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REVIEW 5 major objections 6 minor 60 references

Emergence of Bogoliubov Fermi Surfaces in hybrid Al/InAs heterostructures

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Pith's one-line read An in-plane magnetic field drives an Al/InAs hybrid into a gapless superconducting phase whose zero-energy quasiparticles form anisotropic 'banana-shaped' Bogoliubov Fermi surfaces, detected through non-monotonic, angle-dependent…

desk verdict New angle-resolved stiffness measurement with clean controls points to Bogoliubov Fermi surfaces in Al/InAs, but the interpretation leans on fitted parameters and the excluded 90-degree resonator, so the paper overstates its certainty. read the letter →

arxiv 2608.06553 v1 pith:L3E5R7LJ submitted 2026-08-06 cond-mat.supr-con cond-mat.mtrl-sci

S. Feyrer , V. Dimic , I. Lobato , A. Kirchner , P. Drexler , L. Rupp , D. Bougeard , T. Lindemann
show 10 more authors
S. Gronin G. Gardner M. J. Manfra G. F. R. Ruiz C. A. Balseiro L. Arrachea M. Aprili N. Paradiso C. Strunk L. Tosi
This is my paper · ORCID
classification cond-mat.supr-concond-mat.mtrl-sci
keywords BogoliubovFermisurfacessuperfluidstiffnesskineticinductanceproximity-inducedsuperconductivityAl/InAsheterostructureorbitalFulde-FerrelleffectRashbaspin-orbitcouplingmicrowaveresonators
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 claims that in an epitaxial Al/InAs hybrid superconductor–semiconductor platform, an in-plane magnetic field drives the proximitized two-dimensional electron gas into a gapless superconducting phase whose zero-energy quasiparticles form anisotropic 'banana-shaped' Bogoliubov Fermi surfaces. The evidence comes from lumped-element microwave resonators: measuring the kinetic inductance gives the superfluid stiffness, and as the field increases the stiffness drops non-monotonically and most strongly when the field is perpendicular to the supercurrent. The paper argues this anisotropic hook-shaped response cannot be explained by orbital pair breaking in a control Al/GaAs sample, but is reproduced by a two-layer microscopic model that combines Zeeman spin splitting, Rashba spin-orbit coupling, and an orbital Fulde-Ferrell Doppler shift. If correct, microwave stiffness becomes a direct thermodynamic probe of finite-momentum, anisotropic gapless superconductivity in hybrid systems.

What carries the argument

The load-bearing object is the superfluid stiffness tensor $D_{ij}(B)$, defined as one quarter of the curvature of the free energy $F(q)$ with respect to Cooper-pair momentum $q$ at its minimum, and measured experimentally through the kinetic inductance of narrow inductor wires. The theory is a two-layer weak-coupling model: a clean 2DEG with Rashba spin-orbit coupling coupled by interlayer tunneling to a thin Al layer with BCS pairing. The in-plane field enters through a Zeeman term $V_Z$ and an orbital Doppler momentum $\hbar q_D = e d\,(\hat{z}\times B)$; the proximity effect is treated to second order in tunneling, producing a uniform induced gap $\Delta_0$. When $V_Z$ or $V_D$ crosses $\Delta_0$, the ground-state energy as a function of $q$ changes from a parabola to a double well, the equilibrium $q_0$ becomes nonzero in the Al layer, and the stiffness exhibits a sudden drop followed by a shallow 'hook'. The model yields a phase diagram with gapped, one-BFS, and two-BFS phases, traversed successively as the field increases.

What would settle it

A direct test is to vary the induced gap $\Delta_0$ (for example by gating or by changing Al thickness) and check that the critical field at which the hook appears shifts as $B_c = 2\Delta_0/(g\mu_B)$ in the Zeeman-dominated case; if the non-monotonic stiffness suppression survives with an unchanged critical field, the Bogoliubov-Fermi-surface interpretation would be in doubt.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the superfluid stiffness tensor of the Al/InAs two-dimensional electron gas becomes sharply anisotropic and non-monotonic under an in-plane magnetic field, with the strongest suppression when the field lies perpendicular to the inductor wire. The authors identify this macroscopic response as a fingerprint of Bogoliubov Fermi surfaces: once the Zeeman energy $V_Z = g\mu_B B/2$ or the Doppler energy $V_D = \eta_D e v_F d B$ exceeds the induced gap $\Delta_0$, zero-energy Bogoliubov quasiparticles appear on banana-shaped segments of the Fermi surface, and these segments selectively suppress supercurrent flow when the current crosses them. Stabilization of the gapless phase requires finite Cooper-pair momentum in the parent Al layer, and the combined Zeeman-plus-orbital-Fulde-Ferrell model reproduces the measured hook-like drop and its angular broadening from 50 to 100 mT. The paper also argues that directional scattering of these zero-energy quasiparticles off mesoscopic disorder amplifies the anisotropy beyond the clean-limit value, and accounts for it with a phenomenological factor $\gamma$.

