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Landau levels in the mixed state of two-dimensional nodal superconductors: models, featured magneto-optical response and quantized thermal Hall effect

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Pith's one-line read A two-dimensional topological-insulator–superconductor heterostructure is claimed to host Landau levels with Chern number ±1 in its vortex lattice, with quantized thermal Hall conductivity and peaked magneto-optical response as…

desk verdict Plausible symmetry-based route to Bogoliubov Landau levels in a 2D TI-SC heterostructure, but the central numbers are in a missing supplement. read the letter →

arxiv 2501.13168 v1 pith:WP5G2EUR submitted 2025-01-22 cond-mat.supr-con

Zhihai Liu , Bin Wang , Luyang Wang This is my paper · ORCID
classification cond-mat.supr-con
keywords LandaulevelsmixedstatevortexlatticeBogoliubovquasiparticlesgeneralizedchiralsymmetrythermalHalleffecttopologicalinsulator-superconductorheterostructuremagneto-opticalconductivity
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 argues that, unlike the usual fate of quasiparticles in the vortex lattice of a gapless superconductor, a two-dimensional topological-insulator–superconductor heterostructure can show genuine Landau quantization of its Bogoliubov quasiparticles when a perpendicular magnetic field creates a vortex lattice. The decisive ingredient is a generalized chiral symmetry of the low-energy Hamiltonian: in the heterostructure the spatially varying supercurrent couples to quasiparticles as an effective vector potential and does not scatter them into Bloch waves, whereas in an intrinsic Weyl superconductor the same supercurrent breaks the symmetry and no Landau levels form. For the 2D system the paper obtains a Dirac-like ladder $E_n = \sqrt{n}E_1$ with a dispersionless zeroth level, each non-degenerate level carrying Chern number $\pm1$. It follows, the paper claims, that the mixed state of this superconductor shows a peaked magneto-optical conductivity and, in the low-temperature limit, a quantized transverse tensor $\tilde{\sigma}_{xy}$ and quantized thermal Hall ratio $\kappa_{xy}/T$. A sympathetic reader would take this as a concrete route to Landau-quantized Bogoliubov quasiparticles and to quantized thermal Hall physics in a proximitized 2D superconductor.

What carries the argument

The machinery that carries the argument is the generalized chiral symmetry for tilted Dirac/Weyl cones, expressed by $\zeta^\dagger H_{\mathrm{BW}}\zeta = -H_{\mathrm{BW}}$ with $\zeta = (\sigma_z - i\eta_x\sigma_y)/\sqrt{1-\eta_x^2}$. Together with this, the key structural move is to absorb the vortex-lattice phase into an effective vector potential $\mathbf{A}_{\mathrm{eff}} = \mathbf{a} + \kappa m\mathbf{u}_s$ after the Anderson gauge transformation, where $\mathbf{a} = \frac{1}{2}\nabla\phi$ is the superconducting gauge field and $\mathbf{u}_s$ the supercurrent velocity. In the heterostructure Weyl superconductor this effective potential couples to Bogoliubov–Weyl fermions and preserves the generalized chiral symmetry, which protects a dispersionless zeroth Landau level and yields the $\sqrt{n}$ ladder; in the intrinsic Weyl superconductor the external vector potential drops out of the pairing-linearized Hamiltonian, the supercurrent term enters as $\eta_x v_F m v_{s,x}\sigma_z$ and breaks the symmetry, so only Bloch-wave-like quasiparticles survive. For the 2D TI–superconductor heterostructure, the zero-field Hamiltonian has a Bogoliubov–Dirac node at $\beta^2 = 1+\Delta_0^2$, and the numerical vortex-lattice solution shows that the same symmetry logic produces Dirac Landau levels with Chern numbers $\pm1$.

