Dispersion of plasmons in three-dimensional superconductors
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Dispersion of plasmons in three-dimensional superconductors
T. Repplinger, M. Gélédan and H. Kurkjian
Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, 31400, Toulouse, France
S. Klimin and J. Tempere
TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Anvers, Belgique
We study the plasma branch of an homogeneous three-dimensional electron gas in an s-wave
superconducting state. We focus on the regime where the plasma frequency ωp is comparable to
the gap ∆, which is experimentally realized in cuprates. Although a sum rule guarantees that the
arXiv:2201.11421v1 [cond-mat.supr-con] 27 Jan 2022
departure of the plasma branch always coincides with the plasma frequency, the dispersion and
lifetime of the plasmons is strongly affected by the presence of the pair condensate, especially at
energies close to the pair-breaking threshold 2∆. When ωp is above 1.7∆, the level repulsion is
strong enough to give the plasma branch an anomalous, negative dispersion with a minimum at
finite wavelength. At non-zero temperature and at ωp > 2∆, we treat in a non-perturbative way
the coupling of plasmons to the fermionic excitations, and show that a broadened plasma resonance
inside the pair-breaking continuum coexists with an undamped solution in the band gap. This
resonance splitting is associated with the presence of multiple poles in the analytic continuation of
the propagator of the Cooper pairs.
Introduction: Despite being a very mature experimen- damental understanding of superconductivity and long-
tal platform, supporting numerous technical applications, range interactions in many-body physics, those modes
superconductors still hold some of the most fundamental can also be used in plasmonics, or to probe and ma-
open questions of many-body physics. The impressively nipulate superconducting materials [18]. Experimental
high critical temperature (Tc ) and the unconventional research on Josephson plasmons, which describe trans-
Cooper pairing in cuprates and iron-based superconduc- verse plasmonic excitations is superconducting layers is
tors are the most famous of those fascinating questions. still very active today [19].
However, even some properties of conventional Bardeen-
At low energy-momentum, plasmons can be described
Cooper-Schrieffer (BCS) superconductors are still inten-
by phenomenological approaches based on London elec-
sively discussed, such as the existence of an amplitude
trodynamics [20], but a microscopic theory is needed
collective mode [1–3], reminiscent of the Higgs mode in
when the eigenfrequency of plasmons approaches the
high-energy physics.
pair-breaking threshold. In this regime, the theoreti-
In fact, even for such usual behavior as plasma oscil- cal literature is still hesitant, in particular in the cases
lations (the collective modes of the electronic density), where a complex plasma mode describing a damped res-
superconductors are still not fully understood. In a pio- onance is expected. Here, we reveal that remarkable
neering work, Anderson [4] has shown that the phononic phenomena affecting the plasma dispersion in presence
(Goldstone) branch that exists in a neutral fermionic con- of superconducting order have been overlooked. Like
densate acquires a gap corresponding to the plasma fre- Anderson, we consider the reference situation of an
quency ωp in presence of long-range Coulomb interaction. isotropic three-dimensional (3D) s-wave superconductor
This mechanism later became famous due to its analogy but our study can be readily extended to layered ge-
with the phenomenon of mass acquisition in high-energy ometries or anisotropic pairing. Due to the repulsion of
physics. The work of Anderson has then been revisited in the pair-breaking threshold, the plasma branch acquires
the context of high-Tc superconductivity [5–9], and nu- a negative curvature and thus a minimum at non-zero
clear/neutronic matter [10]. While Anderson focused on wavenumber when ωp is between 1.696∆ and 2∆ at zero
the regime of large ωp , the frequency of transverse plas- temperature. At larger wavenumber, nonzero tempera-
mons in layered superconductors such as cuprates often ture or when ωp > 2∆, the plasma branch enters the pair-
lies below the pair-breaking threshold 2∆ [11–13], such breaking continuum but remains observable with a finite
that an undamped plasma branch can be expected. The lifetime that we calculate using recently develop technics
departure of the plasma branch was shown to always co- [21–23] to treat the coupling to the fermionic continuum.
