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REVIEW 4 major objections 4 minor 47 references

Spectroscopic and optical response of odd-frequency superconductors

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Optical conductivity can unmask odd-frequency superconductivity that spectroscopy misses.

desk verdict A careful model calculation that finds new optical signatures for odd-frequency pairing, but the paper overclaims uniqueness and 'transparency windows' without testing a frequency-dependent even-frequency gap. read the letter →

arxiv 1908.08228 v1 pith:MPVIZXO5 submitted 2019-08-22 cond-mat.supr-con cond-mat.str-el

P. O. Sukhachov , A. V. Balatsky This is my paper · ORCID
classification cond-mat.supr-concond-mat.str-el
keywords odd-frequencysuperconductivityBerezinskiipairingopticalconductivityspectralfunctiondensityofstatesKramers-Kronigrelationssuperconductinggapsymmetrytransparencywindows
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

This paper asks whether odd-frequency (Berezinskii) superconductivity—a pairing state odd under time inversion—has an experimentally accessible optical signature. The authors calculate the spectral function, electron density of states, and optical conductivity for a parabolic-band superconductor with two frequency-dependent gap ansatzes, including disorder. They find that spectral and DOS probes cannot tell odd-frequency pairing from ordinary BCS pairing, because both depend only on the absolute value of the gap. In contrast, the optical conductivity contains cross terms $\Delta(\omega)\Delta^\dagger(\omega-\Omega)$ that differ sharply between the two cases: the odd-frequency gap produces sharper absorption peaks, cusp-like features in the imaginary part of the conductivity, and can even drive Im $\sigma$ negative, suggesting an optical transparency window. If correct, optics would provide a direct experimental route to detect Berezinskii pairing, which has so far remained elusive.

What carries the argument

The key object is the optical conductivity expressed through Nambu spectral functions, where the paramagnetic response contains products $\Delta(\omega)\Delta^\dagger(\omega-\Omega)$. Because the odd-frequency ansatzes change sign with $\omega$, these cross terms differ from those of a constant even-frequency gap even when $|\Delta(\omega)|$ matches, which is why optics can distinguish pairing symmetries that spectroscopy cannot. The cusps and negative regions in $\mathrm{Im}\,\sigma$ follow from the Kramers–Kronig relation between the sharp onset in $\mathrm{Re}\,\sigma$ and the imaginary part; a wave-function renormalization factor $Z(\omega,p)$, fixed by requiring the spectral sum rule, is introduced to keep the model consistent.

What would settle it

Search for the cusp and the negative imaginary part of the optical conductivity at frequencies just below the absorption peak in a material expected to host odd-frequency pairing (e.g., a superconductor–ferromagnet bilayer); alternatively, compute the frequency-dependent gap self-consistently from a microscopic pairing model and check whether the gap profile falls in the small-$\beta$ regime where the signatures survive.

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Extended reading notes

Core claim

The central claim is that odd-frequency Berezinskii pairing produces characteristic signatures in the optical conductivity even when the spectral function and density of states look identical to those of a conventional BCS superconductor. For a gap $\Delta(\omega)=\alpha\,\mathrm{sgn}(\omega)$, the real part of the conductivity shows absorption peaks at $|\Omega|\simeq 2|\alpha|$ that are sharper and taller than the Mattis–Bardeen peaks of the constant BCS gap, while the imaginary part develops sharp cusps and can become negative just below the peak onset. The paper further shows that the magnitude and position of these features are controlled by the frequency profile of the gap: for smoother profiles (large $\beta$) the spectral gap closes and all distinctive signatures disappear, leaving a response nearly indistinguishable from the normal state.

Load-bearing premise

The predicted signatures rely on the odd-frequency gap varying rapidly with frequency (the small-$\beta$ regime of the chosen ansatzes); the paper does not derive this profile from a microscopic theory, so the signatures would disappear if a real Berezinskii gap turned out to be nearly frequency-independent.