What would settle it

Direct numerical diagonalization of the full BdG Hamiltonian (8) on a finite vortex lattice, without the low-energy projection, would settle the main claim: if the spectrum shows avoided crossings or fails to follow $E_n=\sqrt{n}E_1$ as the magnetic length $l_B$ is varied, the central claim collapses. Experimentally, measuring $\kappa_{xy}/T$ at low temperature in a TI–s-wave superconductor heterostructure and finding values that do not match integer sums of the level Chern numbers (or plateau values that drift non-trivially with temperature) would falsify the quantized thermal Hall prediction.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that a 2D superconducting system—a thin topological-insulator layer proximitized by an s-wave superconductor, described by a four-band Bogoliubov–de Gennes Hamiltonian with a Dirac node tuned by magnetization and pairing gap—hosts Landau levels of Bogoliubov quasiparticles in the mixed state. At the fine-tuned condition $\beta^2 = 1+\Delta_0^2$ the vortex-lattice spectrum is a doubly degenerate Dirac ladder $E_n=\sqrt{n}E_1$ with a dispersionless zeroth level; when $\beta$ and $\mu$ are tuned the degeneracy splits into massive-Dirac ladders whose individual levels have Chern number $+1$ or $-1$, the same values as the normal-state Landau levels. Because Bogoliubov quasiparticles are charge-neutral, the electric Hall conductivity is not the quantized observable; instead the paper defines a transverse conductivity tensor $\tilde{\sigma}_{xy}(\xi)$ built from Berry curvature of occupied levels and shows it develops integer plateaus, so that $\kappa_{xy}/T \to (\pi^2/3)(k_B/\hbar)^2\tilde{\sigma}_{xy}(0)$ is quantized at low temperature. The paper also predicts that optical transitions between the Bogoliubov levels produce magneto-optical conductivity peaks at $\omega/E_1 = \sqrt{n}+\sqrt{n+1}$, analogous to graphene but with reduced amplitude, and that detuning away from the nodal condition shifts the peak positions. In short, the paper claims that the mixed state of this heterostructure is a genuine Landau-quantized nodal superconductor with measurable topological transport signatures.

Load-bearing premise

The claim rests on the assumption that the vortex-lattice supercurrent, after the Anderson gauge transformation, acts only as an effective vector potential that preserves the generalized chiral symmetry rather than scattering quasiparticles into Bloch waves, and the paper validates this assumption only numerically, with the details placed in the Supplemental Material.

Editorial extensions

If this is right

  • If the 2D heterostructure is realized as a topological-insulator layer on an s-wave superconductor in a perpendicular field, the vortex-lattice spectrum should show a $\sqrt{n}$ Dirac ladder with a dispersionless zeroth level.
  • The magneto-optical conductivity should exhibit peaks at $\omega/E_1 = \sqrt{n}+\sqrt{n+1}$ in the gapless case, with peak positions shifting when the nodal condition $\beta^2 = 1+\Delta_0^2$ is violated.
  • At low temperature, $\kappa_{xy}/T$ should approach an integer multiple of $(\pi^2/3)(k_B/\hbar)^2$ because each filled Bogoliubov level contributes Chern number $\pm1$ to the transverse tensor.
  • The contrast between intrinsic and heterostructure Weyl superconductors implies that Landau quantization in a vortex state depends on the generalized chiral symmetry of the low-energy Hamiltonian, not merely on the nodal structure.
  • Because the zeroth Bogoliubov level has nearly uniform Berry curvature, it is a candidate flat band for interaction-driven fractional quantum thermal Hall states.