incide with ωp [9], and as temperature or excitation mo- We find a rich resonance structure in the density-density
menta were varied, a duplication of the plasma resonance response function, with several peaks both above and be-
was observed, with a low-energy branch at energies below low the pair-breaking threshold, which we relate to the
2∆ and a high-energy one above [14–16]. existence of multiple poles in the analytic continuation
The existence of such low-energy plasmons was cited as [21, 23, 24] of the propagator the density-phase fluctua-
a possible explanation of the critical temperature increase tions. In the quasiphononic regime ωp 2∆ correspond-
in cuprates [17]. Besides their importance for the fun- ing to the experimental situation in cuprates [13, 15], the2
q
plasma eigenfrequency behaves as ωp2 + ωq,n 2 where ω
q,n We give here its generic expression
is the phononic dispersion of the neutral fermion conden-
+ −
sate [14]. Finally, we show that when temperature in- X πij (1 − f+ − f− ) πij (f+ − f− )
Πij (z, q) = −
creases towards Tc , the plasmons gradually recover their 2
z − (+ + − ) 2 z − (+ − − )2
2
k
normal (undamped) dispersion [25], except in a region of (4)
size ∆2 /T around the pair-breaking threshold. All these in terms of the Fermi-Dirac occupation numbers f± =
unusal behaviors should have important consequences on 1/(1 + exp(± /T )), free-fermion ξ± = ξq/2±k and BCS
the electromagnetic and transport properties of super- energies ± = q/2±k with ξk = k 2 /2m − µ and k =
conductors. p ±
ξk2 + ∆2 . The coefficients πij can be deduced from
Linear response within RPA: We study an homoge- Eq. (36) in [28]. The first term in Eq. (4) gives rise to
neous electron gas evolving in a cubic volume V with the pair-breaking continuum {q/2+k + q/2−k }k , gapped
a average density ρ, defining the Fermi wavenumber at low q by the pair-breaking threshold 2∆. The second
ρ = kF3 /3π 2 . Electrons interact through both the long- term exists only at T 6= 0 and gives rise to the gapless
range Coulomb potential VC (r) ∝ 1/r and a short-range quasiparticle-quasihole continuum {q/2+k − q/2−k }k
part, responsible for s-wave Cooper pairing, and mod-
The spectrum of the collective modes is found as the
elled by a contact potential of coupling constant g:
poles of χ, hence as the zeros of M :
V (r1 , r2 ) = gδ(r1 − r2 ) + VC (r1 − r2 ) (1)
detM↓ (zq , q) = 0 (5)
We note that a momentum cutoff at the Debye frequency
should be used to regularize the divergence caused by The ↓ sign recalls that when the collective mode is cou-
the contact potential, although in the following discus- pled to the pair-breaking [21] or quasiparticle-hole con-
sion this cutoff can safely be send to infinity. In terms tinuum [23, 31], it is complex energy is found only after
of the electron mass m and wavenumber q, the Fourier an analytic continuation of M from upper to lower half-
transform of the Coulomb potential is VC (q) = mωp2 /ρq 2 complex plane.
(we use ~ = kB = 1 throughout the article). The present analysis of the plasmon dispersion focuses
We imagine that the system is driven at fixed frequency on the typical weak-coupling regime of superconductors,
ω and wavenumber q by an external field (for example with ∆ much smaller than the Fermi energy F , and the
an electromagnetic field) and we study the collective re- excitation wavelength comparable to the Cooper pair size
sponse within linear response theory. The response func- ξ = kF /2m∆. In this regime, the fluctuation of the mod-
tion can be computed either using a path integral formal- ulus of the order parameter are decoupled from the phase-
ism [26] with both a pair and density auxiliary fields, or density fluctuations:
in the Random Phase Approximation (RPA), neglecting
as in [4] the exchange-scattering diagrams (the effect of detM↓ = 0 ⇐⇒ M11,↓ M33,↓ − M13,↓ 2
= 0 or M22,↓ = 0
those diagrams is discussed in Ref. [27]). In a supercon- (6)
ductor, since the density response δρ is coupled to the The second condition gives rise the “pair-breaking” or
fluctuations of the order parameter, in phase δθ and a “Higgs” modulus mode which in the weak-coupling regime
priori in modulus δ|∆|, the linear response function χ is is insensitive to Coulomb interactions [22, 32]. Here, we
a 3 × 3 matrix: study the density-phase modes, fulfilling the first condi-
tion.
