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REVIEW 3 major objections 4 minor 48 references

Unconventional Hall Effect in Gapless Superconductors: Transverse Supercurrent Converted from Normal Current

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A gapless superconductor converts a longitudinal normal current into a dissipationless transverse supercurrent through quasiparticle Berry curvature.

desk verdict The paper makes a testable prediction for a Hall effect in a recently realized gapless superconductor, but the central claim that the transverse current is a supercurrent is asserted rather than demonstrated, so the paper needs a serious referee and a heavy revision. read the letter →

arxiv 2505.23278 v1 pith:HTYCIASS submitted 2025-05-29 cond-mat.mes-hall

Miaomiao Wei , Longjun Xiang , Fuming Xu , Bin Wang , Jian Wang This is my paper · ORCID
classification cond-mat.mes-hall
keywords superconductingHalleffectgaplesssuperconductorBerrycurvaturesegmentedFermisurfacetransversesupercurrentfinite-momentumCooperpairsproximitysuperconductivityquantumtransport
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 tries to establish that in a gapless superconductor with segmented Fermi surfaces, a longitudinal normal current can be converted into a transverse dissipationless supercurrent, an effect it calls the superconducting Hall effect (ScHE). It argues that the origin is intrinsic: the Berry curvature of Bogoliubov quasiparticles produces an anomalous transverse velocity, captured by the term $-f_n \mathbf{E} \times \boldsymbol{\Omega}_n$ in the current density. The claim is supported by two independent approaches, a thermodynamic free-energy derivation for bulk systems and a four-probe quantum transport calculation. If true, this would be a new kind of Hall effect in which the transverse response is carried by Cooper pairs rather than quasiparticles, observable by rotating the in-plane magnetic field in standard Hall measurements.

What carries the argument

The load-bearing object is the quasiparticle Berry curvature $\boldsymbol{\Omega}_n$ of the Bogoliubov-de Gennes bands, combined with the thermodynamic current formula $\mathbf{J} = \partial F/\partial \mathbf{A}$. In linear response under the relaxation-time approximation, this yields Eq. (7), whose second term, $-f_n \mathbf{E} \times \boldsymbol{\Omega}_n$, is the intrinsic Berry-curvature contribution producing the transverse supercurrent. The segmented Fermi surface, enabled by spin-orbit coupling, an in-plane Zeeman field, and hexagonal warping with coefficient $\lambda$, supplies the anisotropic geometry in which quasiparticle and Cooper-pair transport are allowed in orthogonal directions.

What would settle it

If an angular-dependent Hall measurement on a gapless superconductor with hexagonal warping shows no dissipationless transverse current when the in-plane magnetic field is rotated away from $\theta = 0^\circ$, or if a system with $\lambda = 0$ shows a transverse current, the central claim is falsified. Equivalently, a calculation showing that the $-f_n \mathbf{E} \times \boldsymbol{\Omega}_n$ term is a normal dissipative Hall current rather than a supercurrent would invalidate the ScHE.

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Extended reading notes

Core claim

The central claim is that gapless superconductors with segmented Fermi surfaces host a superconducting Hall effect: a longitudinal electric field drives a quasiparticle current along the gapless direction, while the Berry curvature of those quasiparticles deflects electron and hole motion, and the deflected carriers combine into finite-momentum Cooper pairs that flow as a dissipationless transverse supercurrent. The paper derives a current-density expression, $\mathbf{J} = \sum_{n\mathbf{k}} [\tau \mathbf{E} \cdot \nabla_{\mathbf{k}} f_n \mathbf{v}_n - f_n \mathbf{E} \times \boldsymbol{\Omega}_n]$, where the second term is identified as the intrinsic Berry-curvature contribution to the supercurrent, and confirms the same behavior in a four-probe transport calculation. The effect vanishes when the hexagonal warping vanishes, vanishes for one special magnetic-field orientation, and grows as the field is rotated, matching the symmetry of the quasiparticle Berry curvature.

Load-bearing premise

The argument assumes that the equilibrium thermodynamic relation $\mathbf{J} = \partial F/\partial \mathbf{A}$, with the equilibrium distribution replaced by a nonequilibrium one under the relaxation-time approximation, still gives the correct current, and that the Berry-curvature term represents a Cooper-pair supercurrent rather than an ordinary quasiparticle Hall current.