Reading between the lines

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

  • Extension beyond the paper: the symmetry criterion could be used as a screening rule—for any candidate 2D nodal superconductor, checking whether the vortex-lattice effective Hamiltonian preserves the generalized chiral symmetry would predict whether a $\sqrt{n}$ ladder appears before performing the full vortex-lattice diagonalization.
  • Extension beyond the paper: if the quantized $\kappa_{xy}/T$ plateau is observed, the temperature scale at which quantization sets in would provide an independent measure of the first inter-level spacing $E_1$, cross-checking the magneto-optical peak positions.
  • Extension beyond the paper: because the effective vector potential contains the internal superconducting gauge field $\mathbf{a}$, vortex lattices in these heterostructures might act as synthetic magnetic fields for neutral Bogoliubov quasiparticles, tunable independently of the external flux.
  • Extension beyond the paper: the claim that only the zeroth Bogoliubov level has nearly uniform Berry curvature singles out that level as the place to search for fractional quantum thermal Hall states; computing the full momentum-resolved Berry curvature of the first few levels would sharpen this prediction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies Landau quantization of Bogoliubov quasiparticles in the mixed state of nodal superconductors. It first contrasts intrinsic and heterostructure Weyl superconductors, arguing that the presence or absence of a generalized chiral symmetry determines whether supercurrents act as an effective vector potential without scattering the Landau levels. It then proposes a two-dimensional topological-insulator/s-wave-superconductor heterostructure whose vortex lattice is claimed to host Dirac-like Landau levels En = sqrt(n)E1 with Chern number ±1, leading to peaked magneto-optical conductivity and a quantized thermal Hall conductivity κxy/T in the low-temperature limit. The analytic symmetry argument is coherent, but the quantitative claims—the LL spectrum, Chern numbers, conductivity peaks, and thermal Hall plateaus—are supported only by numerical calculations placed in a Supplemental Material that is not included with the arXiv v1.

Significance. If correct, the central proposition is significant: it would provide a concrete 2D superconducting platform in which the vortex lattice does not destroy Landau quantization, yielding a genuine Landau-level spectrum of Bogoliubov quasiparticles with measurable magneto-optical and thermal Hall signatures, and it would identify a useful symmetry criterion for other nodal superconductors. The paper also makes falsifiable predictions, namely the √n + √(n+1) optical peak positions and the quantized low-temperature values of κxy/T. The analytic symmetry-based distinction between the intrinsic and heterostructure Weyl cases is a valuable conceptual contribution. However, the verification of the central claims currently depends entirely on numerical results whose details are not available in the submitted manuscript, so the significance cannot be fully assessed as submitted.