Editorial extensions

If this is right

  • A measurement of the optical conductivity of a candidate odd-frequency superconductor can look for sharper-than-BCS absorption peaks whose position tracks the gap's frequency scale.
  • The predicted sign change of $\mathrm{Im}\,\sigma$ near the peak onset would appear as a window of reduced reflection—an optical transparency window—at frequencies just below the absorption threshold.
  • Since the DOS and spectral function cannot distinguish $\Delta=\alpha$ from $\Delta=\alpha\,\mathrm{sgn}(\omega)$, any experiment that sees the cusp signature would be direct evidence for the odd-frequency symmetry.
  • The model's negative imaginary part implies the dielectric function changes character near the transparency window, potentially allowing electromagnetic wave propagation where a conventional superconductor would reflect.

Reading between the lines

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

  • A natural test would be to compute the gap function self-consistently for a microscopic model (e.g., a superconductor–ferromagnet interface) and evaluate the effective $\beta$; if realistic gaps are too flat in frequency, the predicted cusps would be suppressed.
  • The same cusp logic may apply to other dynamic pairing states, such as driven or Floquet superconductors, where a time-dependent order parameter could produce analogous conductivity features.
  • Because the effect hinges on $\Delta(\omega)\Delta^\dagger(\omega-\Omega)$ cross terms, similar frequency-sensitive transport probes—such as ac Josephson response or terahertz pump–probe—might be engineered to isolate the odd-frequency component directly.
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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

4 major / 4 minor

Summary. This paper studies the spectroscopic and optical properties of a model superconductor with odd-frequency (Berezinskii) pairing. The authors use a parabolic-band Nambu Green's function, include non-magnetic disorder, and compute the spectral function, density of states, and Kubo optical conductivity for two odd-frequency gap ansatzes (Eqs. (6) and (7)) as well as for the limiting cases Δ=α and Δ=α sgn(ω). They find that for sufficiently steep frequency dependence the spectral function and DOS show a gap and coherence peaks, and that Re σ exhibits peaks that are sharper than in the constant-gap BCS case. The imaginary part shows cusp-like features and can become negative, which the authors interpret as possible optical transparency windows. They conclude that optical conductivity can unambiguously distinguish odd-frequency from even-frequency pairing, whereas the spectral function and DOS cannot.

Significance. The manuscript is a careful application of standard many-body methods and includes a physically sensible consistency check: the spectral sum rule is used to determine the phenomenological renormalization Z(ξ). The explicit Kubo calculation with non-magnetic disorder is a useful reference calculation for a relatively unexplored problem. If the central claim were established, the paper would identify a practical optical probe for odd-frequency pairing, which would be significant. However, the claim of unambiguous distinction is currently not supported: the comparison set lacks a frequency-dependent even-frequency pairing state, and the 'transparency window' interpretation is not justified by the computed positive Re σ. The paper's own limitation section acknowledges that the gap profile must be derived self-consistently, which is exactly where the predicted signatures depend on unverified assumptions.

