{"id":"aa6a22b2-e4a0-47bb-a026-21323272d0c9","arxiv_id":"2608.06553","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Microwave stiffness measurements in Al/InAs heterostructures show an anisotropic, non-monotonic kinetic inductance response under in-plane field, consistent with the emergence of Bogoliubov Fermi surfaces.","lead":"Using lumped-element microwave resonators on an Al/InAs hybrid, the authors observe a non-monotonic, strongly anisotropic drop in superfluid stiffness as an in-plane magnetic field is applied. They attribute the effect to Bogoliubov Fermi surfaces, zero-energy quasiparticle pockets predicted to appear in spin-orbit coupled superconductors.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stiffness 'hook' is reproduced by a fit with two ad hoc anisotropies (gamma and g) and the 90-degree resonator is excluded; the BFS-specific claim needs a direct falsifier.","rationale":"The experiment is a clean and well-controlled observation of non-monotonic anisotropic kinetic inductance suppression, and the control sample (Al/GaAs) rules out ordinary orbital depairing. Yuan-Fu physics is a plausible mechanism. However the manuscript's central claim goes beyond qualitative plausibility: the abstract and main text call the nonmonotonic anisotropic response a 'definitive signature' of BFS. The microscopic calculation captures qualitative features (hook-like drop for B perpendicular to current) but the quantitative agreement is achieved with two phenomenological anisotropies: the disorder anisotropy gamma_perp=4 gamma_parallel, which is introduced ad hoc, and the g-factor which differs by a factor of 6 between scenarios. Most importantly, the 90-degree resonator, which exhibits the strongest anisotropy, is not modeled in Figure 4. This is the weak link in the argument. The reader identified the weak-coupling perturbation theory in SVII as the weakest assumption; I agree that this is unvalidated, but the load-bearing issue is more immediate: even within the authors' own model, the quantitative fit is not constraint-tested for the decisive orientation, and the two anisotropy terms (gamma and g) are free parameters that drive the fit. A direct test: restrict the model to the parameters extracted from the 0-degree and 45-degree fits, and predict the 90-degree resonator response; if the magnitude and field-dependence are off, the 'definitive' claim is overstated. This concern does not invalidate the observation but strengthens the case for a CONDITIONAL verdict.","tokens_in":27278,"tokens_out":1437,"duration_ms":12070,"concrete_test":"Require a fit of the 90-degree resonator (whose measured orientation is unambiguous after recalibrating the capacitor geometry) with the same physical parameters used for the 0 and 45 degree resonators. If the model with fixed g, eta_D, and gamma_perp/gamma_parallel cannot reproduce the larger/sharper suppression, the claim that the effect is quantitative fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that the non-monotonic anisotropic stiffness suppression is a definitive BFS fingerprint. The combined model fits two resonators with five parameters (Bc1, Bc2, Teff, gamma_perp/gamma_parallel, plus the scale), and the anisotropy magnitude is absorbed by setting gamma_perp = 4 gamma_parallel. This is a phenomenological anisotropy parameter, not a derived BFS consequence, and it contributes substantially to the anisotropic signal. Meanwhile the 90-degree resonator, which shows the largest anisotropy, is excluded from the fits due to a lithography error and is not modeled. The failure of the BFS model to capture the amplitude without a disorder anisotropy, plus the unexplained g-factor inconsistency (g=35 or 39.5 in Zeeman-only vs g=5.5 in combined model), makes the interpretation weaker than claimed. For BFS, the predicted signature should be testable by a quantitative relation between the three resonator orientations and the field-angle dependence, but the fits are not shown for the 90-degree resonator, so the decisive orientation is unverified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27634,"tokens_out":4973,"duration_ms":45579,"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":[{"comment":"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.","section":"Fig. 4; Fig. 2(e-f); Table S1"},{"comment":"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.","section":"Main text, parameter discussion after Fig. 4; Fig. 3(h)"},{"comment":"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.","section":"Supplementary