Negative superfluid density and spatial instabilities in driven superconductors

Andrey Grankin; Vadim Oganesyan; Victor Galitski

REVIEW 4 major objections 3 minor 2 cited by

Negative superfluid density and spatial instabilities in driven superconductors

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Periodically driven superconductors can develop a negative superfluid density, which the authors read as an instability of the homogeneous state toward spatial order-parameter textures.

desk verdict A genuinely new, internally consistent prediction of a divergent negative superfluid density under resonant Higgs drive, but the central scaling rests on a simplified number-nonconserving bath and the 'generally unstable' conclusion runs ahead of the evidence. read the letter →

arxiv 2501.08216 v1 pith:IRDLYJOV submitted 2025-01-14 cond-mat.supr-con

Andrey Grankin , Victor Galitski , Vadim Oganesyan This is my paper · ORCID

classification cond-mat.supr-con

keywords superfluiddensityHiggsmodeFloquet-Bogoliubovbandstime-dependentsuperconductivityUsadel-KeldyshequationsphaseslipsparamagneticMeissnerresponsequenchdynamics

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 a superconducting condensate whose pairing amplitude is periodically modulated, exciting the Higgs mode, can lose its phase stiffness while the drive is on. Near a drive frequency of twice the gap, the static correction to the superfluid density is negative and grows as $-\theta^2/(\Delta\gamma_{\mathrm{inel}})$, where $\theta$ is the modulation amplitude and $\gamma_{\mathrm{inel}}$ the inelastic scattering rate, so in the weak-dissipation limit the homogeneous state has $n_s<0$. A negative superfluid density means the current responds paramagnetically rather than diamagnetically, which the authors interpret as a thermodynamic instability of the uniform solution. Their spatially resolved dynamical Bogoliubov-de Gennes simulations support this reading: order-parameter inhomogeneities and phase slips appear at the same times that the uniform calculation yields negative $n_s$. If correct, strongly driven Higgs oscillations in conventional superconductors, and analogous amplitude-driven Fermi gases, should break up into a patterned state instead of oscillating homogeneously.

What carries the argument

The machinery is the Floquet problem of the time-periodic Bogoliubov-de Gennes Hamiltonian, solved in quasiclassical Usadel-Keldysh form. The central object is the set of Floquet-Bogoliubov quasienergy bands: when the drive frequency $\omega_0$ approaches $2\Delta$, the first Floquet band overlaps the zeroth band near $\xi_k=0$, producing a divergent density of Bogoliubov quasiparticle states. The superfluid density is extracted from the Keldysh current kernel, and expanding in the modulation amplitude $\theta$ to second order, the static piece picks up the divergence and scales as $n_s^{(2)}/n_s^{(0)}\propto -\theta^2/(\Delta\gamma_{\mathrm{inel}})$ at $\omega_0\approx2\Delta$, with $\gamma_{\mathrm{inel}}\to\delta$ for finite detuning $\delta=2\Delta-\omega_0>0$.

What would settle it

Solve the same Floquet-Usadel equations with a microscopically computed, frequency-dependent phonon self-energy, such as the full Eliashberg function, instead of Eq. (10), and check whether the static correction $n_s^{(2)}/n_s^{(0)}$ still diverges as $-\theta^2/(\Delta\gamma_{\mathrm{inel}})$ at $\omega_0=2\Delta$; if it saturates or changes sign, the instability is an artifact of the reservoir model. Experimentally, drive a thin-film superconductor at $\omega_0\simeq2\Delta$ and look for a paramagnetic Meissner response or spontaneous phase-slip textures while the drive is on.

Watch

Extended reading notes

Core claim

The central claim is that in a superconductor driven at or near the Higgs resonance $\omega_0\approx 2\Delta$, the static superfluid density acquires a negative correction that diverges as $-\theta^2/(\Delta\gamma_{\mathrm{inel}})$ in the limit of weak inelastic dissipation. The authors argue this is not a heating artifact: the heating scale is $\theta^2/\sqrt{\Delta\gamma_{\mathrm{inel}}}$, whereas the static correction behaves as $n_s^{(2)}/n_s^{(0)}\approx -E_{\mathrm{ex}}/\sqrt{\Delta\gamma_{\mathrm{inel}}}$, so the negative sign tracks a divergent Floquet-Bogoliubov density of states rather than quasiparticle heating. They conclude that the homogeneous time-dependent superconducting solution is generally unstable when the Floquet bands overlap, and that the system relaxes by forming a complicated spatial landscape of topological excitations, specifically phase slips. This is new because negative superfluid density is usually associated with exotic states such as Fulde-Ferrell-Larkin-Ovchinnikov phases or odd-frequency pairing; the paper finds it in an amplitude-driven conventional $s$-wave superconductor.