2i∆δθ(q, ω) uθ (q)
Anomalous dispersion of long wavelength plasmons:
2δ|∆(q, ω)| = χ(ω, q) u|∆| (q) , (2)
We first study analytically the plasmon dispersion in the
gδρ(q, ω) uρ (q)
limit q 1/ξ, where, by analogy with the normal case
[25], one can expect the quadratic law [7]:
where uθ , u|∆| and uρ are respectively the phase, modulus
and density driving fields [28]. The response matrix χ
is expressed in terms of the bare propagator Π as χ = q2
zq = ω0 + α + O(q 4 ) (7)
−M −1 Π with 2m
The expansion in powers of q is more easily performed
V /g 0 0
M = Π − D and D ≡ 0 V /g 0 (3) using the recombined matrix:
0 0 V /2VC (q)
M11 zM13 + 2∆M11
M
f=
Remark that due to the Coulomb potential M and Π zM13 + 2∆M11 z 2 M33 + 4∆zM13 + 4∆2 M11
do not commute, such that χ is not a symmetric matrix. (8)
The matrix Π was computed for instance in Refs. [28–30]. In particular the origin ω0 of the plasma branch is found3
simply by solving M
f33 = 0 to lowest order in q. We have 0 1.696
10 2
2
z2 8 1.5
ρV q
∆Im α/F
M33 =
f 1− 2 1
2m ωp 6
∆Re α/F
+ 0.5
Xm e 33 (1 − f+ − f− ) me − (f+ − f− ) 4
+ 2 2
− 2 33 (9) 0
z − (+ + − ) z − (+ − − )2 2 2 3 4 5 6
k ωp /∆
0
where we have used a sum rule1 to sim- −2
T =0
T /∆ = 10
plify the first line, and we set m e± 33 = T → Tc
[ξ+ − ξ− ] [+ ± − ] + − ∓ ξ+ ξ− ∓ ∆ /2+ − .
2 2
The −4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
summation on the second line is of order q 4 and will ωp /∆
only affect the expression of the dispersion parameter
α. Thus, the origin ω0 of the plasma branch always FIG. 1: Dispersion parameter Re α (multiplied by ∆/F to
coincides with the plasma frequency [9] have a finite weak-coupling limit), in function of the plasma
frequency at zero (black curve) and high temperature (red
ω0 = ωp (10) curve). The normal dispersion 6F /5ωp is shown by the red
dotted curve. The value where α changes sign at T = 0 is indi-
This expected results shows that superconductivity does cated by the black dotted line. Inset: the damping parameter
Im α, which becomes nonzero inside the pair-breaking contin-
not affect the departure of the plasma branch. As we uum [2∆, +∞[ (red area).
now explain, the situation is quite different for the low-q
dispersion and lifetime.
To extract the curvature α of the plasma branch, we ωp 2∆, the second term in (12) becomes negligi-
expand M f33 to subleading order in q, and consider the 2
ble, such that we recover the normal plasmon dispersion
other matrix elements α → 6F /5ωp [25]. In the opposite « quasiphononic »
X m+ (1 − f+ − f− ) limit ωp 2∆, which corresponds to the experimental
m− 11 (f+ − f− )
V
M11 = 11
2 2
− 2 2
− situation of Refs. [13, 15], rather than expanding for fixed
z − (+ + − ) z − (+ − − ) g z as prescribed by (7), one should expand for q → 0 while
k
+
e 13 (1 − f+ − f− ) −
e (f+ − f− ) keeping z/vF comparable to q [14]. This yields2
X m
m
M
f13 =
2 2
− 2 13 (11)
z − (+ + − ) z − (+ − − )2 q
k
zq −→ ωp2 + c2 q 2 (13)
q→0
with m±11 = (+ ± − )(+ − ± ξ+ ξ− ± ∆ )/2+ − and
2 cq/ωp fixed
±
e 13 = ±(+ ± − )(2ξ+ ξ− + 2∆ − + − − )∆/2+ − . At
m 2 2 2
√
zero temperature, this yields the fully analytic expression where c = vF / 3 is the speed-of-sound of the weakly-
of α: interacting condensate of neutral fermions.