Load-bearing premise

The central interpretation assumes the proximity effect between the Al and the two-dimensional electron gas is weak enough to be described by second-order interlayer tunneling with a uniform induced gap $\Delta_0$; if the real interface is in a stronger-coupling regime, higher-order Andreev processes would change the quasiparticle spectrum and the stiffness signature.

Editorial extensions

If this is right

  • In-plane fields of only tens of millitesla drive the Al/InAs hybrid into a gapless phase, so gapped-proximity descriptions of these platforms are incomplete under modest fields.
  • Finite Cooper-pair momentum in the Al layer is required to stabilize the Bogoliubov-Fermi-surface phase, so the equilibrium superfluid at nonzero field carries a nonzero pairing momentum.
  • Kinetic-inductance microwave resonators can be used as a general thermodynamic probe of anisotropic gapless superconductivity in layered and moiré superconductors with finite-momentum pairing.
  • Disorder scattering of zero-energy Bogoliubov quasiparticles is highly directional, so transport and diode-effect experiments in these phases will be strongly affected by mesoscopic disorder.
  • The sizable orbital Fulde-Ferrell contribution means that designs and interpretations of Majorana experiments on Al/InAs nanowires and planar junctions must include this orbital term.

Reading between the lines

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

  • One testable extension is to fit the full angular map $\Delta D(\theta)$ at fixed field to infer the angular width $\phi_c = \arccos(\Delta_0/E_{q'})$ of the Bogoliubov Fermi surfaces, which would map their shape without spectroscopy.
  • Reducing the wire width or improving the interface quality should shrink the phenomenological disorder factor $\gamma$, separating intrinsic Bogoliubov-Fermi-surface physics from disorder-enhanced suppression.
  • The two-BFS regime predicts a second, shallower drop in $D(B)$ at higher fields; resolving this feature at lower microwave power or temperature would confirm the 1-BFS to 2-BFS transition.
  • The same measurement protocol could be applied to a gate-tunable Al/InAs device: changing the chemical potential or induced gap should move the critical fields in a prescribed way, providing a parameter-free consistency check.
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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

5 major / 6 minor

Summary. The paper reports microwave kinetic-inductance measurements on three lumped-element resonators fabricated in an Al/InAs proximitized heterostructure. Under an in-plane magnetic field the resonance frequency shows a non-monotonic and strongly anisotropic suppression of the superfluid stiffness, strongest when the field is perpendicular to the inductor current. A control sample with Al on GaAs shows only a weak B^2 dependence, and a careful out-of-plane field compensation is used to exclude vortex effects. The authors attribute the anomaly to the emergence of Bogoliubov Fermi surfaces (BFS) induced by a combination of Zeeman spin-splitting and orbital Fulde-Ferrell (Doppler) effects, and support this with microscopic BdG calculations of the stiffness tensor. The data are compared with three scenarios: Zeeman-only, orbital-only, and combined Zeeman plus orbital, with fitted parameters g, eta_D, T_eff, and a disorder anisotropy factor gamma.

Significance. If the interpretation holds, the paper would establish a macroscopic thermodynamic fingerprint of Bogoliubov Fermi surfaces in a proximitized semiconductor 2DEG, a result of substantial interest for topological superconductivity and finite-momentum pairing physics. The strengths of the manuscript are the high-quality control experiments (Al/GaAs devices), the careful compensation of residual out-of-plane fields, the angle-resolved data, and the microscopic stiffness-tensor framework grounded in a thermodynamic definition of D_ij. However, the current evidence is not as definitive as the text claims: the theory curves are fits with several free parameters, the 90-degree resonator that shows the largest anisotropy is excluded from the quantitative comparison, and a phenomenological disorder anisotropy parameter absorbs part of the anisotropic signal. The paper is a valuable experimental contribution, but the central claim needs substantial strengthening or reframing.