Editorial extensions

If this is right

  • If the superconducting Hall effect is real, rotating the in-plane magnetic field in a gapless superconductor should produce a transverse supercurrent that is zero at $\theta = 0^\circ$ and grows as $\theta$ deviates.
  • The transverse supercurrent should vanish when hexagonal warping is absent, because the quasiparticle Berry curvature also vanishes for $\lambda = 0$.
  • Two-probe normal-metal/superconductor transport should show Andreev reflection dominating when the gap opens along the transverse direction, reproducing the segmented Fermi surface geometry.
  • Four-probe quantum transport should reproduce the thermodynamic prediction qualitatively, with the transverse current carried as a supercurrent rather than a normal dissipative current.
  • Angular-dependent Hall measurements on a circular disc of a proximitized gapless superconductor, such as a Bi2Te3/NbSe2 heterostructure, should reveal the signatures of both the tilted gap and the ScHE.

Reading between the lines

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

  • The ScHE may turn angle-resolved Hall measurements on gapless superconductors into a direct probe of the quasiparticle Berry curvature distribution, in the same way that anomalous Hall measurements map band geometry in metals.
  • A dissipationless transverse supercurrent that can be switched by rotating the magnetic field suggests a possible mechanism for lossless current routing controlled by field orientation, although the paper does not explore device applications.
  • The thermodynamic-plus-Berry-curvature framework could extend naturally to other partially gapped superconducting states, including finite-momentum or altermagnet-based superconductors, to predict similar transverse supercurrent responses.
  • If the transverse response is truly dissipationless, it should be distinguishable from a normal Hall current by measuring the absence of transverse voltage drop, a testable experimental signature.
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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 proposes a new 'superconducting Hall effect' (ScHE) in a two-dimensional gapless superconductor described by the BdG Hamiltonian in Eq. (2), with Rashba spin-orbit coupling, an in-plane Zeeman field, and hexagonal warping. It argues that in the segmented Fermi-surface regime, a longitudinal normal current driven by a small electric field is converted into a transverse dissipationless supercurrent via the quasiparticle Berry curvature. The central formula is Eq. (7), whose second term, -fn E × Ωn, is interpreted as the intrinsic supercurrent contribution. The claim is supported by (i) a thermodynamic derivation from J = ∂F/∂A, (ii) two-probe N-S transport calculations reproducing the segmented Fermi-surface transmission features, and (iii) a four-probe NEGF calculation showing a transverse current that depends on the magnetic-field orientation and the warping strength. An experimental realization in Bi2Te3/NbSe2 heterostructures is proposed.

Significance. If the identification of the Berry-curvature term as a Cooper-pair supercurrent is correct, the ScHE would be a conceptually new transport phenomenon, distinct from previously studied superconducting Hall and diode effects. It would also be a concrete case where the band geometry of Bogoliubov quasiparticles controls the condensate current, and it yields a falsifiable angular dependence (JH = 0 at θ = 0 and for λ = 0). The paper makes a useful connection to the experimentally realized gapless superconductor Bi2Te3/NbSe2 and proposes a specific measurement geometry. However, the central claim is presently an interpretation layered on Eq. (7), and the key derivation is not contained in the main text; the significance is therefore conditional on the authors closing that gap.