At nonzero temperature, α depends on the dimen-
6F 32F ∆2 Arcsin (ωp /2∆) sionless temperature T̄ = T /∆ and plasma frequency
α= − (12)
ω̄p = ωp /∆. We find
q
5ωp 15ωp ω 4∆2 − ω 2
p p
F 6ω̄p 8
which is shown as a black curve on Fig. 1. This expression α= (I3 + J0 − J2 ) − I1 (14)
∆ 5 3ω̄p
remains valid when ωp > 2∆ and the plasma branch is
embedded in the pair-breaking continuum. In this case, in terms of the dimensionless integrals In =
one should use ωp → ωp + i0+ and Im α < 0 describes R +∞ th(/2T̄ ) R +∞
dξ n (ω̄2 −42 ) and Jn = 2T̄ ω̄2 0
1 dξ
with
0 n ch2 (/2T̄ )
the nonzero damping rate of plasmons. We note that p p
ξ + 1. The red curve in Fig. 1 shows α in
p
= 2
the repulsion of the pair-breaking threshold leads to a
squareroot divergence of Re α and Im α when approach- the vicinity of the critical temperature T /Tc = 0.9989
ing the pair-breaking threshold respectively from below (T /∆ = 10). We observe that α tends to its normal
and above. This opens an interval ωp ∈ [1.696∆, 2∆[ limit 6F /5ωp uniformly except in a neighborhood of size
where plasmons have an anomalous negative dispersion ≈ ∆2 /T around the pair-breaking threshold 2∆. There,
at the origin (Reα < 0). In the conventional limit the divergence of the real and imaginary part is preserved
whenever T < Tc , showing that a regime of anomalous
1
P
Explicitely, we have used k [(1 − f+ − f− ) (+ + − )
+ − − ξ+ ξ− − ∆ −(f+ − f− ) (+ − − ) + − + ξ+ ξ− + ∆2 ]
2 2 Note that this is consistent with the behavior of α in the limit
/2+ − = ρV q 2 /2m. ωp /∆ → 0.4
plasmon dispersion subsists until the transition to the 2.2 2.5
normal phase. In usual situations, ωp is fixed in units
2.1
of the Fermi energy F , but the ratio ωp /∆(T ) can still 2
be adjusted by varying the temperature. The negative 2
plasma dispersion will thus eventually occur when in- 1.5
ωqmin /∆
qmin ξ
creasing the temperature provided ωp < 2∆(T = 0). 1.9
1
Near Tc , we note that the plasma branch may also in- 1.8
teract with the phononic Carlson-Goldman excitations
0.5
[7, 29, 33] describing the motion of the superconduct- 1.7
ing electrons embedded in a majority of normal carri-
1.6 0
ers. A convincing description of this phenomenon re- 1.6 1.70 1.8 2 2.2 2.4 2.6 2.8
quires going beyond the collisionless regime of undamped ωp /∆
fermionic quasiparticles [34], which is beyond the scope
of this work. FIG. 2: The dispersion minimum ωqmin and the wavenumber
One could be surprised than plasmons remain un- qmin at which it is reached in function of the plasma frequency.
damped (Im α = 0) for ωp < 2∆ despite the nonzero For ωp < 1.696, qmin is identically 0 and ωqmin coincides with
temperature, which provides a decay channel through ωp (oblique dotted line). In the limit ωp → +∞, qmin diverges
linearly and ωqmin tends to 2∆.