major comments (5)
  1. [Fig. 4; Fig. 2(e-f); Table S1] The central claim rests on the anisotropic suppression being largest when B is perpendicular to J, yet the device that realizes this configuration most directly, the 90-degree resonator, is excluded from the fits because of a lithography error noted in Table S1. Fig. 4 shows fits only for the 0-degree and 45-degree resonators. Since the decisive orientation is unverified, the manuscript should either provide a quantitative model of the 90-degree data (even with reduced accuracy) or explicitly state and justify why it cannot be modeled; without this, the claim that the response is a definitive BFS signature is not supported by the presented quantitative comparison.
  2. [Main text, parameter discussion after Fig. 4; Fig. 3(h)] The fitted g-factor is inconsistent across scenarios: the Zeeman-only fit requires g=35 (resonator a) and g=39.5 (resonator b), while the combined model gives g=5.47 and 5.88, and the phase diagram in Fig. 3(h) is drawn for g=11.5. These values are not mutually consistent, and the large Zeeman-only values are unusual for an InAs 2DEG. The authors should provide a consistency check, such as an independent g-factor measurement on the same wafer, or explicitly discuss why the Zeeman-only scenario, despite giving a similar visual fit, should be discarded. As it stands, the fitted parameters do not uniquely identify the combined Zeeman plus orbital scenario.
  3. [Supplementary SVIII, Eq. (S48)] The quantitative agreement is achieved by introducing the phenomenological anisotropy factor gamma with gamma_perp = 4 gamma_parallel, which multiplies only the perpendicular component of the 2DEG stiffness in Eq. (S48). Since this parameter directly enhances the anisotropic signal, the claim that the BFS model 'captures' the observed anisotropy is weakened: part of the anisotropy is put in by hand. The paper should provide a microscopic estimate or an independent constraint on gamma (for example, from a disorder-scattering calculation for gapless quasiparticles), or show the fits for gamma_perp = gamma_parallel as a baseline to make clear what fraction of the anisotropy is genuinely model-derived.
  4. [Supplementary SVII, Eqs. (S11)-(S13)] The entire BdG spectrum of the 2DEG relies on treating the Al-2DEG coupling in second-order perturbation theory in the interlayer tunneling t_c, yielding the local self-energy of Eq. (S11) and the uniform induced gap Delta_0 of Eq. (S13). No independent check of t_c or of the validity of this weak-coupling limit for this particular Al/InAs interface is provided. Higher-order Andreev processes would modify the quasiparticle dispersion and hence the stiffness signature. The authors should state this limitation in the main text or provide a quantitative estimate of t_c from normal-state or tunneling data to justify the weak-coupling expansion.
  5. [Abstract; 'Comparison with theory' in main text] The manuscript calls the non-monotonic anisotropic response 'a definitive signature of BFS', but the data are compared with a family of models (Zeeman-only, orbital-only, combined) that all produce hook-like non-monotonic features, and the best fit is selected among them. The control experiment rules out the simplest vortex and orbital-depairing pictures, but it does not by itself rule out other gapless or anisotropic pairing states that could produce a similar stiffness suppression. The authors should either demonstrate a model-independent quantitative prediction that discriminates BFS from these alternatives, or soften the claim to 'consistent with the emergence of BFS'.
minor comments (6)
  1. [Main text, Eq. (3)] In Eq. (3), alpha = L_k/L is used as the kinetic inductance fraction, but the symbol L is not explicitly defined as L_geo + L_k in the main text; please define it to avoid ambiguity.
  2. [Main text, section on ground-state energy] The word 'subtructed' appears in the sentence 'the contribution of the bands associated to the BFS has to be subtructed'; it should be 'subtracted'.
  3. [Fig. 2 caption] The angle labels in Fig. 2(i,j) contain apparent formatting artifacts (for example, 'A ngle dependence' and stray degree symbols); the caption should be cleaned and the definition of theta restated explicitly.
  4. [Table S1 and main text, sample overview] The lithography error that reduced the number of fingers in the 90-degree resonator is only mentioned in the supplementary information; it should also be stated in the main text, since it directly affects which data are compared with theory in Fig. 4.
  5. [Supplementary Eq. (S20)] The momentum cutoff Lambda in Eq. (S20) is never specified; please state its value or explain why the results are cutoff-independent.
  6. [References] Reference [26] is missing the volume and article number (it appears as 'Sci. Adv. 10, 10.1126/sciadv.adr4817'); please complete the bibliographic details.