major comments (3)
  1. [Theoretical framework, Eq. (7)] Equation (7) is the load-bearing result, but its derivation is not in the main text; the text says 'The derivation detail is provided in the Supplementary Material.' The main text also states that Eq. (3) is suitable for equilibrium systems, then replaces fn by gn to capture the driven system. This out-of-equilibrium step is asserted rather than derived. Because the second term of Eq. (7) is the entire basis for the ScHE, the authors must either present the derivation in the main text or provide a rigorous justification for applying the equilibrium thermodynamic expression to a steady state with a finite electric field. Please clarify whether the Berry term originates from the dependence of the quasiparticle spectrum on the Cooper-pair momentum or from the ordinary band geometry; these two origins lead to different physical interpretations.
  2. [Eq. (7), second term] The identification of -fn E × Ωn as a 'dissipationless transverse supercurrent carried by finite-momentum Cooper pairs' is not established. In the Δ→0 limit, the same expression is the standard intrinsic anomalous Hall current of quasiparticles in a time-reversal-broken metal, and a dc electric field in a superconductor normally produces the inductive London response rather than a finite dc Berry-curvature conductivity. The paper needs to show that the transverse current is carried by the pair (off-diagonal) component of the BdG current operator. A concrete way would be to derive Eq. (7) from the microscopic current operator and separate the normal (electron-hole diagonal) and anomalous (Andreev off-diagonal) contributions. Without such a separation, the central claim is unsupported.
  3. [Quantum transport formalism, Eq. (9), Fig. 3(f)] The four-probe calculation computes the total transverse charge current I3 (or I4) through the Landauer-Büttiker formula; it does not decompose this current into quasiparticle and Cooper-pair contributions. The sentence 'Eq. (9) can also calculate both normal current and supercurrent' is therefore not demonstrated by the presented results. Figure 3(f) shows that a transverse current appears with the expected angular dependence, but it does not establish that this current is a supercurrent. Please provide a channel-resolved decomposition, or a complementary calculation such as the dependence of IH on Δ at fixed θ, to distinguish a supercurrent from a quasiparticle Hall current.
minor comments (4)
  1. [Throughout] Typos include 'suppercurrent' in the Fig. 3(f) caption, 'B filed' below Eq. (1), 'gaped' in the introduction, 'the the' in the Mechanism section, 'tranverse' in the four-probe paragraph, and 'identity' for 'identify' in the Conclusion.
  2. [Fig. 1(f)] The main text does not specify the integration domain, the temperature, or the definition of t for the JH calculation, so the quantitative JH values cannot be reproduced from the main text alone. These parameters should be stated explicitly.
  3. [Supplementary Material references] The text repeatedly refers to the Supplementary Material for essential details, including the derivation of Eq. (7), the symmetry argument for Fig. 2, and the circular-disc measurement setup. For a self-contained manuscript, please ensure that all load-bearing material appears in the main text or is available to the reviewers.
  4. [Eq. (7)] The electron charge e is not shown in Eq. (7); please state the convention (e = 1) or restore the explicit charge factors in the Drude and Berry terms.

Circularity Check

2 steps flagged · score 4.0 of 10

The quantitative Berry-curvature calculation is independent, but the central identification of the Berry term as a Cooper-pair supercurrent is definitional/renaming rather than derived.

  1. self definitional [Theoretical framework for describing ScHE, Eq. (7) and following paragraph]
    "In the dc limit (ω→0) and under the relaxation time approximation, we finally obtain the current density in vector notation: J = Σ_nk [τ E · ∇k fn vn − fn E × Ωn] ... The second term is the intrinsic contribution from quasiparticle Berry curvature, inducing a dissipationless transverse supercurrent and hence the ScHE."

    The ScHE is introduced as a transverse supercurrent whose intrinsic origin is quasiparticle Berry curvature, and Eq. (7)'s second term is exactly that Berry-curvature transverse current. By equating the Berry term with the ScHE, the paper's demonstration that the ScHE exists reduces to evaluating the very term already identified as the ScHE. The load-bearing physical identification — that this Berry term is a Cooper-pair supercurrent rather than a quasiparticle anomalous Hall current — is asserted, not derived. Without that identification, the computation shows only a known quasiparticle Hall current, so the central claim is true by definition rather than by derivation.

  2. renaming known result [Quantum transport formalism, Eq. (9) and Fig. 3(f)]
    "Similar to the thermodynamic approach in Eq. (7), Eq. (9) can also calculate both normal current and supercurrent. ... Figure 3(f) presents the dependence of the transverse supercurrent IH (equal to I3 or I4) on the magnetic field orientation angle θ and the warping strength λ."

