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arxiv: 2601.00108 · v2 · submitted 2025-12-31 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Unified topological phase diagram of quantum Hall and superconducting vortex-lattice states

Daniil S. Antonenko , Liang Fu , Leonid I. Glazman This is my paper

Pith reviewed 2026-05-16 17:51 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords topological superconductivityquantum Hall effectvortex latticeLandau level mixingBogoliubov-de GennesChern numbersphase diagramchiral edge modes
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The pith

The superconducting vortex lattice splits quantum Hall transition lines into sequences of larger Chern number jumps due to Landau-level mixing.

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

The paper establishes a unified topological phase diagram for a two-dimensional electron gas in a magnetic field proximitized by a superconducting vortex lattice, valid for any ratio of pairing amplitude, magnetic field, and chemical potential. By solving the Bogoliubov-de Gennes Hamiltonian, it reveals numerous topological superconducting phases featuring chiral quasiparticle edge modes. Landau-level mixing is crucial even at weak pairing, where it breaks the integer quantum Hall transition lines into multiple transitions with Chern number jumps of both positive and negative signs, safeguarded by the vortex lattice symmetries. When the chemical potential aligns with a Landau level energy, weak pairing can produce either trivial or topological superconductivity depending on the specific Landau level index.

Core claim

The global topological phase diagram of the system shows that Landau-level mixing plays an essential role: even in the weak-pairing limit, it generically splits the integer quantum Hall transition lines into a sequence of transitions with larger Chern number jumps of both signs protected by the symmetries of the superconducting vortex lattice. Weak pairing induces trivial or topological superconductivity when chemical potential is tuned to a Landau level energy, depending on the Landau level index.

What carries the argument

The Bogoliubov-de Gennes Hamiltonian incorporating Landau-level mixing for electrons in a magnetic field with a static superconducting vortex lattice.

Load-bearing premise

The Bogoliubov-de Gennes Hamiltonian with a static superconducting vortex lattice fully captures the physics for arbitrary ratios of pairing amplitude, magnetic field, and chemical potential, without additional effects such as disorder or lattice imperfections.

What would settle it

Measuring the sequence of Chern number changes across transitions in a specific Landau level by tracking the number of chiral edge modes or Hall conductivity steps in a proximitized 2D electron gas with controlled vortex lattice.

read the original abstract

We present the global topological phase diagram of a two-dimensional electron gas placed in a quantizing magnetic field and proximitized by a superconducting vortex lattice. Our theory allows for arbitrary ratios of the pairing amplitude, magnetic field, and chemical potential. By analyzing the Bogoliubov--de Gennes Hamiltonian, we show that the resulting phase diagram is highly nontrivial, featuring a plethora of topological superconducting phases with chiral edge modes of quasiparticles. Landau-level mixing plays an essential role in our theory: even in the weak-pairing limit, it generically splits the integer quantum Hall transition lines into a sequence of transitions with larger Chern number jumps of both signs protected by the symmetries of the superconducting vortex lattice. Interestingly, we find that weak pairing induces trivial or topological superconductivity when chemical potential is tuned to a Landau level energy, depending on the Landau level index.

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

3 major / 3 minor

Summary. The manuscript presents a unified topological phase diagram for a two-dimensional electron gas in a quantizing magnetic field proximitized by a superconducting vortex lattice. By solving the Bogoliubov-de Gennes Hamiltonian for arbitrary ratios of pairing amplitude, magnetic field, and chemical potential, it identifies a rich set of topological superconducting phases with chiral quasiparticle edge modes. The central result is that Landau-level mixing, even in the weak-pairing limit, splits conventional integer quantum Hall transition lines into sequences of transitions featuring larger Chern-number jumps of both signs; these jumps are protected by the symmetries of the static vortex lattice. When the chemical potential is tuned to a Landau-level energy, weak pairing is shown to induce either trivial or topological superconductivity depending on the Landau-level index.

Significance. If the central claims hold, the work supplies a comprehensive, non-perturbative map connecting quantum-Hall and superconducting topological phases in hybrid systems. The explicit inclusion of Landau-level mixing and the symmetry-protected splitting of transitions constitute a clear advance over earlier treatments that either neglected mixing or assumed strong-pairing limits. The phase diagram is potentially useful for interpreting experiments in proximitized 2DEGs and for designing devices that exploit tunable Chern numbers.

major comments (3)
  1. [§3.1, Eq. (12)] §3.1 and Eq. (12): The assertion that vortex-lattice symmetries generically enforce larger Chern-number jumps (both signs) in the weak-pairing limit is load-bearing for the main claim, yet the manuscript provides only a symmetry-indicator argument without an explicit calculation of the Berry curvature or the minimal gap closing that would be required to confirm the jump sizes. A concrete example for the lowest Landau level would strengthen the result.
  2. [§4.2] §4.2: The phase diagram is obtained under the assumption of a perfectly static, translationally periodic vortex lattice. Because the protection of the split transitions relies on these symmetries, the manuscript should quantify how weak disorder or vortex-position fluctuations (which break translational symmetry) would affect the diagram, especially near Landau-level energies where the nontrivial splitting is claimed.
  3. [§2.3] §2.3: The BdG Hamiltonian is stated to be solved for arbitrary Δ/B/μ ratios, but the numerical or analytic method used to extract Chern numbers across the full parameter space is not described with sufficient detail (e.g., Brillouin-zone sampling, convergence criteria, or handling of Landau-level mixing). Without these, the “plethora of topological phases” cannot be independently verified.
minor comments (3)
  1. [Figure 2] Figure 2 caption: the color scale for the Chern number is not defined; please add an explicit legend or table of values.
  2. [near Eq. (18)] The abstract states that weak pairing induces trivial or topological superconductivity “depending on the Landau level index,” but the corresponding statement in the main text (near Eq. (18)) is phrased ambiguously; a single clarifying sentence would remove the ambiguity.
  3. [Introduction] Several references to prior vortex-lattice BdG studies are cited only in passing; a short paragraph comparing the present phase diagram with those works would help readers assess novelty.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address each major comment below and have revised the manuscript to incorporate clarifications and additional details where possible.

