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In a clean d-wave superconductor with Coulomb interaction, the transverse phase mode matches the s-wave plasma mode at zero temperature but softens and broadens at finite temperature, while the longitudinal amplitude mode is a direction-dep

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2026-08-03 10:29 UTC pith:DD4FA7LB

load-bearing objection Solid, cross-checked derivation of finite-q collective-mode dispersions in a clean d-wave superconductor; the headline finite-T damping claim is asserted, not computed, and the clean-model caveat is real but openly acknowledged. the 3 major comments →

arxiv 2601.09782 v1 pith:DD4FA7LB submitted 2026-01-14 cond-mat.supr-con cond-mat.str-el

Spatially resolved collective modes in d-wave superconductors

Kazi Ranjibul Islam , Samuel Awelewa , Andrey V. Chubukov , Maxim Dzero This is my paper
classification cond-mat.supr-con cond-mat.str-el
keywords d-wave superconductivitycollective modesplasma modeamplitude (Higgs) modeCarlson-Goldman modeCoulomb screeningnodal quasiparticlespair susceptibility
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.

This paper aims to establish how the two collective excitations of a d-wave superconductor—the phase (transverse) and amplitude (longitudinal) modes—behave when long-range Coulomb interaction is included. The authors show that at zero temperature the transverse mode is exactly the s-wave plasma mode, but at finite temperature it becomes softer and much broader because nodal quasiparticles partially screen the Coulomb field. The longitudinal mode, a resonance inside the quasiparticle continuum, has a peak frequency that depends on the direction of the wavevector relative to the d-wave nodes, while its decay in time is 1/t² independent of both the magnitude and direction of momentum. A sympathetic reader would care because these are concrete, momentum-resolved signatures that distinguish d-wave from s-wave superconductors and can be compared with pump-probe and Raman experiments on cuprates.

Core claim

The central claim is that in a clean d-wave superconductor the pair susceptibility separates into a Coulomb-dressed transverse (phase) part and a longitudinal (amplitude) part untouched by Coulomb. At T=0 the phase mode is the same plasma mode as in s-wave: ω ∝ √q in 2D and ω = √(4πne²/m) in 3D. At finite T it softens and broadens because nodal quasiparticles partially screen the Coulomb potential; the effective velocity falls linearly with T. The amplitude mode is a resonance at 2Δ_max at q=0; at finite q its peak frequency depends on momentum direction, while its time-domain decay is 1/t² independent of momentum magnitude and direction.

What carries the argument

The load-bearing object is the pair-pair susceptibility χ(q,Ω), decomposed into transverse and longitudinal polarization bubbles. The transverse bubble is dressed by the screened Coulomb interaction via a bubble insert Π_GF²/(V_q^{-1}-2Π_0); the longitudinal bubble is not. The d-wave form factor γ(θ)=√2 cos 2θ makes the gap vanish at nodal points, so thermally excited nodal quasiparticles sit at low energy and partially screen the Coulomb potential at finite T. Two independent methods—quasiclassical Eilenberger equations in Keldysh-Nambu space and diagrammatic ladder sums—yield the same mode dispersions.

Load-bearing premise

The calculation assumes a clean, single-band, parabolic-band d-wave superconductor with only BCS pairing plus Coulomb interaction; if electron correlations or disorder change the screening or the decay exponent, as the paper itself concedes, the quantitative predictions would not transfer to real cuprates.

What would settle it

Measure the time-domain decay of the coherent amplitude-mode oscillations in a clean d-wave superconductor (e.g., via THz pump-probe); if the envelope decays as t^{-α} with α≠2, or if α depends on the pump wavevector direction, the central claim is falsified. Alternatively, measure the Carlson-Goldman mode frequency near Tc; if its softening with T does not follow the linear-in-T reduction of ζ(T) relative to an s-wave reference, the finite-T screening mechanism is wrong.

