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REVIEW 2 major objections 5 minor 33 references

Repulsive interactions in a superconductor can bind Bogolyubov quasiparticles into subgap excitons that act as intrinsic two-level systems.

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2026-07-10 17:33 UTC pith:BX2VPIPM

load-bearing objection Solid BCS+Bethe–Salpeter construction of repulsion-driven Bogolyubov excitons as a TLS candidate; the microwave-scale identification is still an uncomputed extrapolation. the 2 major comments →

arxiv 2607.07812 v1 pith:BX2VPIPM submitted 2026-07-08 cond-mat.supr-con

Bogolyubov excitons as a microscopic origin of two-level systems

Andrey Grankin , Victor Galitski This is my paper
classification cond-mat.supr-con
keywords two-level systemsBogolyubov excitonsJosephson qubitssuperconducting decoherenceBardasis–Schrieffer modesproximity effectdensity-density responsesurface TLS
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Josephson qubits lose coherence to two-level systems whose microscopic identity is still unsettled. This paper argues that the culprit can be purely electronic: bound states of Bogolyubov quasiparticles formed when a conventional superconductor has attraction only in the s-wave channel and repulsion in higher angular-momentum channels. That repulsion produces an effective attraction between quasielectrons and quasiholes, yielding sharp subgap collective modes (Bogolyubov excitons). In the bulk they stay largely dark to ordinary probes; at a surface or in a proximity-coupled repulsive metal layer they acquire a charge dipole, couple to an external electric field, and produce avoided crossings with a microwave resonator—exactly the spectroscopic signature of TLS. Because the mechanism needs no structural defects and cannot be annealed away, the authors conclude that TLS-like excitations may be intrinsic to superconductors themselves.

Core claim

In a conventional superconductor with s-wave attraction and repulsion in higher-l channels, the repulsive interactions bind Bogolyubov quasiparticles into sharp subgap excitons. Surface or proximity versions of these excitons carry an electric dipole; the vacuum and single-exciton states form a two-level system that hybridizes with a resonator and produces the avoided crossings observed for TLS.

What carries the argument

The density–density polarization Π_q(ω) obtained from the gauge-invariant Bethe–Salpeter equation for the charge vertex Γ_3. Its poles give the exciton energies (analytic weak-coupling formulas for both bulk higher-l and surface s-wave cases); the residue of that pole sets the dipole coupling strength that appears in the resonator spectral function R(ω).

Load-bearing premise

That realistic screened Coulomb interactions, or a strongly reduced surface minigap, can push the exciton frequency down from near 2Δ into the few-gigahertz window where TLS are actually observed.

What would settle it

Build a controlled superconductor–repulsive-metal interface with tunable screening and measure whether discrete subgap resonances appear that anti-cross with a coplanar resonator; absence of such resonances when the higher-l repulsion or the minigap is varied would rule the mechanism out.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes that subgap Bogolyubov excitons—bound states of Bogolyubov quasielectrons and quasiholes induced by repulsive interactions in higher angular-momentum channels of a conventional s-wave superconductor—can serve as an intrinsic, defect-free microscopic origin of two-level systems (TLS) in superconducting devices. Starting from a 2D Hamiltonian with channel-decomposed interaction V_{k-k'}, the authors solve the gauge-invariant Bethe–Salpeter equation for the density vertex and obtain analytic weak-coupling poles for l eq0 excitons (Eq. 8) and, in a proximity-coupled repulsive boundary layer, for an s-wave exciton (Eq. 9) together with the Josephson plasmon. They show that surface/boundary excitons acquire a dipole response (via finite-q fields or weak mass anisotropy), couple capacitively to a resonator mode, and produce avoided crossings in the spectral function R(ω) (Eq. 11, Fig. 1d) that match TLS signatures (i)–(iii) and (v). Appendices supply controlled analytic continuations, residues, and the linear-response TLS matching.

Significance. If the energy-scale and localization arguments hold, the work would supply a purely electronic, annealing- and screening-resistant TLS candidate that is intrinsic to superconductors and interfaces, with a concrete smoking-gun geometry (proximitized repulsive layer + resonator) that can be fabricated and tuned. Strengths that should be credited include the controlled analytic poles of the Bethe–Salpeter equation (Josephson, s-wave and l≠0 excitons, Hartree shift, residue Z_ex), the consistent linear-response resonator formula that reproduces the TLS susceptibility near resonance, and the falsifiable prediction of avoided crossings whose size is set by microscopic parameters rather than by ad-hoc defect densities. The proposal therefore opens a well-defined experimental and theoretical program even if the present static-contact model is only a minimal starting point.

