Post-quench gap dynamics of two-band superconductors

Michael Sch\"utt; Peter P. Orth; Rafael M. Fernandes; Tianbai Cui

REVIEW 3 major objections 3 minor 55 references

Post-quench gap dynamics of two-band superconductors

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that after a quench of the pairing interaction, a two-band BCS superconductor's gap oscillations damp as $t^{-3/2}$ rather than the single-band $t^{-1/2}$, and beat at two frequencies set by the two asymptotic gap values.

desk verdict A solid analytic result—two-band quench gap tails decay as t^-3/2 instead of t^-1/2—with a real caveat: the key steady-state ansatz is unproven and numerically tested only in a narrow regime. read the letter →

arxiv 1908.06125 v1 pith:WQ7RWVZJ submitted 2019-08-16 cond-mat.supr-con cond-mat.str-el

Tianbai Cui , Michael Schütt , Peter P. Orth , Rafael M. Fernandes This is my paper · ORCID

classification cond-mat.supr-concond-mat.str-el

keywords two-bandsuperconductorspost-quenchdynamicsBCSgaposcillationscollisionlessdampingpower-lawdecaypseudospinLaplace-spaceperturbationtheorybeatingin

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 what happens to the superconducting gap immediately after the pairing interaction is suddenly changed ('quenched') in a two-band superconductor, before any coupling to the lattice or quasiparticle scattering has had time to act. The paper claims that in the collisionless regime the gap oscillations relax with a power-law envelope $t^{-3/2}$, significantly faster than the well-known $t^{-1/2}$ tail of a single-band BCS superconductor, and that they beat because the two bands settle at two different asymptotic gap values $\Delta_{1,\infty}$ and $\Delta_{2,\infty}$. This matters because two-band superconductivity is realized in MgB$_2$, iron-based superconductors, and SrTiO$_3$, so the decay law and the beating pattern are concrete signatures of multi-band pairing in ultrafast pump-probe experiments.

What carries the argument

The argument is carried by the pseudospin representation of the BCS model, in which each Cooper pair is a spin precessing in a self-consistent magnetic field set by the gap, and by a Laplace-space perturbation theory that expands the equations of motion around the long-time non-equilibrium steady state rather than around equilibrium. The central analytic object is the function $\Upsilon(\Delta,x)=v_f\frac{\sqrt{x^2+1/\Delta^2}\,\arccos(\sqrt{x^2+1/\Delta^2})}{\sqrt{1-(1+x^2)/\Delta^2}}$, whose branch cuts in the complex-frequency plane control the long-time decay. In the single-band case the Laplace-space response has a $1/\sqrt{\epsilon}$ branch point at $s=2i\Delta_\infty$, which produces the $t^{-1/2}$ tail; in the two-band case, inter-band coupling converts this to a $\sqrt{\epsilon}$ singularity at two branch points, $s=2i\Delta_{1,\infty}$ and $s=2i|\Delta_{2,\infty}|$, producing the faster $t^{-3/2}$ tail. To close the equations, the paper introduces an ansatz for the steady-state pseudospin distribution, $S^x_{\alpha,\infty}/\Delta_{\alpha,\infty} = (\tilde\Delta_{\alpha,f}/\tilde\Delta_{\bar\alpha,f})/(2\sqrt{\varepsilon^2+\Delta_{\alpha,f}^2})$, which satisfies the gap constraint and is used with the Laplace final-value theorem to determine $\Delta_{\alpha,\infty}$ self-consistently.

What would settle it

Solve the full time-dependent two-band BCS equations for a case with unequal densities of states and non-zero intra-band pairing, for example $\eta=0.5$ and $r=1$, over times $t \gtrsim 50/\Delta_{1,\infty}$, and fit the envelope of $\Delta_\alpha(t)-\Delta_{\alpha,\infty}$: a $t^{-3/2}$ decay confirms the claim, whereas a $t^{-1/2}$ or different power law refutes it. An experimental counterpart would be time-resolved terahertz spectroscopy of a two-band superconductor film, where the asymptotic envelope of the coherent gap oscillation should fall as $t^{-3/2}$ rather than $t^{-1/2}$.

Watch

Extended reading notes

Core claim

The central claim is that the long-time gap dynamics after a weak interaction quench is not a sum of two independent single-band responses. For a fully gapped $s$-wave two-band superconductor, the deviation of each gap from its asymptotic value has the form $\delta_\alpha(t) \sim A_\alpha \sin(2\Delta_{1,\infty}t+\pi/4)/(\Delta_{1,\infty}t)^{3/2} + B_\alpha \sin(2|\Delta_{2,\infty}|t-\pi/4)/(|\Delta_{2,\infty}|t)^{3/2}$, with analogous terms in the companion band (Eqs. 53a, 53b). The paper argues that the exponent change from $t^{-1/2}$ to $t^{-3/2}$ is generic for two-band systems and independent of quench amplitude within the damped-oscillation regime: inter-band coupling splits the single branch point of the Laplace-space response into two square-root branch points, which shifts the asymptotic decay by one power of time. The same analysis reproduces the exact single-band solution in the $\eta\to 1$ limit and matches numerical solutions of the two-band equations for weak quenches with $\eta=0.8$.

