without continuous rotational symmetry which can still be treated using the effective-band picture. The Hamilt...

    work page

  24. [24]

    ˜B−1]c n + ˜γ p ˜B√ndn−1 + ˜W dn, dn → − ˜β′[(n+ 1

    work page

  25. [25]

    ˜B−1]d n + ˜W cn + ˜γ p ˜B √ n+ 1c n+1 (S48) With the states arranged as (c, d, c 0, d0, c1, d1,· · ·, c n, dn,· · ·) T , the Hamiltonian matrix is given by H(+1) =   ˜B 2 −1 ˜W· · · ˜W− ˜β′( ˜B 2 −1) ... ... ˜γ p ˜Bn(n+ 1

    work page

  26. [26]

    ˜B−1 ˜W ˜W− ˜β′[(n+ 1

    work page

  27. [27]

    ˜B−1] ˜γ q ˜B(n+ 1) ... ...   .(S49) For numerical studies, we truncate the states to include onlyc, c 0, c1,· · ·, c N−2 andd, d 0, d1,· · ·, d N−2. Hence the Hamiltonian matrix (S49) becomes a 2N×2Nfinite matrix and can be diagonalized. A test forN= 1000 for the caseβ ′ = 0 shows that the numerical LLs agree well with the exact solutions,...

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