major comments (4)
  1. [LLs in the mixed state of 2D superconductors, Fig. 1(b)] The central numerical result—the Dirac-LL ladder En = sqrt(n)E1 in the vortex lattice of Hamiltonian (8)—is stated to follow from calculations described only by reference [17], 'See the Supplemental Material'. The arXiv v1 contains no Supplemental Material, no code, and no convergence checks. Because this figure is the sole direct evidence that the vortex lattice does not destroy Landau quantization in the proposed 2D system, the manuscript as submitted does not allow a referee or reader to verify the main claim. Please include the Supplemental Material or an equivalent appendix with the full real-space BdG Hamiltonian, the vortex phase profile, the magnetic unit cell construction, boundary conditions, k-point sampling, the values of L and l_B used, and finite-size scaling checks.
  2. [Text after Eq. (3)] The statement that the supercurrent velocity 'does not scatter the LLs' is load-bearing but is not established. From a = (1/2)∇φ and m u_s = (1/2)∇φ − eA, the effective vector potential A_eff = a + κ m u_s has curl ∇×A_eff = (1+κ)π Σ_i δ(r−r_i) − κ eB. Unless κ = −1, which corresponds to Δ0 = 0, the effective magnetic field is spatially nonuniform with flux-line singularities, and a periodic nonuniform field generically converts would-be Landau levels into Bloch bands, as shown in Ref. [5]. The generalized chiral symmetry protects only the zero mode, not the full √n ladder or the Chern numbers of every level. Please provide a quantitative argument, such as an estimate of the vortex-lattice scattering potential relative to the level spacing or a controlled limit, showing that this scattering is suppressed in the parameters used in Fig. 1.
  3. [Footnote [19], second-order-in-v_s terms] The paper explicitly neglects virtual transitions of second order in the supercurrent velocity v_s. Near each vortex core v_s diverges as 1/r, so the neglected terms are not uniformly small over the magnetic unit cell. Please estimate the magnitude of these terms and their effect on the Landau-level spectrum and Chern numbers. If the numerical calculation already includes the full BdG Hamiltonian without this second-order truncation, that should be stated explicitly, and a comparison between the truncated effective theory and the full numerics should be shown.
  4. [Equations (10)–(13)] The magneto-optical and thermal Hall conductivities are the main observable predictions, but the operator HM appearing in the Landau-level basis is not defined in the main text, and the velocity operator of Eq. (11) is written in a form whose gauge invariance with respect to the vortex phase and the Peierls substitution is not demonstrated. Please provide HM in the magnetic unit cell, define the current operator used, and give a derivation or reference showing that the Kubo formula in the form of Eq. (10) yields a gauge-invariant conductivity when evaluated in the magnetic Brillouin zone.
minor comments (5)
  1. [Fig. 1 caption] In Fig. 1(a), the two-band model is plotted as E/E1 while the single-band model is plotted as E/(0.6E1), but both curves appear on the same vertical axis; this mixing of scales should be clarified to avoid a misleading visual comparison.
  2. [Reference [17]] The reference to the Supplemental Material contains only the phrase 'See the Supplemental Material'; a detailed document should be provided with the submission, and the arXiv version should include it as an ancillary file.
  3. [Fig. 1 parameters] The captions quote l_B = 45 and l_B = 46 without stating the unit; since the lattice constant is set to 1, these values appear to be in units of a0, and the corresponding magnetic unit cell sizes and total flux should be stated explicitly.
  4. [Fig. 3] For the green dashed curve, the parameters β = 1.5 and Δ0 = 1 do not satisfy the gapless condition β^2 = 1 + Δ0^2, and the text states that the conductivity peaks shift, but no formula or numerical estimate is given for the shifted peak positions.
  5. [Eq. (12)] The low-temperature limit in Eq. (14) assumes that σ̃xy(ξ) is evaluated at ξ = 0 and that no other energy scale enters; please state explicitly the condition that the chemical potential lies in a gap of the Bogoliubov spectrum, since this is required for the quantization of κxy/T.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the 2D Landau-level claim rests on independent prior results (Refs. 7, 9, 10) plus new numerical diagonalization; the single self-citation (Ref. 8) is not load-bearing.

full rationale

The derivation is self-contained with respect to circularity. The effective-vector-potential reduction for the heterostructure Weyl superconductor (Eq. (3), with A_eff = a + kappa m u_s) is obtained by gauge transformation and block diagonalization 'following Refs. 7 and 18'; the Landau quantization and the generalized chiral symmetry are credited to independent prior work (Refs. 7, 9, 10). The only self-citation, Ref. [8] (Liu and Wang, PRB 110, 174520), is used to describe squeezed LLs for tilted cones in the previously studied heterostructure Weyl superconductor and is not needed for the new 2D claim. The 2D model Eq. (8) is new, and its LL spectra, Chern numbers +/-1, magneto-optical conductivity, and thermal Hall conductivity are computed from that model (Figs. 1, 3, 4) rather than fitted to the claimed outputs; the gapless condition beta^2 = 1 + Delta0^2 is a parameter choice, not a fit. The real weaknesses are verification gaps: the numerical construction is relegated to Supplemental Material [17], footnote [19] limits Eq. (3) to second order in v_s, and the nonuniform effective field a = (1/2) grad phi means the Franz-Tesanovic Bloch-wave alternative is not excluded by construction. These are correctness risks, not circular steps, because the paper does not define the sqrt(n) ladder into the model or rename a fitted quantity as a prediction. Score 1 reflects one minor, non-load-bearing self-citation and otherwise independent content.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The claim depends on the mean-field BdG framework, a low-energy approximation that drops second-order supercurrent terms, and the assumption that the vortex-lattice supercurrent behaves as a clean vector potential. Several model parameters (β, μ, Δ0, lB, L, ε) are chosen by hand to realize the gapless condition and place the Fermi level in gaps. No new physical entities are introduced.