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

If this is right

  • The Carlson-Goldman/plasma mode in a d-wave superconductor near Tc should appear at lower frequency and with broader width than in s-wave, providing a direct finite-T signature.
  • The longitudinal (amplitude) susceptibility is nonzero at all frequencies due to nodal quasiparticles, unlike s-wave where it vanishes below 2Δ; pump-probe oscillations should show a momentum-direction-dependent period.
  • The decay of the amplitude mode's oscillations in the time domain is 1/t² independent of both magnitude and direction of momentum, a universal signature in clean d-wave superconductors.
  • At zero temperature the plasma-mode dispersion matches s-wave, so any deviation in experiment at low T would indicate physics beyond the clean single-band model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If electron correlations renormalize nodal quasiparticle damping, the 1/t² decay is likely to become a different exponent; the paper itself notes this, so measuring the exponent in cuprates could quantify correlation strength.
  • The directional dependence of the amplitude-mode resonance suggests that angle-resolved experimental probes, such as directional THz pump-probe or momentum-resolved Raman, should observe different oscillation periods for excitations along nodal versus antinodal directions.
  • The finite-T plasma-mode softening could be tested by near-field or transmission experiments that tune T/Tc and look for the linear-in-T reduction of the mode velocity predicted here.
  • Since pair-breaking disorder suppresses d-wave superconductivity, extending the calculation to include disorder may replace the clean 1/t² decay with a faster decay, offering another experimental handle.

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

3 major / 5 minor

Summary. The paper studies collective excitations of a clean d-wave superconductor with long-range Coulomb interaction using two complementary methods: quasiclassical Keldysh-Nambu theory and diagrammatic ladder/bubble resummation. The transverse (phase/plasma) mode is found to have the same dispersion as in an s-wave superconductor at T=0, while at finite T its velocity is reduced by a factor set by ζ(T), which decreases linearly with T because of nodal quasiparticles. The longitudinal (amplitude) mode is analyzed at zero temperature; it is a resonance in the continuum whose peak frequency depends on the direction of q for q in the nodal versus antinodal directions, while the time-domain decay is claimed to be 1/t² independent of |q| and direction. The two formalisms are shown to agree for the dispersion relations, and the gap equation eliminates the coupling constant. The paper also discusses possible connections to recent pump-probe and Raman experiments on cuprates.

Significance. If the central claims hold, the paper provides concrete, falsifiable predictions for the momentum-resolved collective-mode spectrum of a d-wave superconductor, especially the direction-dependent amplitude-mode dispersion and the softening of the Carlson-Goldman mode. The manuscript has clear strengths: the two independent derivations agree; the dispersion formulas are derived in detail, including the low-temperature asymptotic ζ(T) in Eqs. (37)-(38); and the results are parameter-free in the sense that the pairing coupling is eliminated via the gap equation. The formal machinery is therefore largely sound. However, two headline claims — the finite-T transverse-mode damping and the q-independent longitudinal decay — are asserted rather than derived, and the paper itself concedes in Sec. V that correlations may change the predicted power-law decay, limiting the direct experimental transferability.

major comments (3)
  1. [Abstract, §III.B.2, §V] The central claim that the transverse mode at finite T has a 'much larger decay rate' is not supported by the calculation. The text computes only real parts of the relevant functions: Re χ_AB in Eq. (34) and Fig. 3, and the dispersion from the determinant of Eq. (39), leading to Eq. (40). No imaginary part of χ_T, Π_T, Π_GF, or Π_0 is evaluated, plotted, or used to obtain a pole width. A broader peak in Re χ_AB is not a quantitative damping rate. Since this enhanced damping is a headline result and motivates the CG-mode discussion in Sec. V, it must either be derived by computing Im χ_T at finite T or the claim must be removed/weakened.
  2. [§IV.C, Conclusions] The claim that the longitudinal mode decays as 1/t² independent of |q| and direction is based on Fig. 7 and a sentence in Sec. IV.C, but no analytic derivation or quantitative fit is provided. The abstract states 'the decay rate of this mode does not depend on momentum,' yet the width of Im χ_L is never extracted or compared. Given that this is the other headline result, the authors should provide either an analytic argument (e.g., from the ω→0 behavior of Im χ_L) or a careful numerical demonstration, including the extraction of the exponent.
  3. [§V] The discussion concedes that electronic correlations 'may change the exponent in the power law decay' relative to the clean parabolic-band result. This is an honest limitation, but it undercuts the direct experimental application to cuprates stated in the same section. The formal result for the clean model is still valuable; however, the manuscript should more carefully separate the model prediction from the experimental expectation, especially in the Abstract and Conclusions, where the predictions are stated without this caveat.
minor comments (5)
  1. [Introduction, p. 4] The phrase 'the longitudinal mode in a d-wave superconductor (CG mode at T≤T_c)' appears to be a typo: the Carlson-Goldman mode is a transverse phase mode, not a longitudinal mode.
  2. [Fig. 4] In the right (s-wave) panel, the legend entry 'v_F q = Δ' appears to be missing a numerical value; several entries are incomplete. Please check all legend labels.
  3. [Eq. (76)] The expressions contain log(W²/ν) and log(W²/|ν|) with ν later defined as ν=2ν/Δ_max. Please clarify the dimensionless normalization of the argument throughout, since logarithms of dimensionful quantities are ambiguous.
  4. [§III.B.2] The finite-T calculation approximates Δ(T)≈Δ(0) while keeping thermal factors from nodal quasiparticles. This is reasonable for T≪Δ, but the approximation should be stated more explicitly with the expected size of the correction.
  5. [§VIII] The data availability statement ('not publicly available upon publication because it is not technically feasible') is unusual; the paper contains no data files, so it might be clearer to state that no experimental data were generated.