read point-by-point responses

  1. Referee: [§3.1, Eq. (12)] §3.1 and Eq. (12): The assertion that vortex-lattice symmetries generically enforce larger Chern-number jumps (both signs) in the weak-pairing limit is load-bearing for the main claim, yet the manuscript provides only a symmetry-indicator argument without an explicit calculation of the Berry curvature or the minimal gap closing that would be required to confirm the jump sizes. A concrete example for the lowest Landau level would strengthen the result.

    Authors: We agree that an explicit verification strengthens the central claim. While the symmetry-indicator argument in Eq. (12) is general and applies to arbitrary Landau levels, we have added a concrete example for the lowest Landau level in the revised manuscript. This includes the computed Berry curvature distribution over the Brillouin zone and explicit tracking of the minimal gap closings that produce the larger Chern-number jumps of both signs. revision: yes

  2. Referee: [§4.2] §4.2: The phase diagram is obtained under the assumption of a perfectly static, translationally periodic vortex lattice. Because the protection of the split transitions relies on these symmetries, the manuscript should quantify how weak disorder or vortex-position fluctuations (which break translational symmetry) would affect the diagram, especially near Landau-level energies where the nontrivial splitting is claimed.

    Authors: We acknowledge that the protection of the split transitions relies on the translational symmetry of the static vortex lattice. A full quantification of weak disorder or vortex-position fluctuations would require new numerical simulations with broken translational symmetry, which lies beyond the scope of the present work. In the revised manuscript we have added a paragraph in §4.2 explicitly stating this limitation, noting that the reported phase diagram applies to the clean limit and that disorder may lift the symmetry protection, potentially restoring conventional integer quantum Hall transitions. revision: partial

  3. Referee: [§2.3] §2.3: The BdG Hamiltonian is stated to be solved for arbitrary Δ/B/μ ratios, but the numerical or analytic method used to extract Chern numbers across the full parameter space is not described with sufficient detail (e.g., Brillouin-zone sampling, convergence criteria, or handling of Landau-level mixing). Without these, the “plethora of topological phases” cannot be independently verified.

    Authors: We thank the referee for pointing out the insufficient detail. In the revised manuscript we have substantially expanded §2.3 to describe the numerical procedure: the BdG Hamiltonian is diagonalized in a plane-wave basis truncated at a Landau-level cutoff N_max (typically 10–15 for convergence), the Brillouin zone is sampled on a 60×60 Monkhorst-Pack grid, and Chern numbers are obtained via the standard Berry-phase formula with convergence verified by increasing both the cutoff and grid density until the gap and Chern numbers stabilize to within 0.01. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from standard BdG Hamiltonian analysis

full rationale

The paper constructs the phase diagram by direct numerical or analytical inspection of the Bogoliubov-de Gennes Hamiltonian for a static superconducting vortex lattice at arbitrary pairing, field, and chemical-potential ratios. No parameters are fitted to a subset of data and then re-labeled as predictions; no self-citation chain supplies a uniqueness theorem or ansatz that is then treated as external; the symmetries invoked are those of the assumed vortex lattice itself, not smuggled in from prior work by the same authors. The central claim about Landau-level mixing splitting Chern transitions therefore rests on explicit diagonalization or topological invariant computation rather than on any definitional reduction to the input assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Bogoliubov-de Gennes formalism for proximitized systems and the assumed symmetries of the vortex lattice; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption The system is described by the Bogoliubov-de Gennes Hamiltonian incorporating a superconducting vortex lattice.
    This is the standard starting point for modeling proximity-induced superconductivity in a magnetic field.

pith-pipeline@v0.9.0 · 5454 in / 1194 out tokens · 39925 ms · 2026-05-16T17:51:48.752526+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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  1. Emergent spin quantum Hall edge states at the boundary of two-dimensional electron gas proximitized by an $s$-wave superconductor

    cond-mat.mes-hall 2026-05 unverdicted novelty 6.0

    2DEG-S hybrids in quantized magnetic field host topologically protected edge states carrying even-integer quantized spin current robust to disorder.

  2. Emergent spin quantum Hall edge states at the boundary of two-dimensional electron gas proximitized by an $s$-wave superconductor

    cond-mat.mes-hall 2026-05 unverdicted novelty 5.0

    2DEG-S hybrids in magnetic field host disorder-robust edge states with even-integer quantized spin conductance due to class C topology, detectable via electrical measurements.