Load-bearing premise

The load-bearing premise is that a phonon bath can be represented by an energy-independent fermionic reservoir with one inelastic rate $\gamma_{\mathrm{inel}}$ (Eq. 10); that model explicitly breaks number conservation, and if a realistic phonon bath does not produce the same divergence at the Floquet resonance, the negative superfluid density could disappear.

Editorial extensions

If this is right

Near $\omega_0\simeq 2\Delta$, a driven homogeneous superconductor cannot persist; the uniform amplitude-oscillating state should spontaneously develop spatial inhomogeneities and phase slips.
The instability is strongest in the weak-dissipation limit $\gamma_{\mathrm{inel}}\to 0$, where the negative static correction to $n_s$ diverges, and increasing inelastic scattering suppresses it.
Finite detuning gives the same scaling with $\gamma_{\mathrm{inel}}\to\delta$, so the effect survives off resonance.
In ultracold Fermi gases with interaction-modulation-induced Higgs oscillations, analogous spatial patterning should occur rather than only uniform amplitude dynamics.
The sign of the static $n_s$ correction is a diagnostic for the stability of the time-dependent mean-field state: when it turns negative, the homogeneous solution breaks down.

Reading between the lines

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

If this mechanism is generic, other time-dependent mean-field ordered phases with resonant amplitude modes, such as charge-density waves or spin-density waves, could show an analogous Floquet-band-overlap instability; the paper studies only $s$-wave superconductors, so this is an extrapolation.
The Floquet analysis fixes $\Delta(t)=\Delta+\theta\cos\omega_0 t$ while the spatially resolved numerics are self-consistent; a full self-consistent two-dimensional treatment could reveal whether the instability selects periodic stripe patterns or proceeds as a disordered avalanche of phase slips.
Replacing the phonon bath by a frequency-independent fermionic reservoir is the main modeling step, and computing $n_s$ with a microscopically derived, frequency-dependent self-energy would test whether the $1/\gamma_{\mathrm{inel}}$ divergence survives in a realistic phonon bath.
Appendix B itself flags an unphysical surface contribution to the polarization from the simplified parabolic band with a hard cutoff, removed by renormalizing the velocity vertices; repeating the calculation with a lattice dispersion would confirm the quoted $n_s$ scaling.

Share X Bluesky LinkedIn Reddit HN

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

Two things you should know before reading. The core claim is genuinely new: at drive frequency ω0 ≈ 2Δ, second-order Floquet-Usadel theory gives a static correction n_s^(2)/n_s^(0) ∝ −θ²/(Δγ_inel), diverging as the inelastic rate goes to zero. So the homogeneous driven state can acquire a negative superfluid density — a paramagnetic response and a spatial stiffness instability. Prior work had oscillatory n_s or parametric photon emission; this divergent static component is not there. The second thing: the result is only as solid as the bath it comes from, and the authors know it. Their Eq. (10) is a number-nonconserving fermionic-reservoir self-energy, not the physical phonon self-energy of Eq. (5); they say so explicitly in the text. The 1/γ_inel divergence is a direct consequence of that Dynes-like broadening.

Credit where due. The calculation is internally consistent; the response functions are computed independently of the self-consistently determined order parameter, with no fitting to the target result. The Appendix D heating estimate (Δ^(2) ∝ θ²/√(Δγ_inel)) shows that within the model the negative n_s is not a heating artifact, and it does leave a window where the correction is small while θ²/(Δγ) ~ 1. The quench numerics using the physical phonon self-energy (Fig. 1b) also show negative n_s when 2Δ > ω_D, which is independent evidence the phenomenon is not purely a construction of Eq. (10). The 1D phase-slip simulation is a nice numerical experiment, and its time-correlation with negative n_s is suggestive. Minor: no code or data is shipped, so the 1D results are not independently checkable from the paper.