major comments (2)
  1. The central identification with observed TLS (4–12 GHz) is not yet supported by a calculation inside the same framework used for the poles. With bulk Al-scale 2Δ ≃ 90 GHz, the weak-coupling expressions (Eqs. 8–9 and App. B) place ω_ex only a small fraction below 2Δ unless |λ_l ν0| is O(1) or the local minigap is strongly suppressed. The manuscript itself notes (discussion after Eq. 11) that the static contact V_q is only a minimal parametrization and that the true screened Coulomb interaction is frequency-dependent and can be enhanced or singular at finite frequency (Ref. [24]), yet that dynamical kernel is never inserted into the Bethe–Salpeter equation that determines the poles or the residue Z_ex that sets the avoided-crossing size. Without at least a model evaluation of the dynamical interaction (or a quantitative estimate of the surface minigap reduction needed for realistic proximi
  2. The linear-response treatment (App. A and Eq. 11) correctly reproduces the resonant electric susceptibility of a TLS near ω_ex, but the manuscript claims consistency with the full set of TLS signatures, including saturability (signature iv). Saturation requires anharmonicity (Pauli blocking, exciton–exciton interactions, or localization into a finite volume). The text acknowledges this and defers a microscopic nonlinear calculation, yet still presents the modes as TLS candidates that match the experimental phenomenology. Either a minimal estimate of the anharmonicity scale (e.g., from the localization volume implied by a disordered minigap) or a clear restriction of the claim to the linear-response signatures (i)–(iii) and (v) is needed for the identification to be load-bearing.
minor comments (5)
  1. Figure 1(a) caption and the surrounding text should state the precise definition of the high-energy cutoff E0 used in the analytic estimate Eq. (8), since the binding energy depends logarithmically on it.
  2. Notation for the Hartree vertex V^H_q versus the channel couplings λ_l is introduced cleanly, but the gate-screening approximation V^H_q ≈ −2π e^{2} d is used both as a constant and as a free parameter V^H ν0; a single consistent symbol and a brief remark on when the constant approximation remains valid would help.
  3. In App. B the regularization of M_{2,2} via the BCS gap equation is standard, but the intermediate step that isolates the 1/(λ0 ν0 Δ) term could be flagged more explicitly for readers less familiar with the Schrieffer formalism.
  4. The phrase “cannot be eliminated by screening or annealing” in the abstract and conclusion is strong; a short qualifier that the modes may still be suppressed by stronger surface screening or by engineering the local density of states would avoid overstatement.
  5. Several arXiv identifiers and journal citations are given; a quick consistency check that all cited TLS experiments (Refs. [1–11]) are correctly linked to the signatures (i)–(v) would improve readability.

Circularity Check

0 steps flagged

No significant circularity: exciton poles and TLS-like resonator response follow from BCS Green functions plus Bethe–Salpeter, not from fits or self-definition.

full rationale

The load-bearing chain is: (i) Hamiltonian with angular-channel interactions (Eqs. 1–2); (ii) mean-field Nambu Green functions and BCS gap equation (Eqs. 3–4); (iii) density vertex from the Bethe–Salpeter equation (Eqs. 5–7, App. B); (iv) analytic subgap poles for l≠0 and for repulsive s-wave proximity (Eqs. 8–9, App. B); (v) capacitive coupling to a resonator mode yielding R(ω) with an avoided crossing when Π has a pole (Eqs. 10–11, Fig. 1d). None of these steps defines the exciton energy in terms of measured TLS frequencies, nor fits λ_l, Δ_ind, or V_H to TLS spectra and then re-labels the fit as a prediction. Figure parameters are stated as illustrative (e.g. coupling chosen to give a reasonable avoided-crossing scale). Appendix A only shows that a sharp pole in Π produces the same linear electric susceptibility as a TLS—an identification of response form, not a circular derivation of the pole. Self-citations ([24], [25]) supply screening/plasma background and are not uniqueness theorems that force the TLS claim. Energy-scale caveats (static contact V_q vs dynamical Coulomb; need for reduced minigap) are correctness/extrapolation issues, not circular reductions. The derivation is self-contained against its own microscopic inputs.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The load-bearing structure is standard BCS plus channel-decomposed interactions and collective-mode response; free parameters set interaction strengths, proximity gap, screening, and resonator coupling for plots. No new particle is postulated beyond naming repulsion-driven Bogolyubov excitons and identifying vacuum/one-exciton as TLS levels. Independent experimental confirmation of these modes as TLS is not provided.