Circularity Check

0 steps flagged

No significant circularity: central dispersions are derived from the microscopic model with g eliminated by the gap equation; the only same-group citation is a non-load-bearing q=0 benchmark.

full rationale

The derivation chain is self-contained. The pairing coupling g enters through the gap equation (Eqs. 4/12) and is eliminated when the susceptibilities are rewritten, e.g. Eqs. (47), (50), and (68); no fitted experimental input is relabeled as a prediction. The T=0 transverse dispersion and plasma frequency are obtained independently from the quasiclassical determinant (Eqs. 36-40) and from the diagrammatic pole condition (Eqs. 49-54), and the finite-T softening is controlled by the analytically evaluated ζ(T) in Eqs. (37)-(38)/(62) and Appendix D. The longitudinal-mode frequency, cusp structure, and 1/t^2 decay are computed from δΠ_L and the Fourier transform (Eqs. 69-78 and Fig. 7), not imposed by construction. The only self-citation is Ref. [87], a same-group preprint used in Sec. IV C for the q=0 longitudinal decay; because the nonzero-q decay is computed here and the q=0 asymptotic behavior is also consistent with the analytically derived Im χ_L ∝ Ω^3 law in Sec. IV B, this citation is not load-bearing. The abstract's finite-T 'much larger decay rate' for the transverse mode is asserted without computing Im χ_T, but that is an unsupported-evidence/correctness concern, not a circular reduction; the dispersion itself is derived. Overall the paper has at most one minor, non-load-bearing self-citation, so the circularity score is 2.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper's results rest on standard BCS/quasiclassical/RPA machinery. No experimental data are fitted, no new particles or mediators are introduced, and the only same-group reference (Ref. [87]) is a q=0 d-wave longitudinal benchmark that is also rederived in this paper.

axioms (6)
  • domain assumption BCS mean-field state with d-wave gap Δ_k = Δ cos 2θ_k, determined by the non-linear gap equation (4).
    Section II.A; the entire calculation starts from this state.
  • domain assumption Quasiclassical Eilenberger equation (5) with normalization (7) is a valid description of low-energy dynamics.
    Section II.B.1; used for the quasiclassical derivation.
  • domain assumption Pair-pair susceptibility is computed in ladder/bubble approximation (Figs. 1-2), with RPA-screened Coulomb interaction (16).
    Section II.B.2; neglects vertex corrections beyond the ladder and RPA density bubbles.
  • domain assumption Separable d-wave pairing interaction V_d = -g γθ γθ' and long-range Coulomb potential V_q = 2πe²/|q| in 2D and 4πe²/q² in 3D.
    Eqs. (2)-(3).
  • standard math Analytic continuation iΩ_m → Ω + iδ from Matsubara to retarded functions is valid.
    Section II.B.2; used to locate poles and branch cuts in χ.
  • domain assumption Small-q and small-Ω/Δ expansions are controlled; the ζ(T) result assumes T ≪ Δ.
    Sections III.B.2, IV.B, Appendix D; the asymptotic ζ(T) is explicitly a low-temperature expansion.

pith-pipeline@v1.3.0-alltime-deepseek · 32951 in / 14427 out tokens · 152382 ms · 2026-08-03T10:29:49.358942+00:00 · methodology

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read the original abstract

We analyze the dispersion of collective modes in a superconductor with $d-$wave symmetry of the order parameter in the presence of long-range Coulomb interaction. We use diagrammatic technique and quasiclassical theory in Keldysh-Nambu formalism to compute longitudinal and transverse pair susceptibilities and extract from them the dispersion of the longitudinal and transverse collective mode. We show that at T=0, the dispersion of the transverse (plasma) mode is the same as in an s-wave superconductor, but at a finite temperature it is softer and has a much larger decay rate due to the partial screening of the Coulomb potential by nodal quasiparticles. We show that the dispersion of the longitudinal mode depends on the direction of momentum with respect to the positions of the nodes of the d-wave gap, while the decay rate of this mode does not depend on momentum. We discuss experimental implications of our results.