free parameters (6)
  • β (mass term) = β=2 in Fig 1(a); β=√2 in Fig 1(b); β=1.5 in Figs 3-4; fine-tuned condition β²=1+Δ0² for gapless node
    Hand-chosen to realize a gapless Dirac/Bogoliubov node; the LL spectrum and the quantization depend on these values.
  • μ (chemical potential) = μ=0.003, 0.081, 0.15, -0.021, etc.
    Chosen to place the Fermi level in LL gaps; the plateaus in σ̃xy and quantization of κxy/T depend on μ.
  • Δ0 (pairing amplitude) = 0.5, 1.0
    Model parameter setting the pairing gap and the gapless condition β²=1+Δ0².
  • lB (magnetic length) = 45, 46, 22
    Defines the vortex lattice spacing and the LL energy scale E1.
  • L (system size) = 22
    Finite size of the magnetic unit cell in the numerics; no convergence study is shown.
  • ε (TI model parameter) = 3
    Chosen to place the TI layer in the strong topological phase.
assumptions (5)
  • domain assumption Mean-field BdG theory with a globally coherent superconducting phase φ(r) and the Anderson gauge transformation is valid in the vortex lattice.
    Used to derive Eqs. (1)-(3) and (5); standard but unproven in the paper.
  • ad hoc to paper Virtual transitions to higher bands can be neglected because they are of second order in the supercurrent velocity vs.
    Stated in footnote [19]; underlies the low-energy two-band Hamiltonian Eq. (3) and the LL argument.
  • domain assumption The spatially varying supercurrent can be represented as an effective vector potential eA = a + κmus that produces Landau quantization.
    Central to the claim that LLs form; adopted from Ref. [7] and assumed for the 2D model.
  • domain assumption The square vortex lattice with two vortices per magnetic unit cell is an appropriate model for the mixed state.
    The numerical spectrum in Fig. 1(b) depends on this configuration; details are in the missing SM [17].
  • domain assumption The clean limit (no disorder) applies to the Kubo formula and the thermal Hall calculation.
    Magneto-optical and thermal Hall predictions assume no impurity scattering.

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Pith. "Pith review of Landau levels in the mixed state of two-dimensional nodal superconductors: models, featured magneto-optical response and quantized thermal Hall effect." pith.science (2026). https://pith.science/paper/WP5G2EUR

@misc{pith2026250113168,
  author       = {Pith},
  title        = {Pith review of: Landau levels in the mixed state of two-dimensional nodal superconductors: models, featured magneto-optical response and quantized thermal Hall effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WP5G2EUR}},
  note         = {Machine review of arXiv:2501.13168}
}
abstract

Landau quantization of low-energy quasiparticles (QPs) in the mixed state of gapless superconductors is a celebrated problem. So far, the only superconducting system that has been shown to host Landau levels (LLs) of Bogoliubov QPs is the Weyl superconductor. Here, we first investigate the QPs in the mixed state of two Weyl superconductors, an intrinsic one and a heterostructure one, and reveal that the QP states in the former do not form LLs, in contrast to those in the latter where LLs of QPs are shown to exist. The key is whether the low-energy Hamiltonian respects a generalized chiral symmetry. Following the analysis, we show that a twodimensional superconducting system -- a topological insulator-superconductor heterostructure respecting the generalized chiral symmetry -- exhibits LLs with Chern number $\pm 1$ in the mixed state. We also show featured responses resulting from the LLs, including peaked magneto-optical conductivity and quantized thermal Hall conductivity, which could be used as experimental probes to detect LLs in superconductors.

Figures

Figures reproduced from arXiv: 2501.13168 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The QP spectra in the vortex lattice of two-band (red [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) A TI-superconductor heterostructure (a thin layer of a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The real part of the magneto-optical conductivity (in units [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The transverse conductivity tensor ˜σ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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