Circularity Check

2 steps flagged · score 6.0 of 10

Quantitative BFS match is a multi-parameter fit with an ad hoc anisotropy factor; qualitative directional selectivity remains independent.

  1. fitted input called prediction [Main text, 'Comparison with theory' (Fig. 4); SI Sec. SVIII]
    "In practice, to fit the data we proceed as follows: for the Zeeman and orbital scenarios we generate a large set of curves D(B) vs B, like Fig. 3(i,j), for different critical fields Bc∈[0.04−0.15] T and temperatures Teff∈[100−500] mK. ... We scale every curve to minimize the error with the experimental data which provides the relative weight of the stiffness of the Al and the 2DEG: ΔD(B)/D(0) = ΔD2D(B)/(D2D+DAl)(0). Our best fit within each scenario is provided by the set of parameters that globally minimize the error."

    The load-bearing 'prediction' of the non-monotonic hook and its field position/broadening is obtained by scanning Bc1, Bc2, Teff, g, η_D and an overall scale, then selecting values that minimize the error against the same experimental data later described as 'captured' by the model. Thus the quantitative agreement is imposed by fitting, not independently predicted. Only the qualitative directionality (stronger suppression for B⊥J) is a parameter-free model consequence; the location and amplitude of the signature are fit inputs.

  2. fitted input called prediction [Main text after Fig. 4; SI Sec. SVIII]
    "To quantitatively account for this anisotropic scattering, we parameterize the disorder-induced momentum relaxation through a phenomenological anisotropy factor γ, defined by D(B⊥)=γD(B∥) (see Supplementary Information). By incorporating this disorder-driven enhancement, the theoretical framework achieves excellent quantitative agreement with the experimental data, underscoring the interplay between emergent gapless states and mesoscopic disorder."

    The strong perpendicular/parallel anisotropy is the very effect the paper calls a 'definitive signature' of BFS, but its amplitude is not derived from the BFS model. The SI makes this explicit: 'Hence, we obtain γ⊥ = 4γ∥.' The measured anisotropy magnitude is therefore inserted by hand, and the subsequent 'excellent quantitative agreement' is circular for that amplitude. The angular selection is genuine BFS content, but the magnitude of the claimed fingerprint is a fitted parameter.

full rationale

The derivation is not circular in the definitional sense: the BdG free-energy curvature calculation (Eq. S19) genuinely yields a non-monotonic 'hook' only when BFS are present, and the directional selectivity (B⊥J suppressing stiffness more than B∥J) is a parameter-free qualitative consequence of the banana-shaped BFS orientation. The Al/GaAs control sample independently shows the effect is absent in a trivial superconductor, so the qualitative claim is externally anchored. No load-bearing self-citation chain was found; the cited prior work by overlapping authors (e.g., Ref. [58]) is not used as the proof of any central premise. However, the paper's quantitative 'capture' of the data is a fit, not an independent prediction. Bc1, Bc2, Teff, g, η_D, the Al/2DEG stiffness scale, and the disorder anisotropy γ (with γ⊥=4γ∥) are all adjusted to minimize error against the same datasets used to claim the fingerprint. The factor-of-four amplitude enhancement is inserted by hand rather than derived from the BFS model. Additionally, the 90° resonator, which shows the largest anisotropy, is excluded from the fits due to a lithography error, so the decisive orientation is unmodeled. The same data can be fit with g≈35 in the Zeeman-only scenario and g≈5.5 in the combined model, further illustrating parameter flexibility. Thus the central claim is partially supported by a self-consistent fit rather than by a parameter-free quantitative prediction; the genuinely predicted directional fingerprint keeps this from being fully circular.

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

The central BFS interpretation rests on a mean-field BdG model of the 2DEG with Rashba SOC, Zeeman and Doppler terms, plus a BCS parent layer. Most material inputs are taken from prior literature, but the key parameters determining where and whether BFS appear (g, eta_D, T_eff) are fitted to the very data being interpreted, and the mismatch in anisotropy magnitude is closed by an ad hoc gamma. No new entities are invented; BFS and FF pairing are prior concepts.