major comments (4)
  1. [Sec. IV B and Sec. V] The central claim that optical conductivity can unambiguously distinguish odd- and even-frequency pairing is not established. The only even-frequency case computed is the constant gap Δ=α. Because Re σ involves products Δ(ω)Δ†(ω−Ω) (see the analogous structure in Eq. (45)), a frequency-dependent even-frequency gap—for example, a strong-coupling gap with phonon structure or a sign-changing even gap with zero crossings—would produce frequency-dependent coherence factors as well. Since the spectral function and DOS depend only on |Δ(ω)|, such an even-frequency gap with the same magnitude profile would reproduce all spectroscopic data while potentially producing similar peaks, cusps, or negative Im σ. The paper contains no calculation or argument that these features are absent for a frequency-dependent even-frequency gap; without that control, the conclusion in Section V is an overclaim.
  2. [Sec. IV C and abstract] The phrase 'optical transparency windows' is an overclaim. The paper itself notes that Re σ remains positive and hence absorptive; negative Im σ changes the reactive (dielectric) response and the refractive index, but it does not remove attenuation. A region with nonzero Re σ is not transparent in the usual sense. Please either compute a relevant quantity such as transmittance or absorbance, or rephrase as 'negative imaginary conductivity' or 'anomalous reactive response'.
  3. [Secs. II B and IV] The predicted signatures are not robust because they rely on the specific strong frequency dependence of the gap ansatzes. As shown in Figs. 4(c), 5(c), 7, 9(c), 10(c), 12(c), and 13(c), for larger β the spectral gap closes and the optical peaks, cusps, and negative Im σ disappear, leaving a near-normal response. The ansatzes in Eqs. (6) and (7) are not derived from a microscopic theory, so the experimental relevance of the claimed signatures rests on an unverified assumption. This limitation is acknowledged in Section V, but it should be reflected in the abstract and conclusions, where the signatures are described as powerful and unambiguous.
  4. [Sec. IV C] The imaginary part of the conductivity is computed in the clean limit (τ→∞), whereas the real-part peaks shown in Figs. 8(c), 9, and 10 are computed for dirty superconductors (τ=10/Λ). The Kramers–Kronig relation connects Re σ and Im σ for the same system and the same parameters, so the claimed correspondence between clean-limit cusps and dirty-limit onset peaks is not directly demonstrated. Please show Im σ for the same τ used in the Re σ calculation, or state explicitly that the cusp sharpness is a clean-limit property that may be broadened by disorder.
minor comments (4)
  1. [Figures 8-13] Several figure labels appear garbled in the manuscript text (for example, axis labels in Fig. 8(c) and the captions of Figs. 9 and 10). Please ensure the final figures have clean, readable mathematical labels.
  2. [Sec. II B] The parameters α, β, and Λ are introduced with little explanation; please state their energy scales and clarify that α has units of energy and β is dimensionless in Eqs. (6) and (7).
  3. [Sec. IV C, Eq. (46)] The sign convention for ε(Ω) should be stated explicitly, and the physical interpretation of negative Im σ in terms of refraction and reflection should be accompanied by a more careful discussion or a reference addressing the conditions under which a negative imaginary part implies propagation.
  4. [References] Reference [26] is listed as 'to appear'; please update it if it has been published, or mark it clearly as an unpublished work.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optical response is computed from stated gap ansatzes with scanned parameters and an independent sum-rule constraint, not from fitted or self-referential inputs.

full rationale

I find no circular step in which a claimed prediction reduces by construction to an input. The spectral function, DOS, and optical conductivity are evaluated from the Kubo formula and the retarded/advanced Green's function built on the explicitly stated gap ansatzes in Eqs. (6) and (7). The parameters α, β, and τ are scanned over ranges rather than fitted to the predicted conductivity, and the renormalization Z(ξ) is fixed by the independent spectral sum rule IA = 1, which is a self-consistency condition and not a restatement of the optical output. The paper's limitation paragraphs candidly state that the gap frequency profile is assumed and that a rigorous self-consistent determination of Z and of the gap itself is outside its scope; this is an acknowledged assumption, not an input disguised as a prediction. Self-citations to Berezinskii, Balatsky–Abrahams, and the Linder–Balatsky review supply historical context and the SP*OT* classification but are not load-bearing for the conductivity derivation, which is carried out within the paper. The skeptic's objection that no frequency-dependent even-frequency gap was used as a control is a valid completeness concern about the strength of the distinguishing claim, but it is not a circularity: the paper never fits the even-frequency comparison to the odd-frequency result, nor does it define odd-frequency pairing in terms of the cusps or negative Im σ it predicts. The central derivation is self-contained, and the predicted signatures follow from the assumed Δ(ω) profile through the equations displayed in the text.

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

The model's conclusions rest on several input choices that the reader must accept: the effective-action Green's function, the specific odd-frequency gap ansatzes, the Z renormalization fixed by the sum rule, and the Born-approximation disorder model. None is derived from a microscopic Hamiltonian in this paper, and the paper itself notes that the qualitative conclusions depend on the frequency profile of the gap in Sec. V.