Soft spots, in proportion. The bath-model worry is real but partial: the physical phonon model does produce negative n_s in the quench, so the fear that the effect disappears with a physical bath is not established — but it is also not refuted for the Floquet steady state, because the divergent scaling is only derived for the constant-γ reservoir. The perturbative expansion in θ is pushed to the regime θ²/(Δγ) ~ 1 where it is not controlled; the sign of the leading divergence is probably right, but the quantitative boundary of the instability is not fixed by this calculation. And the 1D simulation is a quench, not a Floquet steady state, so the abstract's 'generally unstable' runs ahead of the evidence — the paper supports instability in the studied quench regimes, not a general theorem.

This paper is for the ultrafast pump-probe community and people who worry about Floquet steady states in superconductors. It deserves a serious referee: the claim is important and the derivation is checkable. My recommendation: send it to review, and steer the referee toward one question — whether a physical, number-conserving phonon self-energy regularizes the divergence — and ask the authors to scale back the 'generally unstable' phrasing.

Referee Report

4 major / 3 minor

Summary. The paper studies Higgs amplitude oscillations in BCS-type superconductors after a quench or under periodic modulation of the pairing amplitude. Using a diffusive Floquet-Usadel description with a Dynes-like inelastic self-energy, the authors report that for drive frequency ω0 ≈ 2Δ the static superfluid density acquires a negative correction scaling as n_s^(2)/n_s^(0) ∝ -θ²/(Δγ_inel), which they interpret as an instability of the homogeneous solution. They support this with an analytic estimate of heating, a diagrammatic expansion of the current response, and a self-consistent numerical simulation of a quasi-1D disordered BdG superconductor showing proliferation of phase slips when n_s becomes negative.

Significance. If correct, the result would establish a new Floquet mechanism for negative superfluid density in conventionally paired superconductors, with direct implications for time-resolved experiments and for the stability of spatially homogeneous driven superconducting states. The manuscript has genuine strengths: the Floquet-Usadel expansion is presented explicitly (Appendix A), the static correction and its γ_inel scaling are stated as concrete, falsifiable predictions, the response functions are computed without fitting the target instability, and the authors are transparent about the number-nonconserving nature of Eq. (10) and about the simplified band structure. The claimed effect is interesting and would be important for the driven-superconductivity community, but its predictive force is currently tied to a phenomenological reservoir self-energy and to a perturbative regime that may not include the instability threshold.

major comments (4)

[Floquet-Usadel description, Eq. (10)] The central scaling n_s^(2)/n_s^(0) ∝ -θ²/(Δγ_inel) is derived using the frequency-independent reservoir self-energy Eq. (10), which the authors themselves state 'effectively describes coupling to a non-superconducting fermionic reservoir' and 'explicitly breaks the fermion number conservation.' This is not a minor modeling detail: the 1/γ_inel divergence comes from filling states at the Floquet resonance with a constant quasiparticle broadening, whereas the physical phonon self-energy Eq. (5) contains an Eliashberg spectral function α²F(ω) and does not provide a constant, number-nonconserving broadening, especially when 2Δ > ω_D makes single-phonon recombination kinematically forbidden. A concrete and necessary test is to solve Eq. (9) with Eq. (5), or with an energy-dependent γ(ω) extracted from α²F, and to check whether the negative static correction survives; until this is done, the negative-superfluid-density result is a property of the toy reservoir rather than of the phonon-coupled model advertised in the abstract.
[Perturbative expansion, Appendix A] The second-order expansion in θ is internally consistent, but it is used outside its controlled regime. The condition for the static correction to become comparable to the equilibrium value is θ²/(Δγ_inel) ~ 1, i.e. γ_inel ~ θ²/Δ. Near the Floquet resonance the iterative solution of Eq. (A1) produces first-order sideband Green functions of order θ/γ_inel ~ Δ/θ ≫ 1, so the effective expansion parameter is not small even though θ/Δ is small. Thus negative n_s appears precisely where the perturbative series has already broken down. The manuscript needs either a nonperturbative Floquet solution, or a demonstration that a different small parameter controls the series at the instability threshold.
[Appendix D] The heating estimate is not independent evidence for the physical mechanism. The quoted scaling Δ^(2) ∝ θ²/√(Δγ_inel) and E_ex = θ²/√(Δγ_inel) are obtained from the same constant-γ_inel reservoir (Eq. (D1) introduces γ_inel as a pair-breaking parameter), so it cannot exclude a bath-induced regularization of the 1/γ_inel pole. Moreover, in the weak-dissipation limit the heating estimate itself diverges as γ_inel → 0, so the steady state is defined only within the same model. A physical phonon bath could suppress or restructure the singularity differently, and that scenario must be explicitly excluded before the conclusion is drawn.
[Quasi-one-dimensional case, Fig. 3] The 1D simulation in Appendix C is a quench protocol (a sudden change of λ from 1.2 to 2) on a disordered tight-binding superconductor, not a Floquet steady state at ω0 ≈ 2Δ, and it does not include the reservoir self-energy Eq. (10). It therefore demonstrates that a sufficiently strong interaction quench can produce negative n_s and phase slips, but it cannot validate the specific Floquet-resonance mechanism or the abstract's 'generally unstable' conclusion. The extrapolation from a quench to periodic driving should either be stated as a conjecture or supported by a periodically driven spatial simulation.