SVIII, Eq. (S48)"},{"comment":"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.","section":"Supplementary SVII, Eqs. (S11)-(S13)"},{"comment":"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'.","section":"Abstract; 'Comparison with theory' in main text"}],"minor_comments":[{"comment":"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.","section":"Main text, Eq. (3)"},{"comment":"The word 'subtructed' appears in the sentence 'the contribution of the bands associated to the BFS has to be subtructed'; it should be 'subtracted'.","section":"Main text, section on ground-state energy"},{"comment":"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.","section":"Fig. 2 caption"},{"comment":"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.","section":"Table S1 and main text, sample overview"},{"comment":"The momentum cutoff Lambda in Eq. (S20) is never specified; please state its value or explain why the results are cutoff-independent.","section":"Supplementary Eq. (S20)"},{"comment":"Reference [26] is missing the volume and article number (it appears as 'Sci. Adv. 10, 10.1126/sciadv.adr4817'); please complete the bibliographic details.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a strong experimental paper with good controls and a plausible theoretical framework, but the central 'definitive signature' claim is currently not supported by the quantitative analysis: the decisive 90-degree resonator is excluded from the fits, the g-factor values are inconsistent across scenarios, and a phenomenological disorder anisotropy absorbs part of the signal. I recommend major revision focused on reframing the claim and addressing these points. No concerns about scientific misconduct or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious experimental paper with a new measurement and a plausible interpretation; the BFS attribution is not as airtight as the text claims, because the quantitative comparison leans on several fitted parameters and one resonator is left out.\n\nWhat is actually new: the angle-resolved, lumped-element kinetic inductance measurement on Al/InAs, with a clean Al/GaAs control, careful out-of-plane compensation, and reproducible non-monotonic anisotropic stiffness suppression. That is a real addition. The microscopic two-layer model with Zeeman plus orbital FF and the stiffness tensor calculation is honestly built; the hook-like feature follows from the model rather than being baked in by normalization. The quality-factor change around 50 mT and the angular dependence are consistent with a gapless phase, and the control experiment does rule out vortex motion and trivial orbital depairing fairly convincingly.\n\nSoft spots, in order. First, the \"definitive signature\" language in the abstract and text overreaches. The theory curves in Fig. 4 are fits, not parameter-free predictions. Second, the g-factor inconsistency: Zeeman-only fits give g around 35-39.5, while the combined model gives g around 5.5. That is a large jump and it is not explained. Third, the disorder anisotropy gamma_perp = 4 gamma_parallel absorbs a factor-of-four discrepancy; it is a phenomenological parameter and contributes substantially to the anisotropic signal, so it is not a derived BFS consequence. Fourth, the 90-degree resonator, which shows the largest anisotropy, is excluded from the fits due to a lithography error; that is the decisive orientation for a BFS-specific check and it is left unverified. Fifth, there are no error bars and no data deposit, so fit quality is hard to judge independently. The weakest link in the model is the second-order perturbation treatment of the Al/2DEG coupling in Section SVII; if the interface is not in the weak-coupling regime, higher-order Andreev processes would alter the quasiparticle spectrum and the stiffness signature.