Load-bearing premise

The entire derivation rests on an assumed formula for how the two bands share pairing amplitude after the system has relaxed: the long-time steady-state spin distribution is taken to have the same energy dependence as the final equilibrium distribution, with the two gaps rescaled by the ratio of their asymptotic values; if this distribution is not what generic two-band systems actually reach, the predicted $t^{-3/2}$ exponent would be replaced by something else.

Editorial extensions

If this is right

Multi-band gap dynamics cannot be decomposed into independent single-band responses: the faster decay is generated by inter-band dephasing of pairs from the two bands.
A clear spectral fingerprint is two oscillation frequencies $2\Delta_{1,\infty}$ and $2|\Delta_{2,\infty}|$; when the asymptotic gaps are close, the gap trace shows beating at their difference.
The same branch-point splitting argument implies that superconductors or superfluids with more than two bands should also exhibit the $t^{-3/2}$ collisionless decay.
Because the self-consistent Laplace method reproduces the exact single-band solution, it can be applied to quenches in non-integrable fully gapped states such as $s+is$ or $s+id$ pairing, where no exact solution exists.
Within the damped-oscillation regime the exponent is independent of quench amplitude, so observing $t^{-3/2}$ rather than $t^{-1/2}$ would identify multi-band pairing even without knowing the exact quench strength.

Reading between the lines

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

A testable extension: terahertz pump-probe studies of a two-band film, e.g., MgB$_2$ or an iron-based superconductor, could extract the envelope exponent from the delay trace; the steeper $t^{-3/2}$ tail would distinguish two-band pairing from the single-band $t^{-1/2}$ response seen in simple metallic films.
In a cold-atom Fermi gas with two components, crossing a pairing resonance should show the same branch-point splitting, providing a clean platform to vary the density-of-states ratio and quench amplitude continuously.
The steady-state pseudospin distribution behind the ansatz of Eq. (32) could be extracted from the same numerical solutions at long times; checking it directly would show whether the $t^{-3/2}$ exponent survives for generic intra-band couplings or only near the pure inter-band limit.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The headline result is genuine: for two-band s-wave superconductors, the long-time gap oscillation tail is t^-3/2, not the single-band t^-1/2. The mechanism is clean—interband coupling splits the Laplace-space branch point, turning a 1/sqrt singularity into sqrt, and they confirm it numerically with the full prefactors and beating pattern. The methodological step is also worth something: expanding around a self-consistently determined non-equilibrium steady state, rather than the final equilibrium state, and showing it recovers the exact single-band solution. The derivation is explicit and the appendices give all prefactors. Credit where due—this is reproducible, honest work.

The soft spot is exactly what the stress-test flags. The steady-state pseudospin distribution, Eq. (32), is an unproven ansatz. It satisfies the gap constraint and reduces correctly in the eta->1 single-band limit, but it is not derived from the equations of motion. The entire t^-3/2 branch-point structure depends on the small-epsilon behavior of that ansatz. If the true distribution vanishes differently at epsilon=0, the exponent changes. Numerical validation covers eta=0.8, r=0, two weak quenches (vi/vf=0.95 and 0.9), and one r=0.5 trace. That is enough to make the claim plausible for weak quenches in that regime, but not enough for the abstract's "wide range of quench amplitudes." The paper itself restricts to phase II, so the abstract oversells.

Is this fatal? No. The central physics—two-band dephasing gives faster collisionless relaxation and beating at two gap frequencies—is robust. The precise t^-3/2 exponent is conditional on the ansatz, but the numerics strongly support it for the tested regime. I would want the ansatz tested over a broader eta and r range, or better, derived from a controlled approximation, before treating t^-3/2 as a universal two-band law. For phase II, r=0, it holds up.

This paper is for anyone working on pump-probe or quench dynamics in multiband superconductors; it gives a concrete observable that distinguishes multi-band pairing. It deserves a serious referee, and I would accept it for peer review with the expectation that the authors either extend the numerics or temper the claim in the abstract.