    Equation (9) is the standard NEGF formula for the total charge current in a lead; I3 and I4 are the total transverse currents, with no decomposition into normal (electron-hole diagonal) and pair (Andreev/off-diagonal) channels. Labeling the total transverse current as 'transverse supercurrent' assumes the conclusion. The qualitative agreement with Fig. 1(f) therefore confirms only that the same Berry-curvature integral appears in the two formalisms, not that the transverse current is carried by Cooper pairs. The term 'supercurrent' is a renaming of the computed total transverse current rather than a demonstrated outcome.

full rationale

The paper is not circular in the sense of fitting parameters or importing a uniqueness theorem from self-citations; the angular dependence and warping-strength dependence are genuine computed results, and the four-probe NEGF calculation is an independent method. However, the central claim that the Berry-curvature transverse current is a dissipationless Cooper-pair supercurrent is established by definition and assertion, not by derivation. The thermodynamic derivation of Eq. (7) is delegated to a Supplementary Material not present in the reviewed artifact, so the origin of the Berry term (quasiparticle band geometry versus Cooper-pair momentum) cannot be verified. The four-probe calculation computes total transverse currents and labels them supercurrents without a channel decomposition. These issues make the 'supercurrent' identification partially circular/renaming, while the quantitative Berry-curvature transport results retain independent content. The minor self-citation [34] is not load-bearing and does not affect the score.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claim rests on a small number of chosen model parameters (none fitted to experiment) and on four assumptions about the transport formalism and the interpretation of the transverse response. The weakest links are the out-of-equilibrium use of the thermodynamic formula and the assertion that the Berry-curvature term is a supercurrent.

free parameters (6)
  • Rashba SOC strength αR = 0.1t
    Chosen to realize the gapless superconducting phase.
  • Hexagonal warping strength λ = 0.5t
    Required for nonzero Berry curvature; JH vanishes at λ=0.
  • Chemical potential μ = 0.05t
    Sets the Fermi energy to intersect the superconducting gap.
  • Proximity-induced pairing gap Δ = 0.001t
    Chosen much smaller than other scales to create the gapless state.
  • In-plane magnetic field B (with g μB set to 1) = Bx=0, By=2Δ
    Induces the tilted gap and segmented Fermi surface; the field angle θ is the control parameter.
  • Relaxation time τ = not specified
    Appears in the Drude term but does not affect the Berry-curvature Hall term.
assumptions (4)
  • domain assumption The BdG Hamiltonian with proximity-induced s-wave pairing describes the gapless superconducting state.
    The model is taken from prior literature (Yuan & Fu 2018; Zhu et al. 2021) and is not derived in this paper.
  • ad hoc to paper The thermodynamic expression J = ∂F/∂A yields the correct current when the equilibrium distribution is replaced by a nonequilibrium distribution gn.
    This is a nonstandard extension beyond equilibrium, asserted without formal justification.
  • ad hoc to paper The relaxation-time approximation is valid for the driven system.
    Employed to obtain the Drude term; no relaxation mechanism is specified.
  • ad hoc to paper The transverse Berry-curvature term in Eq. (7) is a dissipationless supercurrent carried by finite-momentum Cooper pairs.
    The central interpretive step of the paper; no independent derivation is given.

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Pith. "Pith review of Unconventional Hall Effect in Gapless Superconductors: Transverse Supercurrent Converted from Normal Current." pith.science (2026). https://pith.science/paper/HTYCIASS

@misc{pith2026250523278,
  author       = {Pith},
  title        = {Pith review of: Unconventional Hall Effect in Gapless Superconductors: Transverse Supercurrent Converted from Normal Current},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTYCIASS}},
  note         = {Machine review of arXiv:2505.23278}
}
read the original abstract

A normal metallic system proximitized by a superconductor can exhibit a gapless superconducting state characterized by segmented Fermi surfaces, as confirmed experimentally. In such a state, quasiparticle states remain gapless along one direction, while a superconducting gap opens in the perpendicular direction. This anisotropy enables a novel Hall effect in gapless superconductors, termed the superconducting Hall effect (ScHE), where a longitudinal normal current carried by quasiparticles is converted into a dissipationless transverse supercurrent. Employing both the thermodynamic approach for bulk systems and quantum transport theory for a four-probe setup, we demonstrate the existence of this effect and reveal its intrinsic origin as the quasiparticle Berry curvature. The predicted ScHE can be experimentally verified via the standard angular-dependent Hall measurements performed on gapless superconductors.

Figures

Figures reproduced from arXiv: 2505.23278 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Quasiparticle band structure of the gapless superconduct [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Quasiparticle Berry curvature distribution on the segmented [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Schematic of a two-probe N-S hybrid system. (b)-(d) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.