Figures

Figures reproduced from arXiv: 2601.09782 by Andrey V. Chubukov, Kazi Ranjibul Islam, Maxim Dzero, Samuel Awelewa.

Figure 1
Figure 1. Figure 1: FIG. 1. Dyson equation for the pair-pair susceptibility [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Dyson equation for the renormalized particle-particle vertices (a) Γ [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Left panel: real part of transverse pair susceptibility [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Frequency dependence of the imaginary and real parts of the longitudinal susceptibility, [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Frequency dependence of the imaginary part of the [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Left panel: dependence of the position of the maximum in Im [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Time dependence of the longitudinal susceptibility plotted for different values of momentum [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗

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Cited by 1 Pith paper

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

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Reference graph

Works this paper leans on

112 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [1]

    Quasiclassical theory. The quasiclassical equation for the single-particle propagator ˇgis [94–97] [ϵˇτ3 − ˇ∆n(r, t)◦,ˇgnϵ] + i 2 {ˇτ3, ∂tˇgnϵ}+iv F (n·∇ r)ˇgnϵ = 0.(5) Here ˇτ3 = ˆτ3 ⊗ˆσ0 is a 4×4 matrix defined in Keldysh and Nambu spaces (note that ˇτ 3 is diagonal in Keldysh space) andn=k/k. Quasiclassical propagator ˇgis an 4×4 matrix in Keldysh and ...

  2. [2]

    Diagrammatic approach. Within the diagrammatic technique, we introduce normal and anomalous Green’s func- tions in the superconducting state along the Matsubara axis, G(k, ωm) = i ωm +ξ k (i ωm)2 −E 2 k , F(k, ω m) = ∆k (i ωm)2 −E 2 k ,(13) whereE k = p ξ2 k + ∆2 k is the quasi-particle excitation energy, andω m =πT(2m+ 1) is the fermionic Matsubara frequ...

  3. [3]

    In the charged case, the system of equations is very similar to (39) where we only need to change 2q→q 2 in the last term in the second equation in (39)

    At finite temperatures, we find that the magnitude of Ω (3D) decreases with temperature just like in the 2D case. In the charged case, the system of equations is very similar to (39) where we only need to change 2q→q 2 in the last term in the second equation in (39). Simple algebra gives the following formula for the dispersion of the transverse mode: Ω(3...

  4. [4]

    The same analysis forD= 3 yields Ω =v F |q|/ √

  5. [5]

    We next include the Coulomb interaction into consideration and use the full expression for the transverse polarization bubble ¯ΠT (q,Ω), defined in Eq

    Both expressions are the same as we found quasi-classically and coincide with the corresponding expressions fors−wave superfluids. We next include the Coulomb interaction into consideration and use the full expression for the transverse polarization bubble ¯ΠT (q,Ω), defined in Eq. (43). We use the fact that ¯ΠT (0,0) = Π T (0,0) and express ¯ΠT (q,Ω) as ...

  6. [6]

    finiteT We first discuss charge neutrald-wave superfluid. The mode dispersion is determined from Π T (q,Ω) = 1/g, where we suppress the explicit temperature dependence of Π T for notational simplicity, and has to be obtained by summing over Matsubara frequencies instead of integrating over the frequency. At finiteT, we find the following expression (see A...

  7. [7]

    This agrees with our analytical reasoning

    The resonance frequency in the nodal direction is lower than that in the antinodal direction. This agrees with our analytical reasoning. and nearθ k =π/2 (or 3π/2). Integration near of these two sets of points gives rise to non-analyticϵlogϵbehavior, but for the cuspϵ= Ω−2Ω max while for the peak,ϵ= Ω−(2Ω max +v 2 F |q|2/(2∆max)). For a generic direction ...

