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REVIEW 3 major objections 4 minor 121 references

Repulsive-Interaction-Driven Topological Superconductivity in a Landau Level Coupled to an $s$-Wave Superconductor

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read At half filling, a repulsive interaction turns a proximitized Landau level into a topological superconductor, with fourfold-degenerate odd-parity ground states and p±ip pairing.

desk verdict Careful exact diagonalization finds a credible interaction-stabilized odd-parity phase at half-filled Landau level, but the topological label rests on an unproven extension of the Read-Green parity argument; it still deserves a serious referee. read the letter →

arxiv 2510.04700 v2 pith:3DB3HJPP submitted 2025-10-06 cond-mat.str-el cond-mat.mes-hallcond-mat.supr-con

Koji Kudo , Ryota Nakai , Hiroki Isobe , J. K. Jain , Kentaro Nomura This is my paper
classification cond-mat.str-elcond-mat.mes-hallcond-mat.supr-con
keywords topologicalsuperconductivityrepulsiveinteractionLandaulevelRashbaspin-orbitcouplings-waveproximityeffectcompositeFermiliquidexactdiagonalizationp±ippairing
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

At half filling of the lowest Rashba-split Landau level, proximity to an ordinary s-wave superconductor does nothing topological for noninteracting electrons. This paper shows that adding a screened repulsive interaction between electrons turns the system into a gapped topological superconductor (RID-TSC) for a finite window of interaction-to-pairing ratios. The topological phase has fourfold-degenerate ground states with odd fermion parity, total momentum away from inversion-invariant points, and p±ip pairing correlations. Because the phase appears only at finite interaction strength and vanishes in the noninteracting limit, repulsion is not an obstacle to pairing here but the mechanism that creates topological superconductivity. The proposal is that a composite Fermi liquid proximity-coupled to an s-wave superconductor is the right physical picture.

What carries the argument

The engine is the projected interacting Hamiltonian H̃_hybrid = H̃_int + H̃_Δ − μN, diagonalized exactly on a torus in the full Fock space. The diagnostic is the identity P = (−1)^N, which connects fermion parity to the parity of the Bogoliubov–de Gennes Chern number; with P = −1 it asserts a nonzero Chern number and hence a topological superconductor. The momentum selection rule — ground states at non-inversion-invariant momenta — is what prevents any adiabatic connection to a mean-field BdG ground state, marking the phase as intrinsically interaction-driven.

What would settle it

Thread a flux quantum through the torus for the RID-TSC ground state and compute the many-body Chern number by exact diagonalization; a zero invariant while the parity remains odd would falsify the topological claim despite the observed ground-state structure. A transport falsifier is thermal Hall conductance: the paper's half-odd-integer chiral central charge prediction would show up as a half-odd-integer quantized value, absent for a trivial odd-parity superconductor.

Watch

Extended reading notes

Core claim

On a torus with a square Abrikosov vortex lattice and Rashba spin-orbit coupling, exact diagonalization of the projected half-filled Landau level gives a ground state for Δ0/VC ≈ 0.1 that is fourfold degenerate, has fermion parity P = −1, momentum (π/2, π/2) and (π/2, 3π/2), and is separated by a gap whose finite-size extrapolation stays finite. The same features appear for several system sizes of the form n_x × n_y = 2s × 4t. The state emerges from the critical point at VC/Δ0 = 0 and expands as the interaction grows, before giving way to the composite Fermi liquid at even larger VC. Odd parity combined with the parity–Chern relation P = (−1)^N is used to identify the phase as topological, a

Load-bearing premise

The claim that the interacting P = −1 state is topological rests on assuming that the parity–Chern-number equality P = (−1)^N, which is proven for noninteracting Bogoliubov–de Gennes states, also applies to the interacting ground state; the paper states that a rigorous proof is lacking.