quasiparticle-quasihole excitations. In fact, to absorb
a plasmon (i.e. to satisfy the resonance condition ωp =
q+k/2 − q−k/2 ) quasiparticles need to have a wavenum-
the analytically continued matrix M . At T = 0, the pair-
ber k > 2mωp /q. The plasmon lifetime thus follows an
2 2 breaking threshold 2∆ and the second branching point
activation law Imzq ∝ e−2mωp /q T which is exponentially [14]
suppressed in the limit ∆/F , T /F → 0 with ωp , q of
order ∆, 1/ξ. Intrinsic plasmon damping at ωp < 2∆ is
r
q2
thus essentially a strong-coupling effect. ω2 = 4∆2 + F (16)
2m
We conclude this section by computing the matrix
residue Zq = limz→zq (z − zq )χ(z, q), which quantifies divide the real axis in three analyticity windows (I, II and
the spectral weight of the plasma resonance. Writ- III, see the inset of Fig. 3), each supporting a separate
z→ωp
ing χ = −1 − M −1 D and using d(detM )/dz → complex root (ωqI , zqII , zqIII respectively) of Eq. (5). This
q→0
suggests a splitting of the plasma resonance into 3 peaks.
M11 M22 dMf33 /z 2 dz together with M11 = z 2 M33 /4∆2 =
While the quadratic Eq. (7) give the low-q dispersion of
−zM13 /2∆ to leading order in q, we obtain, in the phase-
ωqI and zqIII (respectively for ωp < 2∆ and ωp > 2∆), the
density sector:
pole of window II starts from 2∆ and departs following
ωp 4m∆
1
! a non-integer power-law:
ρg q 2
Zq = ∆ ωp2 2m + O(q 2 ) (15)
ω /2∆
s 3/2
i±1 4∆2
ρg q 2 p 8 kF q
II 7/4
zq = 2∆ − √ 1 − + O q
Note that the phase and density excitation channels (re- ∆ 3π 2 ωp2 2m
spectively first and second line of Zq ) dominate respec- (17)
tively in the limits ωp → 0 and ωp → +∞. where the sign of the real part of zqII − 2∆ is negative
Dispersion minimum and resonance splitting at non- if ωp < 2∆ and positive otherwise (in either case zqII re-
vanishing wavenumber: Outside the limit qξ 1, we mains outside the natural interval [2∆, ω2 ] of window II).
study the dispersion of plasmons by numerically evaluat- Remarkably, when ωp = 2∆ the quadratic law reemerges
q2
ing the M matrix. We first characterize on Fig. 2 the dis- zqII = 2∆ − (0.0184 + 0.9953i) ∆F 2m + O(q 4 ). Those re-
persion minimum of the plasma branch. For ωp > 1.696, sults are obtained by expanding at low q as prescribed
it is reached at a nonzero wavenumber qmin (blue curve by Eq. (10) in [21].