  8. [8]

    S. N. Artemenko and A. F. Volkov, Collective excitations with a sound spectrum in super- conductors, Sov. Phys. JETP42, 896 (1975)

    1975

  9. [9]

    V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, inOn Supercon- ductivity and Superfluidity: A Scientific Autobiography(Springer Berlin Heidelberg, Berlin, Heidelberg, 2009) pp. 113–137

    2009

  10. [10]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev.108, 1175 (1957)

    1957

  11. [11]

    P. W. Anderson, Random-phase approximation in the theory of superconductivity, Phys. Rev.112, 1900 (1958)

    1900

  12. [12]

    P. W. Anderson, New method in the theory of superconductivity, Phys. Rev.110, 985 (1958)

    1958

  13. [13]

    N. N. Bogoliubov, A new method in the theory of superconductivity., Sov. Phys. JETP34 (1958)

    1958

  14. [14]

    P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev.130, 439 (1963)

    1963

  15. [15]

    R. V. Carlson and A. M. Goldman, Superconducting order-parameter fluctuations belowT c, Phys. Rev. Lett.31, 880 (1973)

    1973

  16. [16]

    Schmid, The approach to equilibrium in a pure superconductor the relaxation of the cooper pair density, Physik der kondensierten Materie8, 129 (1968)

    A. Schmid, The approach to equilibrium in a pure superconductor the relaxation of the cooper pair density, Physik der kondensierten Materie8, 129 (1968)

    1968

  17. [17]

    Schmid and G

    A. Schmid and G. Sch¨ on, Collective oscillations in a dirty superconductor, Phys. Rev. Lett. 34, 941 (1975)

    1975

  18. [18]

    S. N. Artemenko and A. F. Volkov, Electric fields and collective oscillations in superconduc- tors, Sov. Phys. Usp.22, 295 (1979)

    1979

  19. [19]

    Schmid and G

    A. Schmid and G. Sch¨ on, Linearized kinetic equations and relaxation processes of a super- conductor neart c,, J. Low Temp. Phys.20, 1747 (1979)

    1979

  20. [20]

    I. O. Kulik, O. Entin-Wohlman, and R. Orbach, Pair susceptibility and mode propagation in superconductors: A microscopic approach, Journal of Low Temperature Physics43, 591 (1981). 39

    1981

  21. [21]

    Ohashi and S

    Y. Ohashi and S. Takada, Goldstone mode in charged superconductivity: Theoreti- cal studies of the carlson-goldman mode and effects of the landau damping in the superconducting state, Journal of the Physical Society of Japan66, 2437 (1997), https://doi.org/10.1143/JPSJ.66.2437

    work page doi:10.1143/jpsj.66.2437 1997

  22. [22]

    Ohashi and S

    Y. Ohashi and S. Takada, On the plasma oscillation in superconductivity, Journal of the Physical Society of Japan67, 551 (1998), https://doi.org/10.1143/JPSJ.67.551

    work page doi:10.1143/jpsj.67.551 1998

  23. [23]

    Kamenev,Field Theory of Non-Equilibrium Systems(Cambridge University Press, 2011)

    A. Kamenev,Field Theory of Non-Equilibrium Systems(Cambridge University Press, 2011)

    2011

  24. [24]

    R. A. Barankov and L. S. Levitov, Dynamical selection in developing fermionic pairing, Phys. Rev. A73, 033614 (2006)

    2006

  25. [25]

    A. F. Volkov and S. M. Kogan, Collisionless relaxation of the energy gap in superconductors, Zh. Eksp. Teor. Fiz65, 2038 (1974), English translation: Sov. Phys. JETP,38, 1018 (1974)

    2038

  26. [26]

    V. P. Galaiko, Kinetic equation for relaxation processes in superconductors, Sov. Phys. JETP 34, 203 (1972)

    1972

  27. [27]

    Y. M. Galperin, V. I. Kozub, and B. Z. Spivak, Dissipationless bcs dynamics with large branch imbalance, Sov. Phys. JETP54, 1126 (1981)

    1981

  28. [28]

    V. S. Shumeiko,Dynamics of electronic system with off-diagonal order parameter and non- linear resonant phenomena in superconductors(Doctoral Thesis, Institute for Low Temper- ature Physics and Engineering, Kharkov, Ukraine, 1990)

    1990

  29. [29]

    R. A. Barankov, L. S. Levitov, and B. Z. Spivak, Solitons and rabi oscillations in a time- dependent bcs pairing problem, Phys. Rev. Lett.93, 160401 (2004)