Editorial extensions

If this is right

  • A finite interaction strength converts a non-topological proximitized Landau level into a gapped topological superconductor; the extrapolated gap stays finite in the thermodynamic limit, unlike the rapidly collapsing composite-Fermi-liquid gap.
  • At fixed pairing, the RID-TSC phase appears only for VC > 0 and expands as VC/Δ0 grows, while the noninteracting system has only the vacuum and the ν = 1 integer quantum Hall state.
  • The fourfold-degenerate, odd-parity ground states at K = (π/2, π/2) and (π/2, 3π/2) cannot be connected adiabatically to any mean-field BdG superconducting ground state, establishing the phase as intrinsically interaction-driven.
  • Odd fermion parity together with the parity–Chern relation P = (−1)^N implies that the phase is topological with nonzero Chern number and gapless edge modes; the paper expects a half-odd-integer chiral central charge.
  • The p-wave channel of the induced pair amplitude winds once around the origin, giving p±ip pairing correlations in real space that map to momentum-space winding.

Reading between the lines

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

  • Editorial: The mechanism resembles anyon superconductivity in doped fractional Chern insulators: a repulsively correlated parent state (the composite Fermi liquid) plus pairing is converted by interaction into a topological superconductor. Testing this analogy by doping fractional Chern insulators with repulsive interactions could transfer the design principle to lattice systems.
  • Editorial: Comparing square versus triangular vortex lattices would isolate the role of pairing nodes: the square lattice's node positions dictate which non-inversion-invariant momenta are occupied; a triangular lattice with different nodes should shift or suppress the RID-TSC window.
  • Editorial: Because the phase has a finite critical pairing strength, it is not a weak pairing of composite fermions; this suggests the RID-TSC is a distinct state whose topological order, if any, can be probed by ground-state degeneracy on higher-genus surfaces, beyond the fourfold degeneracy observed here.
  • Editorial: If the parity–Chern extension fails, the RID-TSC phase would still be a novel interaction-induced odd-parity superconductor, and the dispute would be only about its topological label, not its existence.
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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

3 major / 4 minor

Summary. The paper studies a model of spinful electrons in a Rashba-coupled Landau level with a screened repulsive Coulomb interaction, proximitized by a type-II s-wave superconductor with a square Abrikosov vortex lattice. The lowest Rashba-split Landau level is projected and treated by full-Fock-space exact diagonalization on a torus. At half filling the authors find a quantum phase transition, as Δ0/VC is increased, from a composite Fermi liquid (even parity, twofold degenerate) to a fourfold-degenerate gapped state with odd fermion parity, total momenta K=(π/2,π/2) and (π/2,3π/2), and p±ip pair-amplitude winding. They interpret this as repulsive-interaction-driven topological superconductivity (RID-TSC), argue that it is not adiabatically connected to any BdG mean-field state, and propose that it can be viewed as proximity coupling a composite Fermi liquid to an s-wave superconductor.

Significance. The strength of the paper is its unbiased numerical approach: exact diagonalization in the full Fock space with all particle-number sectors, a phase diagram as a function of VC/Δ0 and μ, finite-size scaling with an extrapolated finite gap, and direct computation of pair amplitudes. There are no fitted parameters in the ED. If the topological interpretation is correct, the paper would establish a conceptually new route to topological superconductivity—repulsive interactions, not attraction, drive the phase—and would connect the half-filled Landau level problem to recent proposals for chiral superconductivity from fractional Chern insulators. The main weakness is that the topological label rests on an explicitly unproven extension of the Read-Green parity-Chern theorem to an interacting Hamiltonian; the numerical evidence is strong for an odd-parity gapped chiral paired phase, but weaker for the specific claim of nonzero Chern number and topological order.