in Fig. 2), such that the band gap of the plasma branch On Fig. 3, we show the dispersion relation of these
(black curve) is strictly lower than ωp . As visible on solutions for ωp = 1.9∆. In this case ωqI supports the
Fig. 2, this undamped plasma branch at ωq < 2∆ persists main plasma branch departing in ωp (while for ωp > 2∆
even when ωp > 2∆. This is a first sign of the splitting the main branch would be supported by zqIII ), zqII belongs
of the plasma resonance. However, in the normal limit to an indirect region of the analytic continuation (specifi-
ωp /∆ → +∞ the undamped branch tends uniformly to cally RezqII < 2∆) and zqIII follows rather closely the angu-
2∆ with a vanishingly small spectral weight. lar point ω2 . This subtle analytic structure is reflected in
Since we expect the plasma branch to eventually enter frequency behavior of the density-density response func-
the pair-breaking continuum, the characterization of the tion shown on Fig. 4. Besides the Dirac peak below 2∆,
resonance at finite q requires a numerical exploration of one (black curve) then two (grey curves) broadened peaks5
3.5
Sec I 5
Sec II qξ = 0.2
3 Sec II, modulus mode qξ = 0.5
Sec III 4 qξ = 1.0
2∆
2.5
χρρ (ω + i0+ )
ω2
Reωq /∆
3
2
•
Imz
1.5 (I) 2
ωq 2∆ ω2 Rez
× × ×
0
1 I II III 1
×
(II) ×zq(III)
zq
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 1.8 2 2.2 2.4 2.6 2.8 3
qξ ω/∆
FIG. 3: The eigenfrequency Rezq of the plasma branch in FIG. 4: Density-density response of the superconductor in
function of the wave vector q (in unit of the inverse pair radius function of the excitation frequency ω at fixed plasma fre-
ξ = kF /2m∆), with ωp = 1.9∆. The angular points 2∆ and quency ωp = 1.9∆ and excitation wave number q = 0.2/ξ,
ω2 (Eq. (16)) are shown as dotted lines. The analytic windows 0.5ξ and 1.0ξ (corresponding to the vertical dotted lines in
are shown in colors: white for ω < ω1 (window I), blue for Fig. 3). The angular points 2∆ and ω2 (q) are marked by verti-
ω1 < ω < ω2 (window II) and red for ω > ω2 (window III). cal dotted lines. Besides the Dirac peaks below 2∆, broadened
The solution of Eq. (6) in each window in shown as a solid line peaks are visible inside the pair-breaking continuum. Thin
in the corresponding color. The inset shows their schematic dashed lines show the modulus response ρg/F M22 (ω + i0+ ).
trajectories in the complex plane after analytic continuation.
In window II, the pair-breaking mode (solution of M22,↓ = 0)
is shown as a dashed line. The dispersion minimum of the
undamped solution below 2∆ is shown by the black dot. [2] Romain Grasset, Yann Gallais, Alain Sacuto, Maximi-
lien Cazayous, Samuel Mañas Valero, Eugenio Coronado,
and Marie-Aude Méasson. Pressure-Induced Collapse of
the Charge Density Wave and Higgs Mode Visibility in
appear inside the pair-breaking continuum [5] as q is in- 2H-TaS2 . Phys. Rev. Lett., 122:127001, Mar 2019.
creased. In the time domain, this remarkable frequency [3] Ryusuke Matsunaga, Yuki I. Hamada, Kazumasa Makise,
behavior corresponds to an evolution (after e.g. an elec- Yoshinori Uzawa, Hirotaka Terai, Zhen Wang, and Ryo
tromagnetic quench of the electronic density) where the Shimano. Higgs Amplitude Mode in the BCS Supercon-
system oscillated at several frequencies: an exponentially ductors Nb1−x Tix N Induced by Terahertz Pulse Excita-
decaying high-frequency component and a long-lived low- tion. Phys. Rev. Lett., 111:057002, July 2013.
[4] P. W. Anderson. Random-Phase Approximation in the
frequency component.
Theory of Superconductivity. Phys. Rev., 112:1900–1916,
Conclusion: We have described the low-q quadratic December 1958.
dispersion of superconducting plasmons in 3D, and the [5] H. A. Fertig and S. Das Sarma. Collective excitations and
resonance splitting which occurs when the eigenenergy mode coupling in layered superconductors. Phys. Rev. B,
nears the pair-breaking threshold. For a more realistic 44:4480–4494, Sep 1991.
description of plasmons in cuprates, our study should [6] R. Côté and A. Griffin. Cooper-pair-condensate fluctu-
ations and plasmons in layered superconductors. Phys.
be extended to 2D superconductors [35], with possibly
Rev. B, 48:10404–10425, Oct 1993.
Josephson interlayer couplings [16, 36]. Our work may [7] S.N. Artemenko and A.G. Kobel’kov. Collective modes in
also be applied to superfluids of ultracold fermions [37] layered superconductors. Physica C: Superconductivity,
where different kind of long-range interactions can be en- 253(3):373–382, 1995.
gineered with dipolar atoms [38]. [8] D. van der Marel. Collective modes of spin, density,
We acknowledge financial support from the Research phase, and amplitude in exotic superconductors. Phys.