Referee Report

3 major / 3 minor

Summary. This paper analyzes the collisionless post-quench dynamics of two-band fully gapped s-wave BCS superconductors. The authors generalize the Volkov-Kogan Laplace-space method by linearizing the pseudospin equations around the self-consistently determined non-equilibrium steady state rather than the final equilibrium state. They obtain closed-form expressions for the gap deviations in Laplace space, derive long-time asymptotic formulas for Δ1(t) and Δ2(t), and claim that multi-band coupling changes the algebraic decay from t^-1/2 to t^-3/2 and produces beating at frequencies 2Δ1,∞ and 2|Δ2,∞|. The analytic results are compared with Runge-Kutta solutions for η=0.8, r=0 and weak quenches, and the single-band limit is recovered exactly.

Significance. The claimed t^-3/2 Landau-damping tail is a clear, falsifiable prediction that distinguishes two-band from single-band BCS dynamics, and the steady-state expansion is a useful technique for non-integrable quench problems. The manuscript is carefully written: the Laplace-space chain from Eqs. (15) to (25) is logical, the final-value-theorem determination of Δα,∞ is self-consistent, the single-band limit reproduces the exact Lax-vector result, and the analytic formulas in Appendix D give full prefactors. The main concern is that the central power law rests on the unproven steady-state pseudospin ansatz (32), with numerical support only in a limited parameter region.

major comments (3)

[Sec. IV C, Eq. (32)] The steady-state pseudospin distribution S^x_{α,∞}/Δα,∞ is introduced as an ansatz and is not derived from the equations of motion or from conserved quantities. This distribution enters Φ^∞_α(s) via Eq. (33), and the small-ε behavior of Φ^∞ near the branch points is what converts the single-band 1/√(y-1) singularity into the √(y-1) singularity that yields t^-3/2. The numerical justification in Fig. 8 covers r=0, η=0.8 and two weak quenches, which is not enough to establish the generic small-ε form of S^x_{α,∞}. Please either provide a derivation of Eq. (32) or extract the steady-state pseudospin configuration from long-time numerics for a range of η, r, and quench amplitudes to demonstrate that the assumed √ε branch-point structure is generic.
[Appendix B, Eq. (B2)] As printed, the asymptotic expressions for Im[1/D(y)] and Im[Υ(·,y)/D(y)] near y→1 contain 1/√(y-1), which would produce a t^-1/2 tail if used in Eq. (49), in contradiction with the text's statement in Sec. IV D that Im[zδ(z)] ~ √(y-1). Direct expansion of D(y) from Eq. (51) around y=1 gives D = (1/κ)Υ(Δ̃2,f,κ) + O(√(y-1)), hence Im[1/D] ∝ √(y-1). Please correct the appendix and check the prefactor signs in the expansions, since this is the load-bearing step for Eqs. (53).
[Abstract and Sec. V] The claim that the method applies to a 'wide range of quench amplitudes' and that the t^-3/2 damping is 'independent on the quench amplitude' is not supported by the manuscript's own scope, since Sec. IV C restricts the analysis to weak quenches in phase II and Sec. IV B sets r=0, with the r≠0 behavior inferred from a single trace in Fig. 4. Please restate the result as a weak-quench, r=0 finding, with the r dependence as an open question, or provide additional numerical evidence for the claimed generality.

minor comments (3)

[Sec. II B] The statement that quenches of r 'are expected to not lead to qualitative changes' is not demonstrated, since the analytic derivation assumes r=0; this should be explicitly marked as an expectation rather than a result.
[Eq. (49)] The use of cosh(2Δ1,∞zt) in Eq. (49) is potentially confusing because the integration runs along z=iy; writing the factor as cos(2Δ1,∞yt) would make the inversion transparent.
[Fig. 4] Please state in the caption which gap component is plotted in Fig. 4B and confirm that the quench parameters are the same as in Fig. 3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the t^{-3/2} law follows from a stated steady-state ansatz and branch-point analysis, benchmarked numerically and against the exact single-band solution.

full rationale

The central prediction is not obtained by fitting a decay exponent or by renaming an input. The steady-state pseudospin distribution in Eq. (32) is introduced explicitly as an ansatz ('Based on this similarity, we propose the following ansatz') and is constrained by the gap equation, Eq. (29); it contains no free parameter fitted to the target t^{-3/2} decay. The Laplace-space solution, Eqs. (18)-(25), is derived from the linearized equations of motion, and the long-time exponent arises from the asymptotic expansion of Im[z delta(z)] near branch points in Appendix B and the integral in Eq. (52). The exponent is not equivalent to the ansatz by construction: it emerges because the other band's function Upsilon is regular at the first band's branch point, giving square-root rather than inverse-square-root behavior, a two-band structural effect. The ansatz is validated a posteriori against independent numerical solutions in Fig. 8 and against the exact single-band Lax-vector solution in Fig. 5, with the paper itself noting after Eq. (40) that perfect agreement does not necessarily imply that the non-equilibrium distribution function is exact. Self-citations (Refs. 24, 44, 45, 55) are incidental and not load-bearing, and no uniqueness theorem is imported from the authors' prior work. The main limitation, that the ansatz is unproven for generic parameters and the numerics cover mainly eta=0.8, r=0, and weak quenches, is a correctness and robustness concern rather than a circularity; the derivation chain does not reduce to its own inputs.