major comments (3)
  1. [Topological superconductivity section, after Eq. (5)] The central topological claim hinges on the relation P=(-1)^N, Eq. (5), which is proved for noninteracting BdG Hamiltonians. The manuscript explicitly states: 'Although a rigorous proof of this extension to our interacting system is lacking, we expect that the RID-TSC phase is characterized by half-odd integer c.' This is load-bearing: without Eq. (5), the observed P=-1 fourfold-degenerate gapped state could be a topologically trivial odd-parity superconductor, and the boundary argument does not exclude that possibility, because fermion parity is not a topological invariant in an interacting system. I ask for either (i) a proof of Eq. (5) for H̃_hybrid, or (ii) a direct many-body topological invariant: many-body Chern number of the ground-state manifold, modular matrices, or edge/entanglement spectrum. Until then, the abstract and conclusion should say 'evidence for' rather than 'demonst
  2. [Topological superconductivity section, Feature (ii)] The argument that K=(π/2,π/2) and (π/2,3π/2) lie away from inversion-invariant momenta, and hence cannot be adiabatically connected to any mean-field BdG state of Eq. (4), establishes that the phase is interaction-driven, not that it is topological. A trivial odd-parity superconductor in an interacting model could equally have non-IIM total momentum. Thus Features (i)-(iii) plus the momentum argument identify an odd-parity gapped phase with chiral pairing correlations, but the nonzero Chern number and topological order are an additional inference that requires independent support.
  3. [Fig. 5 and pair-amplitude analysis] The winding of the p-wave pair amplitude F_{σ1σ2}(R,r̄) in real space is a correlation function, not a topological order parameter. The mapping to momentum-space winding in SM S6 is formulated for a noninteracting BdG structure; for the interacting ED ground state, a winding of the pair field around the origin does not by itself imply a nonzero Chern number—it could arise from vortex structure or phase texture. This is good supporting evidence for chiral p±ip pairing correlations, but it should not be presented as a direct proof of topological superconductivity.
minor comments (4)
  1. [Fig. 3(b)] The horizontal axis is labelled '1/Ne' while the text says 1/N_phi. Use a single notation consistently.
  2. [Fig. 4 caption] The caption states that the composite Fermi liquid appears at 'VC/Δc ≈ 50'; this should likely be 'VC/Δ0 ≈ 50'.
  3. [Abstract and title] Given the unproven extension of Eq. (5), the phrase 'demonstrate topological superconductivity' is too strong. Consider 'provide numerical evidence for' or 'establish a phase consistent with' topological superconductivity.
  4. [Pair amplitude definition, Eq. (6)] Define R and r̄ explicitly in the main text (center-of-mass and relative coordinates) and state their ranges; the caption of Fig. 5 is the only place where this is indicated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RID-TSC phase is a computed exact-diagonalization result; the topological interpretation rests on an explicitly unproven extension of Eq. (5), which is a stated limitation rather than a circular reduction.

full rationale

The central claim is produced by direct exact diagonalization of the projected Hamiltonian H̃_hybrid = H̃_int + H̃_Δ − μN (Eqs. 2–3) in the full Fock space. The phase diagram, fourfold degeneracy, odd fermion parity, momentum quantum numbers, finite-size gaps, critical pairing strengths, and pair-amplitude windings are numerical outputs, not inputs; no parameter is fitted to produce these results. The topological diagnosis uses Eq. (5), P = (−1)^N, from noninteracting BdG theory, and the paper explicitly states that a rigorous extension to the interacting system is lacking: 'Although a rigorous proof of this extension to our interacting system is lacking, we expect that the RID-TSC phase is characterized by half-odd integer c.' This is a genuine gap in justification, but it is not circular: Eq. (5) is an external theorem invoked to interpret the ED results, not a relation defined in terms of those results. The noninteracting no-TSC baseline is cited from Refs. 64–65 and the Δ=0 CFL identification from Refs. 119–121; these are independent benchmarks, and the paper's new content is precisely the finite-V_C phase that vanishes in the noninteracting limit. Self-citations such as Refs. 68–69 are peripheral and not load-bearing. No equation in the paper reduces by construction to a fitted input or to a self-citation chain. Therefore there is no significant circularity; the main risk is the unproven interacting extension of the parity–Chern correspondence, which is a correctness/limitation issue rather than a circularity issue.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on (i) the ED Hamiltonian with three scanned parameters (V_C, Δ_0, μ), (ii) the single-LL projection, (iii) the square vortex lattice ansatz, and (iv) the unproven extension of the parity-Chern correspondence to the interacting state. No new particles, forces, or conserved quantities are introduced. The most fragile input is (iv), which the paper explicitly flags.