    2004

  30. [30]

    A. V. Andreev, V. Gurarie, and L. Radzihovsky, Nonequilibrium dynamics and thermody- namics of a degenerate fermi gas across a feshbach resonance, Phys. Rev. Lett.93, 130402 (2004)

    2004

  31. [31]

    E. A. Yuzbashyan, B. L. Altshuler, V. B. Kuznetsov, and V. Z. Enolskii, Nonequilibrium cooper pairing in the nonadiabatic regime, Phys. Rev. B72, 220503(R) (2005)

    2005

  32. [32]

    P. W. Anderson, Higgs, anderson and all that, Nature Physics11, 93 (2015)

    2015

  33. [33]

    E. A. Yuzbashyan and M. Dzero, Dynamical vanishing of the order parameter in a fermionic condensate, Phys. Rev. Lett.96, 230404 (2006)

    2006

  34. [34]

    R. A. Barankov and L. S. Levitov, Excitation of the dissipationless higgs mode in a fermionic condensate, arXiv:0704.1292 (2007). 40

    Pith/arXiv arXiv 2007

  35. [35]

    E. A. Yuzbashyan, O. Tsyplyatyev, and B. L. Altshuler, Relaxation and persistent oscillations of the order parameter in the non-stationary bcs theory, Phys. Rev. Lett.96, 097005 (2006), erratum: Phys. Rev. Lett.96, 179905 (2006)

    2006

  36. [36]

    E. A. Yuzbashyan, Normal and anomalous solitons in the theory of dynamical cooper pairing, Phys. Rev. B78, 184507 (2008)

    2008

  37. [37]

    E. A. Yuzbashyan, M. Dzero, V. Gurarie, and M. S. Foster, Quantum quench phase diagrams of ans-wave bcs-bec condensate, Phys. Rev. A91, 033628 (2015)

    2015

  38. [38]

    P. B. Littlewood and C. M. Varma, Gauge-invariant theory of the dynamical interaction of charge density waves and superconductivity, Phys. Rev. Lett.47, 811 (1981)

    1981

  39. [39]

    P. B. Littlewood and C. M. Varma, Amplitude collective modes in superconductors and their coupling to charge-density waves, Phys. Rev. B26, 4883 (1982)

    1982

  40. [40]

    M´ easson, Y

    M.-A. M´ easson, Y. Gallais, M. Cazayous, B. Clair, P. Rodi` ere, L. Cario, and A. Sacuto, Amplitude higgs mode in the 2h−nbse 2 superconductor, Phys. Rev. B89, 060503 (2014)

    2014

  41. [41]

    Pekker and C

    D. Pekker and C. Varma, Amplitude/higgs modes in condensed matter physics, An- nual Review of Condensed Matter Physics6, 269 (2015), https://doi.org/10.1146/annurev- conmatphys-031214-014350

    doi:10.1146/annurev- 2015

  42. [42]

    Matsunaga and R

    R. Matsunaga and R. Shimano, Nonequilibrium bcs state dynamics induced by intense ter- ahertz pulses in a superconducting nbn film, Phys. Rev. Lett.109, 187002 (2012)

    2012

  43. [43]

    Matsunaga, Y

    R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Higgs amplitude mode in the bcs superconductors nb 1-xtixNinduced by terahertz pulse excitation, Phys. Rev. Lett.111, 057002 (2013)

    2013

  44. [44]

    Matsunaga, N

    R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and R. Shimano, Light-induced collective pseudospin preces- sion resonating with higgs mode in a superconductor, Science345, 1145 (2014), https://www.science.org/doi/pdf/10.1126/science.1254697

    doi:10.1126/science.1254697 2014

  45. [45]

    M. Beck, I. Rousseau, M. Klammer, P. Leiderer, M. Mittendorff, S. Winnerl, M. Helm, G. N. Gol’tsman, and J. Demsar, Transient increase of the energy gap of superconducting nbn thin films excited by resonant narrow-band terahertz pulses, Phys. Rev. Lett.110, 267003 (2013)

    2013

  46. [46]

    Sherman, U

    D. Sherman, U. S. Pracht, B. Gorshunov, S. Poran, J. Jesudasan, M. Chand, P. Raychaud- huri, M. Swanson, N. Trivedi, A. Auerbach, M. Scheffler, A. Frydman, and M. Dressel, The higgs mode in disordered superconductors close to a quantum phase transition, Nature Physics11, 188 (2015). 41