free parameters (5)
  • λ_l ν0 (channel couplings, especially λ0, λ2)
    Chosen by hand for figures (e.g. λ0ν0=0.1–0.5 attractive bulk; λ2ν0=−0.5; repulsive surface λ0ν0=−0.25 to −0.5). Control binding energy and whether microwave-scale splitting is plausible.
  • Δ_ind / E0 and δ (proximity and intrinsic gap pieces)
    Set the local gap and Josephson/exciton hybrid frequencies; example Δ_ind=E0/10. Needed for surface-model poles and scale matching.
  • V^H ν0 (Hartree / charging scale)
    Gate-screened interaction strength scanned in Fig. 2; shifts s-wave resonance and can push modes into continuum.
  • δm/m (mass anisotropy)
    Ad hoc weak rotational-symmetry breaking (e.g. 0.025) to activate higher-l modes at q=0 in voltage response.
  • resonator coupling α ∼ (d E_ZPF)^2 ν0 A Z_ex / ω_res^2 and γ_res
    Chosen to produce a visible avoided crossing (∼5×10−6, γ_res=10−3} ω_res); illustrative, not measured.
axioms (5)
  • domain assumption BCS mean-field Nambu Green function with s-wave gap from attractive λ0 and self-consistency Eq. (4).
    Standard weak-coupling superconductivity; used throughout for G_k and collective modes.
  • domain assumption Collective charge response from Bethe–Salpeter vertex Γ3 with channel-decomposed instantaneous V_{k−k′}=∑_l λ_l e^{il(φ_k−φ_{k′})} plus Hartree V^H_q.
    Gauge-invariant density response (Schrieffer; Maiti et al.); poles define excitons/plasmons.
  • ad hoc to paper Gate-screened Coulomb V^H_q ≈ −2π e^2 d constant; interaction otherwise static contact in angular channels.
    Simplifies analytics; authors note real screened interaction is frequency-dependent and may enhance binding.
  • ad hoc to paper Idealized 2D circular Fermi surface; bulk S as constant pair source Δ_ind tunnel-coupled to repulsive boundary layer B.
    Geometry for surface localization and capacitive coupling; represents underscreened interface or external metal.
  • domain assumption Near ω_ex, Π(ω)∼ν0 Z_ex/(ω−ω_ex) so vacuum vs one-exciton subspace matches linear TLS electric susceptibility (App. A).
    Identifies exciton with TLS in linear response only; saturation requires anharmonicity left open.
invented entities (1)
  • Repulsion-induced Bogolyubov excitons as intrinsic TLS (vacuum |0⟩ vs single-exciton |1⟩) no independent evidence
    purpose: Provide a defect-free microscopic two-level system with dipole moment and resonator avoided crossing.
    Closely related to Bardasis–Schrieffer modes but driven by repulsive higher-l channels and applied to TLS; independent experimental evidence that observed TLS are these modes is not given.

pith-pipeline@v1.1.0-grok45 · 32005 in / 3640 out tokens · 48133 ms · 2026-07-10T17:33:18.509499+00:00 · methodology

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read the original abstract

Two-level systems (TLS) are a leading source of decoherence and dielectric loss in Josephson-junction qubits, yet their microscopic origin remains unresolved. We propose an intrinsic, purely electronic TLS candidate: subgap bound states of Bogolyubov quasiparticles. We show that, in a model of a conventional superconductor with electron-electron attraction in the $s$-wave channel and repulsion in higher-angular-momentum channels, the latter produce an effective attraction between Bogolyubov quasiparticles, forming bound states. These Bogolyubov excitons resemble Bardasis--Schrieffer excitons but are driven by repulsive interactions. Bulk Bogolyubov excitons are not easily detectable through single-particle tunneling or other conventional probes, but we show that surface excitons couple to an external electric field and behave as two-level systems. We examine an idealized model of a bulk superconductor proximity-coupled to a two-dimensional repulsive metal, which could represent an external metallic or semiconductor layer or an underscreened region of the superconductor. The two levels correspond to the absence and presence of a single exciton, which carries an electric dipole moment and exhibits an avoided crossing with a resonator, as observed for TLS. Because this mechanism requires no defects and cannot be eliminated by screening or annealing, it suggests that TLS-like excitations may be intrinsic to superconductors.

Figures

Figures reproduced from arXiv: 2607.07812 by Andrey Grankin, Victor Galitski.

Figure 1
Figure 1. Figure 1: Repulsion-induced Bogolyubov exciton formation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Diagrammatic representation of vertex cor [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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    Rotational symmetry breaking We model the rotational symmetry breaking by assuming the electron masses are different alongxandydirections. More specifically, we assume the electronic dispersion is ξk = k2 x 2mx + k2 y 2my −µ= k2 2 cos2 ϕk (m+δm) + sin2 ϕk (m−δm) −µ ≈ k2 2m 1− δm m cos 2ϕk −µ where we assumedδm/m≪1. We note that here we consider the simple...

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