minor comments (3)

[Floquet-Usadel section, figure references] The figure references in the Floquet-Usadel section are inconsistent: the text below Eq. (10) cites Fig. 3(b), Fig. 3(a), and Fig. 3(c) for the time-averaged LDOS, the Floquet bands, and n_s^(1)(ω), but those panels appear in Fig. 2; the in-text references should be corrected to avoid confusion with the quasi-1D simulation in Fig. 3.
[Eq. (10)] The notation in the Keldysh block of Eq. (10) is typeset ambiguously; the argument of the hyperbolic tangent should be written explicitly as tanh(βω/2) rather than as tanh βω/2 with the factor attached to τ3, so that the matrix structure is unambiguous.
[Appendix B] The renormalized velocity vertex θ(k) is introduced to remove the ultraviolet surface term, but the discussion would benefit from one sentence explaining why this renormalization does not change the second-order Floquet response in the energy window that controls the negative n_s correction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the negative superfluid density is a computed consequence of the stated Floquet-Usadel model, not a re-fitted input or a self-citation chain.

full rationale

The central prediction n_s^(2)/n_s^(0) ∝ −θ^2/(Δγ_inel) is obtained by expanding the Floquet-Usadel equation in powers of θ and evaluating the DC supercurrent from the Keldysh Green's functions. The parameter γ_inel is introduced as the inelastic rate in the explicitly stated self-energy of Eq. (10); it is not fitted to the negative superfluid density, and the 1/γ_inel divergence is an algebraic consequence of the Floquet-resonant density of states rather than an input. The 1D BdG simulation in Appendix C is a separate numerical experiment with its own self-consistency equation, and the correspondence between negative n_s and defect proliferation is established numerically rather than imported from a prior result. The only self-citation, ref. [8] (Grankin and Galitski), appears in a list of quench-generation proposals and does not enter the Floquet-Usadel derivation, the self-energy model, or the stability conclusion. The authors themselves flag the number-nonconserving character of the reservoir self-energy in Eq. (10) and the heating estimate in Appendix D uses the same reservoir; this is a legitimate physical-robustness limitation, but it is not circularity because the target result is not assumed in the model. Likewise, the Appendix B band-structure caveat is handled by an explicit renormalization of velocity vertices, not by a hidden redefinition of the result. Overall, the derivation is self-contained with respect to its stated assumptions, and no load-bearing step reduces to its own input by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central result rests on the approximate reservoir self-energy (Eq. 10), the diffusive quasiclassical limit, and the renormalization of high-energy velocity vertices. These are not derived from the original BCS-Holstein model but are imposed to make the calculation tractable; the first is ad hoc, the second is a standard domain assumption, and the third is an unphysical contribution removed by hand.

free parameters (4)