free parameters (5)
  • 2DEG g-factor = 35 (resonator a, Zeeman-only), 39.5 (resonator b, Zeeman-only), 5.47 (a, combined), 5.88 (b, combined)
    Sets the Zeeman energy V_Z = g mu_B B/2. Not independently measured for this sample; the strong model dependence of the fitted value shows the fits are non-unique.
  • Doppler reduction factor eta_D = 0.101 (a) and 0.114 (b) orbital-only; 0.102 (a) and 0.110 (b) combined
    Reduces the ballistic Doppler energy V_D = eta_D e v_F d B to model imperfect screening currents; it effectively sets the orbital critical field.
  • effective temperature T_eff = 176-200 mK
    Controls the broadening of the gapped-to-gapless transition via thermal and microwave-induced quasiparticles; swept during the fit, not independently fixed.
  • disorder anisotropy factor gamma = gamma_perp = 4 gamma_parallel (resonator a), gamma_perp = 2 gamma_parallel (resonator b)
    Phenomenological factor enforcing D(B_perp) = gamma D(B_parallel). It absorbs the factor 4 (2) discrepancy between measured and pristine-model anisotropy and has no microscopic derivation.
  • Al/2DEG stiffness scale Delta_S / gamma_parallel = 115992 (S11 fit)
    Relative weight of the Al and 2DEG stiffness contributions in Eq. S48; fitted to the parallel-field component because the model is assumed more reliable there.
assumptions (6)
  • domain assumption The Al and 2DEG layers are weakly coupled, so the proximity effect can be treated in second-order perturbation theory in the interlayer tunneling t_c (Eqs. S11-S13).
    This self-energy approximation yields the uniform induced gap Delta_0 that underlies every BFS calculation; higher-order Andreev processes are neglected.
  • standard math The superfluid stiffness is given by one quarter of the free-energy curvature with respect to Cooper pair momentum q at the minimum q0 (Eq. 4 of the main text).
    Standard gauge-equivalence / twisted-boundary-condition linear response; requires a well-defined condensate phase.
  • domain assumption The in-plane field enters the 2DEG only through a uniform Doppler momentum q_D = d/ell_B^2 (B x zhat) with a single effective distance d = 15 nm.
    The two-layer model ignores the realistic field profile across the heterostructure and the Al thickness; d is a nominal structural parameter.
  • domain assumption Vortex nucleation inside the inductor is negligible because the wires are 200-400 nm wide and the out-of-plane field is compensated to 5-10 uT.
    Without this, the kinetic inductance would be contaminated by vortex motion; the Al/GaAs control and the pinning sites support the assumption.
  • ad hoc to paper Disorder in the gapless phase can be represented by a single anisotropy factor gamma via D(B_perp) = gamma D(B_parallel).
    Introduced in S11 to close the factor 4 (2) gap between the pristine model and the data; no microscopic derivation is given.
  • standard math The BCS tanh temperature dependence for the induced gap Delta_0(T) is accurate in this regime.
    Used for the finite-temperature free energy; standard in kinetic inductance modeling.

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Pith. "Pith review of Emergence of Bogoliubov Fermi Surfaces in hybrid Al/InAs heterostructures." pith.science (2026). https://pith.science/paper/L3E5R7LJ

@misc{pith2026260806553,
  author       = {Pith},
  title        = {Pith review of: Emergence of Bogoliubov Fermi Surfaces in hybrid Al/InAs heterostructures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3E5R7LJ}},
  note         = {Machine review of arXiv:2608.06553}
}
read the original abstract

We investigate the microwave electrodynamics of a proximitized two-dimensional electron gas in hybrid superconductor/semiconductor heterostructures. Using lumped-element resonators with inductor wires oriented relative to an in-plane magnetic field, we directly probe the superfluid stiffness via the kinetic inductance. As the field increases, the resonance frequency exhibits a non-monotonic and strongly anisotropic evolution that cannot be explained by orbital pair breaking alone. We show that this behavior is consistent with the emergence of Bogoliubov Fermi surfaces, which selectively suppress the supercurrent response depending on the direction of the magnetic field. Microscopic calculations of the stiffness tensor capture the observed anisotropy driven by the interplay of Zeeman and orbital Fulde-Ferrell effects. Our results establish microwave stiffness measurements as a sensitive probe of anisotropic gapless superconductivity in hybrid systems.

Figures

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Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
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Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
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Figure 4
Figure 4. Figure 4: presents the fits of the experimental stiff￾ness data using realistic parameters for the Al/InAs het￾erostructure. To model the system accurately, we ex￾tract the material parameters from literature based on -3 -2 -1 0 0. 00 0. 05 0. 1 0 0. 1 5 0. 20 -4 -3 -2 -1 0 Zeem…

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Pith tools

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