\n\nVerdict: this deserves a serious referee. The measurement technique, the control comparison, and the combined Zeeman/orbital framework are valuable. What would materially strengthen the claim: model the 90-degree resonator, pin down g and gamma by independent measurements, and replace \"definitive\" with the evidence-based language of a consistent scenario. I would cite this for the measurement method and the clean control comparison, less for the BFS claim as established.","headline":"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.","tokens_in":28133,"tokens_out":1687,"would_cite":true,"duration_ms":15293,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Bogoliubov Fermi surfaces","superfluid stiffness","kinetic inductance","proximity-induced superconductivity","Al/InAs heterostructure","orbital Fulde-Ferrell effect","Rashba spin-orbit coupling","microwave resonators"],"falsifier":"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.","tokens_in":27051,"feed_emoji":"📡","tokens_out":7995,"duration_ms":65170,"temperature":0.7,"pith_summary":"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.","feed_headline":"Superfluid stiffness dip reveals Bogoliubov Fermi surfaces","feed_subtitle":"A non-monotonic, angle-dependent drop in superfluid stiffness cannot be explained by ordinary pair breaking.","key_machinery":"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.","core_discovery":"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 $\\gamma$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"introduces the banana-shaped Bogoliubov Fermi surfaces in a proximitized 2DEG with Rashba spin-orbit coupling and in-plane Zeeman field; provides the central theoretical scenario.","marker":"[2]"},{"why":"reports segmented Fermi surfaces induced by Cooper-pair momentum in a hybrid bilayer, the experimental precedent linking finite-momentum pairing to Bogoliubov Fermi surfaces.","marker":"[3]"},{"why":"describes the epitaxial Al/2DEG heterostructure platform and its clean interface, the basis for the proximity-induced superconductivity.","marker":"[15]"},{"why":"gives the measured Rashba/Dresselhaus ratio and material parameters for Al/InAs, plus prior supercurrent rectification data used to set the model parameters.","marker":"[23]"},{"why":"previous Al-InAs microwave resonator study that attributed anomalies to Bogoliubov Fermi surfaces; this paper's design must rule out the vortex-pinning ambiguity it raised.","marker":"[28]"},{"why":"theoretical account of disorder scattering of zero-energy Bogoliubov quasiparticles, used to justify the phenomenological anisotropy factor that amplifies the stiffness suppression.","marker":"[29]"},{"why":"textbook relation between kinetic inductance and superfluid stiffness that links the measured resonance shift to the stiffness tensor.","marker":"[34]"},{"why":"cyclotron resonance measurements supplying effective mass and carrier density for the InAs 2DEG used in the fits.","marker":"[37]"}],"fun_headline_variants":["Anisotropic stiffness drop fingerprints Bogoliubov Fermi surfaces","Bogoliubov Fermi surfaces seen via anisotropic stiffness","Stiffness anisotropy reveals gapless superconductivity in hybrids","Magnetic field unveils Bogoliubov Fermi surfaces in Al/InAs","Angle-dependent stiffness drop marks Bogoliubov Fermi surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Anisotropic stiffness drop fingerprints Bogoliubov Fermi surfaces","Bogoliubov Fermi surfaces seen via anisotropic stiffness","Stiffness anisotropy reveals gapless superconductivity in hybrids","Magnetic field unveils Bogoliubov Fermi surfaces in Al/InAs","Angle-dependent stiffness drop marks Bogoliubov Fermi surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000477,"raw_usage":{"total_tokens":2355,"prompt_tokens":924,"completion_tokens":1431,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":1344}},"tokens_in":540,"tokens_out":1431,"duration_ms":10066,"temperature":1.0,"reasoning_tokens":1344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:32:13.112606+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Shabani, M","cited_arxiv_id":null,"evidence_quote":"describes the epitaxial Al/2DEG heterostructure platform and its clean interface, the basis for the proximity-induced superconductivity."},{"cited_title":"Baumgartner, L","cited_arxiv_id":null,"evidence_quote":"gives the measured Rashba/Dresselhaus ratio and material parameters for Al/InAs, plus prior supercurrent rectification data used to set the model parameters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"previous Al-InAs microwave resonator study that attributed anomalies to Bogoliubov Fermi surfaces; this paper's design must rule out the vortex-pinning ambiguity it raised."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"theoretical account of disorder scattering of zero-energy Bogoliubov quasiparticles, used to justify the phenomenological anisotropy factor that amplifies the stiffness suppression."},{"cited_title":"Chauhan, C","cited_arxiv_id":null,"evidence_quote":"cyclotron resonance measurements supplying effective mass and carrier density for the InAs 2DEG used in the fits."}],"review_version":2}