γ_inel
Effective inelastic scattering rate introduced via the reservoir self-energy Eq. (10); controls the divergence n_s^(2) ~ -θ^2/(Δγ_inel), so the central prediction of negative ns depends on its magnitude.
θ
Amplitude of the gap modulation Δ(t)=Δ+θ cos(ω0 t); the negative correction scales as θ^2, so the effect grows with drive strength.
ωD
Debye cutoff in α2F model; the negative-ns regime occurs when 2Δ > ωD, so the phonon band width is a control parameter.
λ_i, λ_f
Quench parameters in the BCS-Holstein model; chosen to realize the strongly-coupled regime where ns becomes negative.

assumptions (5)

ad hoc to paper The phonon bath can be replaced by an energy-independent fermionic reservoir self-energy (Eq. 10) with rate γ_inel, which breaks fermion number conservation.
This is a strong phenomenological replacement of the full Eliashberg self-energy Eq. (5); the singular Floquet response is computed with this approximate bath. The authors acknowledge it 'effectively describes coupling to a non-superconducting fermionic reservoir'.
domain assumption The quasiclassical (Usadel) limit is valid: τ_el is the shortest timescale and the Green's function depends only on direction n_k.
Needed to reduce Eq. (4) to Eq. (9).
ad hoc to paper A two-dimensional parabolic band with cutoff ξ∈[-E_F,E_F] and a renormalized velocity vertex (θ(k) cutoff) captures the low-energy physics.
Appendix B shows the unrenormalized band produces an unphysical surface contribution to the paramagnetic response, which is removed by hand via a high-energy cutoff function.
domain assumption The phonon propagator remains in equilibrium at the initial temperature and back-action of driven quasiparticles on phonons is neglected.
Stated in the text: 'we ignore the back-action effects on the phonon propagator and assume it is in equilibrium state at the initial temperature.'
standard math The BCS mean-field treatment with self-consistent order parameter is valid for the time-dependent dynamics.
The Gor'kov equations and the self-consistency condition for Δ(t) assume mean-field factorization of the pairing interaction.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Negative superfluid density and spatial instabilities in driven superconductors." pith.science (2026). https://pith.science/paper/IRDLYJOV

@misc{pith2026250108216,
  author       = {Pith},
  title        = {Pith review of: Negative superfluid density and spatial instabilities in driven superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IRDLYJOV}},
  note         = {Machine review of arXiv:2501.08216}
}

read the original abstract

We consider excitation of Higgs modes via the modulation of the BCS coupling within the Migdal-Eliashberg-Keldysh theory of time-dependent superconductivity. Despite the presence of phonons, which break integrability, we observe Higgs amplitude oscillations reminiscent of the integrable case. The dynamics of quasiparticles follows from the effective Bogolyubov-de Gennes equations, which represent a Floquet problem for the Bogoliubov quasiparticles. We find that when the Floquet-Bogoliubov bands overlap, the homogeneous solution formally leads to a negative superfluid density, which is no longer proportional to the amplitude of the order parameter. This result indicates an instability, which we explore using spatially-resolved BdG equations. Spontaneous appearance of spatial inhomogeneities in the order parameter is observed and they first occur when the superfluid density becomes unphysical. We conclude that the homogeneous solution to time-dependent superconductivity is generally unstable and breaks up into a complicated spatial landscape via an avalanche of topological excitations.

Figures

Figures reproduced from arXiv: 2501.08216 by the authors.

**Figure 2.** Dynamic superfluid density within the Usadel [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Quench dynamics a 1-dimensional disordered super [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Diagrams contributing to the dynamical superfluid [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Higgs-mode-induced instability and kinetic inductance in strongly dc-biased dirty-limit superconductors
cond-mat.supr-con 2025-02 conditional novelty 8.0 of 10

In strongly dc-biased dirty-limit superconductors, the Higgs mode creates a frequency window in which the uniform superflow is unstable, and can boost kinetic inductance by nearly two orders of magnitude.
Spatially resolved collective modes in d-wave superconductors
cond-mat.supr-con 2026-01 conditional novelty 6.0 of 10

In d-wave superconductors, the amplitude mode's peak frequency depends on momentum direction but its 1/t² decay does not; the phase mode softens at finite temperature due to nodal quasiparticles.

