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arXiv:cond-mat/9911131v2 [cond-mat.supr-con] 11 Sep 2000

Collective modes and sound propagation in a p-wave superconductor: Sr2RuO4

Hae-Young Kee1, Yong Baek Kim2, and Kazumi Maki3

1Department of Physics, University of California, Los Angeles, CA 90095

2Department of Physics, The Ohio State University, Columbus, OH 43210

3Department of Physics, University of Southern California, Los Angeles, CA 90089

(February 1, 2008)

There are five distinct collective modes in the recently discovered p-wave superconductor Sr2RuO4;

phase and amplitude modes of the order parameter, clapping mode (real and imaginary), and spin

wave. The first two modes also exist in the ordinary s-wave superconductors, while the clapping

mode with the energy

report a theoretical study of the sound propagation in a two dimensional p-wave superconductor. We

identified the clapping mode and study its effects on the longitudinal and transverse sound velocities

in the superconducting state. In contrast to the case of

associated with the collective mode, since in metals ω/(vF|q|) ≪ 1, where vF is the Fermi velocity,

q is the wave vector, and ω is the frequency of the sound wave. However, the velocity change in the

collisionless limit gets modified by the contribution from the coupling to the clapping mode. We

compute this contribution and comment on the visibility of the effect. In the diffusive limit, the

contribution from the collective mode turns out to be negligible. The behaviors of the sound velocity

change and the attenuation coefficient near Tc in the diffusive limit are calculated and compared

with the existing experimental data wherever it is possible. We also present the results for the

attenuation coefficients in both of the collisionless and diffusive limits at finite temperatures.

√2∆(T) is unique to Sr2RuO4 and couples to the sound wave. Here we

3He, there is no resonance absorption

PACS numbers: 74.20.-z, 74.25.Ld, 74.25.-q

I. INTRODUCTION

Shortly after the discovery of superconductivity in Sr2RuO4, the possibility of spin triplet pairing was discussed.

[1] Possible pairing symmetries were also classified based on the crystal symmetry. [2] On the experimental front,

there have been attempts to single out the right pairing symmetry among these possibilities. Recent measurement of

17O-Knight shift in NMR for the magnetic field parallel to the a−b plane showed no change across Tc, which can be

taken as the evidence of the spin triplet pairing withˆd-vector parallel to the c-axis. [3] Hereˆd is called the spin vector

which is perpendicular to the direction of the spin associated with the condensed pair [4]. µSR experiment found

spontaneous magnetic field in the superconducting Sr2RuO4, which seems to indicate broken time reversal symmetry

in the superconducting state. [5] These experiment may be compatible with the following order parameter [2]

ˆ∆(k) = ∆ˆd(k1± ik2), (1)

where ∆ is the magnitude of the superconducting order parameter. Notice that this state is analogous to the A phase

of3He and there is a full gap on the Fermi surface.

On the other hand, there also exist experiments that cannot be explained by a naive application of the order

parameter given by Eq.(1). Earlier specific heat measurement found residual density of states at low temperatures

below Tc[6], which provokesthe ideas of orbital dependent superconductivity [7] and even non-unitary superconducting

state [8]. However, more recent specific heat experiment on a cleaner sample reports no residual density of states

and it was found that the specific heat behaves as T2at low temperatures. [9] This result stimulated a speculation

about different order parameters with line node [10]. However, since there are three bands labeled by α, β, and γ

which cross the Fermi surface, it is not yet clear whether the order parameter given by Eq.(1) is compatible with more

recent specific heat data or not. For example, it is possible that the pairing symmetry associated with the γ band is

still given by Eq.(1) while the order parameter symmetry associated with α and β bands can be quite different. In

this case, the low temperature specific heat will be dominated by the excitations from α and β bands. In order to

resolve the issue, it is important to examine other predictions of the given order parameter and compare the results

with future experiments.

One way of identifying the correct order parameter among possible candidates is to investigate the unique collective

modes supported by the ground state with a given pairing symmetry. The observation of the effects of these collective

modes would provide convincing evidence for a particular order parameter symmetry. If we assume that the order

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parameter of Eq.(1) is realized in Sr2RuO4, the superconducting state would support unique collective modes, the

so-called clapping mode and spin waves as well as the phase and amplitude modes of the order parameter which

exist also in s-wave superconductors. Previously we studied the dynamics of spin waves [11,12]. A possible way to

distinguish the order parameter of the γ band from those of the α (or β) band was also proposed in the context of

spin wave dynamics. [13]

In this paper, we study the dynamics of the sound wave and its coupling to the clapping modes assuming that the

order parameter is given by Eq.(1). As in3He, only the clapping mode can couple to the sound wave and affects its

dynamics. Here we study the sound velocities and attenuation coefficients of the longitudinal and transverse sound

waves. In particular, we identify the clapping mode with the frequency, ω =√2∆(T), and examine the effects of this

mode and disorder on the sound wave propagation.