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

The paper introduces one load-bearing ad hoc assumption, the steady-state pseudospin ansatz (32), plus the numerically motivated equality of gap ratios and the r=0 restriction. No new physical entities or empirically fitted parameters are introduced. The model parameters (vi, vf, η, r) are chosen, not fitted, and the asymptotic gaps are solved self-consistently.

assumptions (5)

domain assumption BCS mean-field pseudospin model exactly captures the collisionless dynamics (Eqs. 5-12).
The paper uses the reduced BCS Hamiltonian and treats the mean-field gap equation as exact in the BCS limit; this is standard in the field but not derived within the paper.
domain assumption Post-quench steady state exists and the Laplace final value theorem applies (lim_{s→0} s δ(s) = 0).
Used in Sec. IV C to fix Δ_{α,∞}; this formally requires the long-time limit to be a constant, i.e. phase II, and the paper notes it breaks down in phase III.
ad hoc to paper Ansatz (32): S^x_{α,∞}/Δ_{α,∞} = (Δ̃_{α,f}/Δ̃_{ᾱ,f})/(2√(ε²+Δ_{α,f}²)) for the steady-state pseudospin distribution.
Proposed in Sec. IV B to satisfy the constraint (29) from the gap equation; validated by numerics at η=0.8 and by the single-band exact solution, but not proven for generic two-band parameters.
ad hoc to paper Δ̃_{1,f}=Δ̃_{2,f} (equal asymptotic-to-final gap ratios) to simplify Eq. (33).
Reported in Sec. IV C 2 as a numerical observation with deviation below 0.01% for r=0; used to set the prefactor of Eq. (33) to 1.
ad hoc to paper r=0 restriction, with behavior for r≠0 inferred from one numerical comparison (Fig. 4).
The analytical derivation sets intra-band interaction to zero; the claim that the t^-3/2 law is insensitive to r rests only on Fig. 4 (r=0.5).

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Cite this review

Pith. "Pith review of Post-quench gap dynamics of two-band superconductors." pith.science (2026). https://pith.science/paper/WQ7RWVZJ

@misc{pith2026190806125,
  author       = {Pith},
  title        = {Pith review of: Post-quench gap dynamics of two-band superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQ7RWVZJ}},
  note         = {Machine review of arXiv:1908.06125}
}

abstract

Recent experimental progress in the fields of cold quantum gases and ultrafast optical spectroscopy of quantum materials allows to controllably induce and probe non-adiabatic dynamics of superconductors and superfluids. The time-evolution of the gap function before relaxation with the lattice is determined by the superposition of coherently evolving individual Cooper pairs within the manifold of the Bardeen-Cooper-Schrieffer (BCS) wavefunction. While dynamics following an abrupt quench of the pairing interaction strength in the single-band BCS model has been exactly solved due to the integrability of the model, the dynamics of post-quench multi-band superconductors remain under scrutiny. Here, we develop a generalization of the Volkov-Kogan Laplace-space perturbative method that allows us to determine the non-adiabatic gap dynamics of two-band fully gapped superconductors for a wide range of quench amplitudes. Our approach expands the long-time dynamics around the steady-state asymptotic value of the gap, which is self-consistently determined, rather than around the equilibrium value of the gap. We explicitly demonstrate that this method recovers the exact solution of the long-time gap dynamics in the single-band case and perfectly agrees with a numerical solution of the two-band model. We discover that dephasing of Cooper pairs from different bands leads to faster collisionless relaxation of the gap oscillation with a power-law of $t^{-3/2}$ instead of the well-known $t^{-1/2}$ behavior found in the single-band case. Furthermore, the gap oscillations display beating patterns arising from the existence of two different asymptotic gap values. Our results have important implications to a variety of two-band superconductors driven out of equilibrium, such as iron-based superconductors, MgB$_{2}$, and SrTiO$_{3}$.

Figures

Figures reproduced from arXiv: 1908.06125 by the authors.

**Figure 2.** FIG. 2. (A) Schematics of the two bands and the interactions [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Numerical results for the gap oscillations in two-band superconductors. (A)-(D) are the results for an interaction [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. (A) Gap oscillations for the case of inter-band pairing [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Comparison of ∆ [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 7.** Figure 7: FIG. 7. Non-analyticity of the gaps in Laplace-space along the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Comparison between the numerical solution of the gap dynamics and the analytical approximation in Eqs. (53a) and [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. Integration contour in the complex Laplace space. [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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