free parameters (4)
  • V_C (electron-electron interaction strength)
    Model parameter scanned; the RID-TSC phase occupies a finite region of the V_C/Δ0–(μ−μ_{1/2})/Δ0 phase diagram (Fig. 4) and vanishes at V_C=0.
  • Δ_0 (proximity pair amplitude)
    Model parameter scanned; the transition occurs near Δ0/VC≈0.02 (Fig. 2a) and the critical pairing strength Δc/VC is extrapolated to the thermodynamic limit (Fig. 3e).
  • μ (chemical potential)
    Tuned to half-filling ν=1/2; the degeneracy structure and phase boundaries depend on this tuning (Fig. 4).
  • Screening length of the interaction = ξ = l_B
    Screened Coulomb potential V(r)=V_C e^{−r/l_B}/(r/l_B) with the screening length set to the magnetic length 'for simplicity' (footnote 71); this choice is an input, not derived.
assumptions (4)
  • domain assumption Parity-Chern correspondence P=(−1)^N (Eq. 5), proven for noninteracting BdG systems, remains valid for the interacting ground state of H̃_hybrid.
    This is the inference rule that turns the observed odd fermion parity into a topological label. Paper: 'Although a rigorous proof of this extension to our interacting system is lacking, we expect that the RID-TSC phase is characterized by half-odd integer c.' (Topological superconductivity section).
  • domain assumption The lowest Rashba-split LL (ϵ_{1,−1}) is isolated: interaction and pairing energies are weak compared to the LL spacing, justifying projection onto this single level.
    Model section: 'We project the Hamiltonian onto the lowest Rashba-coupled LL with energy ϵ_{1,−1}, assuming that μ is tuned near this LL and both the interaction energy and the pair potential are weak compared to the LL spacing.'
  • domain assumption The Abrikosov vortex lattice is square and the pair potential is a sum of lowest-LL wavefunctions of charge-2e Cooper pairs in the chosen rectangular magnetic unit cell.
    Model section: 'Here, we assume a square vortex lattice as Δ(r) = (Δ_0/√2) Σ_j φ^{LLL}_j(r).' The lattice symmetry and unit-cell shape are inputs that select which momenta are included in the ED mesh.
  • domain assumption Interaction matrix elements are evaluated in the g_R → ∞ limit of the Rashba coupling.
    Footnote 75: 'When evaluating the interaction matrix elements V_{k1k2k'1k'2} in Eq. (2), we take the g_R → ∞ limit for simplicity.' Quantitative phase boundaries depend on this limit.

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Cite this review

Pith. "Pith review of Repulsive-Interaction-Driven Topological Superconductivity in a Landau Level Coupled to an $s$-Wave Superconductor." pith.science (2026). https://pith.science/paper/3DB3HJPP

@misc{pith2026251004700,
  author       = {Pith},
  title        = {Pith review of: Repulsive-Interaction-Driven Topological Superconductivity in a Landau Level Coupled to an $s$-Wave Superconductor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DB3HJPP}},
  note         = {Machine review of arXiv:2510.04700}
}
abstract

A two-dimensional topologically nontrivial state of noninteracting electrons, such as the surface state of a three-dimensional topological insulator, is predicted to realize a topological superconductor when proximity-coupled to an ordinary $s$-wave superconductor. In contrast, noninteracting electrons partially occupying a Landau level, with Rashba spin-orbit coupling that lifts the spin degeneracy, fail to develop topological superconductivity under similar proximity coupling in the presence of the conventional Abrikosov vortex lattice. We demonstrate, through exact diagonalization, that introducing in this model a repulsive interaction between electrons induces topological superconductivity at half-filled Landau level for a range of parameters. This appears rather surprising because a repulsive interaction is expected to inhibit, not promote, pairing, but suggests an appealing principle for realizing topological superconductivity: proximity-coupling a composite Fermi liquid to an ordinary $s$-wave superconductor.

Figures

Figures reproduced from arXiv: 2510.04700 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustration of the hybrid system con [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Low-energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. (b), we plot the energy gaps of the RID-TSC state and the CFL state as functions of 1/Nϕ. Unlike the CFL, whose gap collapses rapidly with the system size, the RID-TSC gap decreases more slowly, with extrapolation suggesting a finite value in the thermodynamic limit. To address SC, we compute the s-wave channel of the pair amplitude ⟨c † ↑ (r)c † ↓ (r)⟩, which shows the Abrikosov vortex [ [PITH_FULL_IMAGE:figures/f… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Filling factor [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Modulus of the pair amplitude [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.