Reference graph

Works this paper leans on

28 extracted references · 23 canonical work pages · cited by 2 Pith papers

[1]

Meissner and R

W. Meissner and R. Ochsenfeld, Naturwissenschaften21, 787 (1933)

work page 1933
[2]

Altland and B

A. Altland and B. D. Simons, Condensed matter field theory (Cambridge university press, 2010)

2010
[3]

jt6bTI5axhpMtjk3vP5pQp9zalI=

and Larkin-Ovchinnikov [4] states, which occur in su- perconductors in the presence of a magnetic field or spin imbalance in neutral superfluids. Negative values ofns were also theoretically predicted for the odd-frequency superconductingstates, implyingtheyarethermodynam- ically unstable [5]. Many such scenarios are character- ized by emergent Bogolyubov...

arXiv 2025
[4]

A. I. Larkin, Zh. Eksp. Teor. Fiz47, 1136 (1964)

work page 1964
[5]

Coleman, E

P. Coleman, E. Miranda, and A. Tsvelik, Physical review letters 70, 2960 (1993)

work page 1993
[6]

/O1t5EH2yQYOmNnalDNVkBzdlf0=

account for such a mechanism leading to the time- dependent ns as shown in Fig. 1 (b, c). We first discuss the phonon-induced scattering case. By varying the Debye cut-off frequency for a fixed electron-phonon coupling strength λel-ph, we observe a qualitatively different behavior ofns. In particular, when ωD is sufficiently large, both ns(t) and ∆ (t) ar...

work page
[7]

Fulde and R

P. Fulde and R. A. Ferrell, Physical Review135, A550 (1964)

1964
[8]

The time-dependent superfluid density canbefoundfromEqs.(7, 8) j (t) ≡ −ns(t) m A

for simplicity. The time-dependent superfluid density canbefoundfromEqs.(7, 8) j (t) ≡ −ns(t) m A. Beforedis- cussing the results of a complete numerical calculation, we note that in the absence of disorder or phonon scatter- ings, the superfluid density does not depend on time to lowest order in∆/EF. This is follows from the analysis of the BdG Hamiltoni...

work page

Show all 28 references

[9]

E. A. Yuzbashyan, O. Tsyplyatyev, and B. L. Altshuler, Physical review letters96, 097005 (2006)

2006
[10]

Barankov, L

R. Barankov, L. Levitov, and B. Spivak, Physical review letters 93, 160401 (2004)

2004
[11]

Grankin and V

A. Grankin and V. Galitski, arXiv preprint arXiv:2312.13391 (2023)

2023 arXiv
[12]

Kemper, M

A. Kemper, M. Sentef, B. Moritz, J. Freericks, and T. Devereaux, Physical Review B92, 224517 (2015)

2015
[13]

Matsunaga, N

R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and R. Shimano, Science345, 1145 (2014)

2014
[14]

Buzzi, G

M. Buzzi, G. Jotzu, A. Cavalleri, J. I. Cirac, E. A. Dem- ler, B. I. Halperin, M. D. Lukin, T. Shi, Y. Wang, and D. Podolsky, Physical Review X11, 011055 (2021)

2021
[15]

Dzero, E

M. Dzero, E. Yuzbashyan, and B. Altshuler, Europhysics Letters 85, 20004 (2009)

2009
[16]

Chattopadhyay, C

S. Chattopadhyay, C. J. Eckhardt, D. M. Kennes, M. A. Sentef, D. Shin, A. Rubio, A. Cavalleri, E. A. Dem- ler, and M. H. Michael, arXiv preprint arXiv:2303.15355 (2023)

2023 arXiv
[17]

Schüler, D

M. Schüler, D. Golež, Y. Murakami, N. Bittner, A. Her- rmann, H. U. Strand, P. Werner, and M. Eckstein, Com- puter Physics Communications257, 107484 (2020)

2020
[18]

Barankov and L

R. Barankov and L. Levitov, Physical Review A— Atomic, Molecular, and Optical Physics 73, 033614 (2006)

2006
[19]

Qin and W

T. Qin and W. Hofstetter, Physical Review B96, 075134 (2017)

2017
[20]

A. I. Larkin and Y. N. Ovchinnikov, Sov Phys JETP28, 1200 (1969)

1969
[21]

Larkin and Y

A. Larkin and Y. Ovchinnikov, Sov. Phys. JETP41, 960 (1975)

1975
[22]

Silaev, Physical Review B102, 180502 (2020)

M. Silaev, Physical Review B102, 180502 (2020)

2020
[23]