free parameters (4)
  • α (gap strength) = 0.025Λ, 0.05Λ, 0.1Λ, 0.2Λ, 0.3Λ (scanned)
    Sets the magnitude of the odd-frequency gap; determines the position (≈2|α|) and height of the conductivity peaks in Sec. IV B.
  • β (frequency-profile control) = 0.01, 0.09, 0.11, 0.3, 0.35, 1 (scanned)
    Controls the frequency dependence of the gap ansatzes; the spectral gap closes and all optical signatures disappear above β≈α/Λ (first ansatz) or β≈sqrt(α/Λ) (second ansatz).
  • τ (relaxation time) = 10/Λ, 20/Λ, 30/Λ (scanned)
    Sets the disorder scattering rate; broadens the Drude and superconducting peaks in Figs. 8 through 13.
  • Z(ξ) (renormalization factor) = determined numerically from sum rule IA=1
    Introduced to repair the sum-rule violation of the OF gap ansatzes in Appendix A; all predicted optical signatures depend on this function, and only the ξ-dependent solution is considered.
assumptions (6)
  • domain assumption Effective-action Green's function (Eq. 2) is the proper starting point for odd-frequency pairing because no mean-field Hamiltonian exists.
    The paper states in Sec. II A that 'the corresponding mean-field Hamiltonian cannot be formulated [36,37]' and adopts the effective-action retarded/advanced Green function. If this framework is wrong, all subsequent results fail.
  • ad hoc to paper The gap ansatzes in Eqs. (6) and (7) capture the frequency dependence of the odd-frequency order parameter.
    These are chosen phenomenological forms; the paper provides no microscopic derivation. The predicted signatures are shown in Secs. IV and V to disappear for large β, so the strength of the frequency dependence is load-bearing.
  • domain assumption The non-magnetic disorder self-energy is treated in the first-order Born approximation with a large Fermi surface (Eqs. 9 through 11).
    Standard for transport calculations; the paper notes it should be correct at least qualitatively because τ is a free parameter.
  • standard math The spectral sum rule IA=1 (Eq. 29) is a valid constraint and can be used to determine Z(ω,p).
    The sum rule follows from the representation of the Green function via the spectral function (Refs. 42 through 44). Using it to fix Z is a legitimate self-consistency condition, but Z is not unique and the approximation Z≈Z(ξ) is made.
  • domain assumption The Anderson theorem (Eq. 23) holds for the renormalized gap and frequency in the presence of non-magnetic disorder.
    The paper verifies that the self-energy corrections preserve the ratio ω/Δ; this is standard but assumed rather than derived microscopically.
  • standard math Kubo linear response and the spectral representation of the polarization operator (Eqs. 33 and 37) describe the optical conductivity.
    Standard linear response theory; the derivation of Re σ and Im σ follows textbook steps.

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Cite this review

Pith. "Pith review of Spectroscopic and optical response of odd-frequency superconductors." pith.science (2026). https://pith.science/paper/MPVIZXO5

@misc{pith2026190808228,
  author       = {Pith},
  title        = {Pith review of: Spectroscopic and optical response of odd-frequency superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPVIZXO5}},
  note         = {Machine review of arXiv:1908.08228}
}
read the original abstract

The optical response of superconductors with odd-frequency Berezinskii pairing is studied. By using a simple model with a parabolic dispersion law and a non-magnetic disorder, the spectral function, the electron density of states, and the optical conductivity are calculated for a few gap ansatzes. The spectral function and the electron density of states clearly reveal the gap for the Berezinskii pairing for the sufficiently strong frequency dependence of the order parameters. It is found that, similarly to the conventional BCS pairing, the odd-frequency gaps induce peaks in the real part of the conductivity, which, however, are sharper than in the BCS case. The magnitude and position of these peaks are determined by the frequency profile of the gap. The imaginary part of the optical conductivity for the Berezinskii pairing demonstrates sharp cusps that are absent in the case of the BCS superconductors. The corresponding results suggest that the Berezinskii pairing might allow for the optical transparency windows related to the onsets of the attenuation peaks in the real part of the conductivity. Thus, the study of the optical response not only provides an alternative way to probe the odd-frequency gaps but can reveal also additional features of the dynamic superconducting pairing.

Figures

Figures reproduced from arXiv: 1908.08228 by the authors.

Figure 1
Figure 1. FIG. 1: The dependence of the s-wave OF gaps ∆ = [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The coefficient [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The trace of the spectral function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The trace of the spectral function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The trace of the spectral function [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The electron DOS [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The electron DOS [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The real part of the optical conductivity for ∆ = 0 at a fe [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The real part of the optical conductivity for ∆ = [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The real part of the optical conductivity for ∆ = [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The imaginary part of the optical conductivity Im [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: The imaginary part of the optical conductivity Im [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: The imaginary part of the optical conductivity Im [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: The integral over the frequency from the spectral fu [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: The coefficient [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]

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