Foundation-Flanders (FWO- Vlaanderen) Grant No. Rev. B, 51:1147–1160, January 1995.
[9] Yoji Ohashi and Satoshi Takada. On the plasma oscilla-
G.0618.20.N, and from the research council of the Uni- tion in superconductivity. Journal of the Physical Society
versity of Antwerp. of Japan, 67(2):551–559, 1998.
[10] M. Baldo and C. Ducoin. Plasmons in strong supercon-
ductors. Physics of Atomic Nuclei, 74(10):1508, 2011.
[11] K. Tamasaku, Y. Nakamura, and S. Uchida. Charge dy-
namics across the CuO2 planes in La2−x Srx CuO4 . Phys.
[1] M.-A. Méasson, Y. Gallais, M. Cazayous, B. Clair, Rev. Lett., 69:1455–1458, Aug 1992.
P. Rodière, L. Cario, and A. Sacuto. Amplitude Higgs [12] O. Buisson, P. Xavier, and J. Richard. Observation of
mode in the 2H − NbSe2 superconductor. Phys. Rev. B, Propagating Plasma Modes in a Thin Superconducting
89:060503(R), February 2014. Film. Phys. Rev. Lett., 73:3153–3156, December 1994.6
[13] K. Kadowaki, I. Kakeya, M. B. Gaifullin, T. Mochiku, the Carlson-Goldman Mode and Effects of the Landau
S. Takahashi, T. Koyama, and M. Tachiki. Longitudi- Damping in the Superconducting State. Journal of the
nal Josephson-plasma excitation in Bi2 Sr2 CaCu2 O8+δ : Physical Society of Japan, 66(8):2437–2458, 1997.
Direct observation of the Nambu-Goldstone mode in a [30] Y. Ohashi and A. Griffin. Superfluidity and collective
superconductor. Phys. Rev. B, 56:5617–5621, September modes in a uniform gas of Fermi atoms with a Feshbach
1997. resonance. Phys. Rev. A, 67:063612, June 2003.
[14] E. H. Hwang and S. Das Sarma. Collective modes and [31] Hadrien Kurkjian and Jacques Tempere. Absorption and
their coupling to pair-breaking excitations in layered d- emission of a collective excitation by a fermionic quasi-
wave superconductors. Phys. Rev. B, 52:R7010–R7013, particle in a Fermi superfluid. New Journal of Physics,
September 1995. 19(11):113045, 2017.
[15] F. J. Dunmore, D. Z. Liu, H. D. Drew, S. Das Sarma, [32] V. A. Andrianov and V. N. Popov. Gidrodi-
Qi Li, and D. B. Fenner. Observation of below-gap plas- namičeskoe dejstvie i Boze-spektr sverhtekučih Fermi-
mon excitations in superconducting YBa2 Cu3 O7 films. sistem. Teoreticheskaya i Matematicheskaya Fizika,
Phys. Rev. B, 52:R731–R734, July 1995. 28:341–352, 1976. [English translation: Theoretical and
[16] H. Shibata and T. Yamada. Double Josephson Plasma Mathematical Physics, 1976, 28:3, 829–837].
Resonance in T ∗ Phase SmLa1−x Srx CuO4−δ . Phys. Rev. [33] R. V. Carlson and A. M. Goldman. Superconducting
Lett., 81:3519–3522, October 1998. Order-Parameter Fluctuations below Tc . Phys. Rev.
[17] E. A. Pashitskii and V. I. Pentegov. On the plasmon Lett., 31:880–883, October 1973.
mechanism of high-Tc superconductivity in layered crys- [34] S.N. Artemenko and A.F. Volkov. Collective oscillations
tals and two-dimensional systems. Low Temperature in superconductors revisited. arXiv:cond-mat/9712086,
Physics, 34(2):113–122, 2008. 1997.