    2015

  47. [47]

    Katsumi, J

    K. Katsumi, J. Fiore, M. Udina, R. Romero, D. Barbalas, J. Jesudasan, P. Raychaudhuri, G. Seibold, L. Benfatto, and N. P. Armitage, Revealing novel aspects of light-matter coupling by terahertz two-dimensional coherent spectroscopy: The case of the amplitude mode in superconductors, Phys. Rev. Lett.132, 256903 (2024)

    2024

  48. [48]

    Papenkort, T

    T. Papenkort, T. Kuhn, and V. M. Axt, Nonequilibrium dynamics and coherent control of bcs superconductors driven by ultrashort thz pulses, Journal of Physics193, 012050 (2009)

    2009

  49. [49]

    Behrle, T

    A. Behrle, T. Harrison, J. Kombe, K. Gao, M. Link, J. S. Bernier, C. Kollath, and M. K¨ ohl, Higgs mode in a strongly interacting fermionic superfluid, Nature Physics14, 781 (2018)

    2018

  50. [50]

    Grasset, Y

    R. Grasset, Y. Gallais, A. Sacuto, M. Cazayous, S. Ma˜ nas Valero, E. Coronado, and M.-A. M´ easson, Pressure-induced collapse of the charge density wave and higgs mode visibility in 2h−tas2, Phys. Rev. Lett.122, 127001 (2019)

    2019

  51. [51]

    Shimano and N

    R. Shimano and N. Tsuji, Higgs mode in superconductors, Annual Review of Condensed Mat- ter Physics11, 103 (2020), https://doi.org/10.1146/annurev-conmatphys-031119-050813

    work page doi:10.1146/annurev-conmatphys-031119-050813 2020

  52. [52]

    Nakamura, K

    S. Nakamura, K. Katsumi, H. Terai, and R. Shimano, Nonreciprocal terahertz second- harmonic generation in superconducting nbn under supercurrent injection, Phys. Rev. Lett. 125, 097004 (2020)

    2020

  53. [55]

    Papenkort, V

    T. Papenkort, V. M. Axt, and T. Kuhn, Coherent dynamics and pump-probe spectra of bcs superconductors, Phys. Rev. B76, 224522 (2007)

    2007

  54. [56]

    A. Moor, A. F. Volkov, and K. B. Efetov, Amplitude higgs mode and admittance in super- conductors with a moving condensate, Phys. Rev. Lett.118, 047001 (2017)

    2017

  55. [57]

    Podolsky, A

    D. Podolsky, A. Auerbach, and D. P. Arovas, Visibility of the amplitude (higgs) mode in condensed matter, Phys. Rev. B84, 174522 (2011)

    2011

  56. [58]

    Podolsky and S

    D. Podolsky and S. Sachdev, Spectral functions of the higgs mode near two-dimensional quantum critical points, Phys. Rev. B86, 054508 (2012)

    2012

  57. [59]

    Gazit, D

    S. Gazit, D. Podolsky, and A. Auerbach, Fate of the higgs mode near quantum criticality, Phys. Rev. Lett.110, 140401 (2013). 42

    2013

  58. [60]

    G. E. Volovik and M. A. Zubkov, Higgs bosons in particle physics and in condensed matter, Journal of Low Temperature Physics175, 486 (2014)

    2014

  59. [61]

    Ran¸ con and N

    A. Ran¸ con and N. Dupuis, Higgs amplitude mode in the vicinity of a (2 + 1)-dimensional quantum critical point, Phys. Rev. B89, 180501 (2014)

    2014

  60. [62]

    Krull, D

    H. Krull, D. Manske, G. S. Uhrig, and A. P. Schnyder, Signatures of nonadiabatic bcs state dynamics in pump-probe conductivity, Phys. Rev. B90, 014515 (2014)

    2014

  61. [63]

    T. Cea, C. Castellani, and L. Benfatto, Nonlinear optical effects and third-harmonic gener- ation in superconductors: Cooper pairs versus higgs mode contribution, Phys. Rev. B93, 180507 (2016)

    2016

  62. [64]

    Krull, N

    H. Krull, N. Bittner, G. S. Uhrig, D. Manske, and A. P. Schnyder, Coupling of higgs and leggett modes in non-equilibrium superconductors, Nature Communications7, 11921 (2016)

    2016

  63. [65]