Virtanen, T

P. Virtanen, T. T. Heikkilä, F. S. Bergeret, and J. C. Cuevas, Physical review letters104, 247003 (2010)

2010
[24]

jt6bTI5axhpMtjk3vP5pQp9zalI=

R. Dynes, J. Garno, G. Hertel, and T. Orlando, Physical Review Letters 53, 2437 (1984). <latexit sha1_base64="jt6bTI5axhpMtjk3vP5pQp9zalI=">AAAB6XicbVDLSgNBEOz1GeMr6tHLYBD0EnYDPo5BLx6jmAckS5idzCZDZmeXmV4hLPkDLx4U8eofefNvnCR70MSChqKqm+6uIJHCoOt+Oyura+sbm4Wt4vbO7t5+6eCwaeJUM95gs...

1984
[25]

Ojajärvi, T

R. Ojajärvi, T. T. Heikkilä, P. Virtanen, and M. Silaev, Physical Review B103, 224524 (2021)

2021
[26]

Belzig, F

W. Belzig, F. K. Wilhelm, C. Bruder, G. Schön, and A. D. Zaikin, Superlattices and microstructures25, 1251 (1999)

1999
[27]

Appendix A: Floquet-Usadel equations In this section we provide details on the solution of the Usadel equation

A.Moor, A.F.Volkov, andK.B.Efetov,Physicalreview letters 118, 047001 (2017). Appendix A: Floquet-Usadel equations In this section we provide details on the solution of the Usadel equation. By transforming Eq. (9) to Fourier space we get − iωˆτ3ˇgω,ω ′ + iω′ˇgω,ω ′ ˆτ3 = Z dΩ {...

2017
[28]

jt = i σN 8 A Z dsTr{ˆτ3ˆgt,s ˆτ3ˆgs,t}K , (A2) 6 where we used the normalization condition of the quasi- classical GF

Supercurrent Once the Green’s function is determined, we can find its effect on the supercurrent [23]. jt = i σN 8 A Z dsTr{ˆτ3ˆgt,s ˆτ3ˆgs,t}K , (A2) 6 where we used the normalization condition of the quasi- classical GF. We can explicitly write the Keldysh com- ponent of the...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Meissner and R

W. Meissner and R. Ochsenfeld, Naturwissenschaften21, 787 (1933)

work page 1933

[2] [2]

Altland and B

A. Altland and B. D. Simons, Condensed matter field theory (Cambridge university press, 2010)

2010

[3] [3]

jt6bTI5axhpMtjk3vP5pQp9zalI=

and Larkin-Ovchinnikov [4] states, which occur in su- perconductors in the presence of a magnetic field or spin imbalance in neutral superfluids. Negative values ofns were also theoretically predicted for the odd-frequency superconductingstates, implyingtheyarethermodynam- ically unstable [5]. Many such scenarios are character- ized by emergent Bogolyubov...

arXiv 2025

[4] [4]

A. I. Larkin, Zh. Eksp. Teor. Fiz47, 1136 (1964)

work page 1964

[5] [5]

Coleman, E

P. Coleman, E. Miranda, and A. Tsvelik, Physical review letters 70, 2960 (1993)

work page 1993

[6] [6]

/O1t5EH2yQYOmNnalDNVkBzdlf0=

account for such a mechanism leading to the time- dependent ns as shown in Fig. 1 (b, c). We first discuss the phonon-induced scattering case. By varying the Debye cut-off frequency for a fixed electron-phonon coupling strength λel-ph, we observe a qualitatively different behavior ofns. In particular, when ωD is sufficiently large, both ns(t) and ∆ (t) ar...

work page

[7] [7]

Fulde and R

P. Fulde and R. A. Ferrell, Physical Review135, A550 (1964)

1964

[8] [8]

The time-dependent superfluid density canbefoundfromEqs.(7, 8) j (t) ≡ −ns(t) m A

for simplicity. The time-dependent superfluid density canbefoundfromEqs.(7, 8) j (t) ≡ −ns(t) m A. Beforedis- cussing the results of a complete numerical calculation, we note that in the absence of disorder or phonon scatter- ings, the superfluid density does not depend on time to lowest order in∆/EF. This is follows from the analysis of the BdG Hamiltoni...

work page