[18] A. Dienst, E. Casandruc, D. Fausti, L. Zhang, M. Eck- [35] Zhiyuan Sun, M. M. Fogler, D. N. Basov, and Andrew J.
stein, M. Hoffmann, V. Khanna, N. Dean, M. Gensch, Millis. Collective modes and terahertz near-field response
S. Winnerl, W. Seidel, S. Pyon, T. Takayama, H. Takagi, of superconductors. Phys. Rev. Research, 2:023413, June
and A. Cavalleri. Optical excitation of Josephson plasma 2020.
solitons in a cuprate superconductor. Nature Materials, [36] Rong Guo, Siyu Duan, Ze He, Xuelian Liang, Zi-
12(6):535–541, 2013. han Niu, Ming He, Yushun Jiang, Jingbo Wu,
[19] H. T. Stinson, J. S. Wu, B. Y. Jiang, Z. Fei, A. S. Rodin, Lu Ji, Biaobing Jin, Jian Chen, Huabing Wang,
B. C. Chapler, A. S. McLeod, A. Castro Neto, Y. S. and Peiheng Wu. Josephson Plasmon Resonance
Lee, M. M. Fogler, and D. N. Basov. Infrared nanospec- in Tl2 Ba2 CaCu2 O8 High-Temperature Superconductor
troscopy and imaging of collective superfluid excitations Tunable Terahertz Metamaterials. Advanced Functional
in anisotropic superconductors. Phys. Rev. B, 90:014502, Materials, n/a(n/a):2106891, 2021.
Jul 2014. [37] Wilhelm Zwerger, editor.
[20] Mauro M. Doria, Gilberto Hollauer, F. Parage, and The BCS-BEC Crossover and the Unitary Fermi Gas.
O. Buisson. Plasma waves in anisotropic superconduct- Springer Verlag, 2012.
ing films below and above the plasma frequency. Phys. [38] L. Chomaz, R. M. W. van Bijnen, D. Petter, G. Faraoni,
Rev. B, 56:2722–2731, August 1997. S. Baier, J. H. Becher, M. J. Mark, F. Wächtler, L. San-
[21] H. Kurkjian, S. N. Klimin, J. Tempere, and Y. Castin. tos, and F. Ferlaino. Observation of roton mode pop-
Pair-Breaking Collective Branch in BCS Superconduc- ulation in a dipolar quantum gas. Nature Physics,
tors and Superfluid Fermi Gases. Phys. Rev. Lett., 14(5):442–446, 2018.
122:093403, March 2019.
[22] Yvan Castin and Hadrien Kurkjian. Branche d’excitation
collective du continuum dans les gaz de fermions con-
densés par paires : étude analytique et lois d’échelle.
Comptes Rendus. Physique, 21(3):253–310, 2020.
[23] S. N. Klimin, J. Tempere, and H. Kurkjian. Phononic
collective excitations in superfluid Fermi gases at nonzero
temperatures. Phys. Rev. A, 100:063634, December 2019.
[24] Philippe Nozières. Le problème à N corps : propriétés
générales des gaz de fermions. Dunod, Paris, 1963.
[25] Alexander L. Fetter and John Dirk Walecka.
Quantum theory of many-particle systems. McGraw-
Hill, San Francisco, 1971.
[26] Jan R. Engelbrecht, Mohit Randeria, and C. A. R. Sá de
Melo. BCS to Bose crossover: Broken-symmetry state.
Phys. Rev. B, 55:15153–15156, June 1997.
[27] A. Bardasis and J. R. Schrieffer. Excitons and Plasmons
in Superconductors. Phys. Rev., 121:1050–1062, Febru-
ary 1961.
[28] H. Kurkjian, J. Tempere, and S. N. Klimin. Linear re-
sponse of a superfluid Fermi gas inside its pair-breaking
continuum. Scientific Reports, 10(1):11591, 2020.
[29] Yoji Ohashi and Satoshi Takada. Goldstone Mode
in Charged Superconductivity: Theoretical Studies ofYou can also read