    Fischer, M

    S. Fischer, M. Hecker, M. Hoyer, and J. Schmalian, Short-distance breakdown of the higgs mechanism and the robustness of the bcs theory for charged superconductors, Phys. Rev. B 97, 054510 (2018)

    2018

  64. [66]

    Z. Sun, M. M. Fogler, D. N. Basov, and A. J. Millis, Collective modes and terahertz near-field response of superconductors, Phys. Rev. Res.2, 023413 (2020)

    2020

  65. [67]

    Barresi, A

    A. Barresi, A. Boulet, G. Wlaz lowski, and P. Magierski, Generation and decay of higgs mode in a strongly interacting fermi gas, Scientific Reports13, 11285 (2023)

    2023

  66. [68]

    Derendorf, A

    P. Derendorf, A. F. Volkov, and I. M. Eremin, Nonlinear response of diffusive superconductors to ac electromagnetic fields, Phys. Rev. B109, 024510 (2024)

    2024

  67. [69]

    H. P. O. Collado, N. Defenu, and J. Lorenzana, Engineering higgs dynamics by spectral singularities, Phys. Rev. Res.5, 023011 (2023)

    2023

  68. [70]

    Phan and A

    D. Phan and A. V. Chubukov, Following the higgs mode across the bcs-bec crossover in two dimensions, Phys. Rev. B107, 134519 (2023)

    2023

  69. [71]

    Li and M

    Y. Li and M. Dzero, Collective modes in terahertz field response of disordered superconduc- tors, Journal of Physics: Condensed Matter37, 115602 (2025)

    2025

  70. [72]

    Haenel, P

    R. Haenel, P. Froese, D. Manske, and L. Schwarz, Time-resolved optical conductivity and higgs oscillations in two-band dirty superconductors, Phys. Rev. B104, 134504 (2021)

    2021

  71. [73]

    Dzero, Inverse faraday effect in superconductors with potential impurities, Phys

    M. Dzero, Inverse faraday effect in superconductors with potential impurities, Phys. Rev. B 110, 054506 (2024). 43

    2024

  72. [74]

    Li and M

    Y. Li and M. Dzero, Amplitude higgs mode in superconductors with magnetic impurities, Phys. Rev. B109, 054520 (2024)

    2024

  73. [75]

    Alth¨ user and G

    J. Alth¨ user and G. S. Uhrig, Collective modes in superconductors including Coulomb repul- sion, SciPost Phys.19, 067 (2025)

    2025

  74. [76]

    Dzero and A

    M. Dzero and A. Kamenev, Schmid-higgs mode in the presence of pair-breaking interactions, Phys. Rev. B111, 174502 (2025)

    2025

  75. [77]

    P. A. Nosov, E. S. Andriyakhina, and I. S. Burmistrov, Spatially resolved dynamics of the amplitude schmid-higgs mode in disordered superconductors, Phys. Rev. Lett.135, 056001 (2025)

    2025

  76. [78]

    Silaev, Nonlinear electromagnetic response and higgs-mode excitation in bcs supercon- ductors with impurities, Phys

    M. Silaev, Nonlinear electromagnetic response and higgs-mode excitation in bcs supercon- ductors with impurities, Phys. Rev. B99, 224511 (2019)

    2019

  77. [79]

    Seibold, M

    G. Seibold, M. Udina, C. Castellani, and L. Benfatto, Third harmonic generation from col- lective modes in disordered superconductors, Phys. Rev. B103, 014512 (2021)

    2021

  78. [80]

    Katsumi, N

    K. Katsumi, N. Tsuji, Y. I. Hamada, R. Matsunaga, J. Schneeloch, R. D. Zhong, G. D. Gu, H. Aoki, Y. Gallais, and R. Shimano, Higgs mode in thed-wave superconductor bi2sr2cacu2o8+x driven by an intense terahertz pulse, Phys. Rev. Lett.120, 117001 (2018)

    2018

  79. [81]

    Yang and M

    F. Yang and M. W. Wu, Impurity scattering in superconductors revisited: Diagrammatic formulation of the supercurrent-supercurrent correlation and higgs-mode damping, Phys. Rev. B106, 144509 (2022)

    2022

  80. [82]

    Yang and M

    F. Yang and M. W. Wu, Influence of scattering on the optical response of superconductors, Phys. Rev. B102, 144508 (2020)

    2020

Showing first 80 references.