In a recent paper, Higashitani and Nagai [14] obtained the clapping mode with the frequency, ω =

discussed the possible coupling to the sound wave independent of us. It is, however, important to realize that the

coupling to the sound wave is extermely small because C/vF ≪ 1 in metals, where C is the sound velocity. Indeed

the recent measurement of the sound velocity in the normal and superconducting states of Sr2RuO4reported in [15]

shows that C/vF ≪ 1. They measured the sound velocities of the longitudinal modes, C11 (q,u ? [100]) and C33

(q,u ? [001]), and the transverse modes, C44 (q ? [100], u ? [001]) and C66 (q ? [100], u ? [010]), where q and u

are the directions of propagation and the polarization of ultrasound, respectively. They found that the longitudinal

sound velocities, C11and C33, decrease with a kink at T = Tc, while the transverse sound velocities do not exhibit

any effect of the onset of superconductivity. We estimate from their experimental data that Cl/vF∼ 10−2, where Cl

is the longitudinal sound velocity. It can be also seen that the transverse sound velocity, Ct, is much smaller than the

longitudinal one [15].

Incorporating the correct limit, C/vF ≪ 1, we obtained the sound velocities and attenuation coefficients for both

collisionless and diffusive limits. In the diffusive limit, the quasi-particle scattering due to impurities should be

properly taken into account. One can show that, in a metal like Sr2RuO4, the collisionless limit is rather difficult

to reach because it can be realized only for ω ∼ O(1) GHz. For more practical range of frequencies, kHz − MHz,

the diffusive limit may be easier to achieve. On the other hand, we found that it is much easier to see the effects of

the coupling between the sound waves and the clapping mode in the collisionless limit. Therefore it is worthwhile to

study both regimes.

Here we summarize our main results.

√2∆, and

A. Collisionless limit

In the absence of the coupling to the clapping mode, the longitudinal sound velocity decreases in the supercon-

ducting state because the effect of the screening of the Coulomb potential increases, which happens in the s-wave

superconductors as well. However, one of the important features of the p-wave order parameter in consideration is

that the sound wave can now couple to the clapping mode. This effect is absent in s-wave superconductors. One can

show that, among longitudinal waves, C11mode can couple to the clapping mode, but C33mode cannot. We found

that the longitudinal sound velocity C11decreases as

δC11

C11

l

l

= −λ11

l

?

1

2− 2

?C11

l

vF

?2

(1 − f −

f

4{1 + (2∆(T)/vFq)2})

?

.(2)

where λl is the couping constant and f is the superfluid density. δCl/Clis the relative shift in the sound velocity.

We estimated the frequency regime where one can observe the effect of the clapping mode and found that the effect

is visible if vF|q| ∼ 2 − 3∆(0). This implies that ω ∼ O(1) GHz. Since C33does not couple to the clapping mode,

the velocity change is simply given by

δC33

C33

l

l

= −λ33

l

?

1

2− 2

?C33

l

vF

?2

(1 − f)

?

. (3)

Since the velocity of the transverse wave is much smaller than that of the longitudinal one and the coupling to

the electron system is weaker than the longitudinal case as well, we expect that the change of the transverse sound

velocity is hard to observe. In order to complete the discussion, we also present the results for these small changes in

transverse velocities. Here only C66mode couples to the clapping mode, and C44mode does not. We found

δC66

C66

t

t

= −λ66

t

?

1

2+ 2

?C66

t

vF

?2?

1 − f +

f

4{1 + (2∆/vFq)2}

??

, (4)

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δC44

C44

t

t

= −λ44

t

?

1

2+ 2

?C44

t

vF

?2

(1 − f)

?

, (5)

where the λtis the transverse coupling constant.

We also found that the leading contribution to the attenuation coefficient is the same as that of s-wave supercon-

ductors in the collisionless limit.

B. Diffusive limit

This case corresponds to ω,vF|q| ≪ Γ, where Γ is the scattering rate due to impurities. As in the case of the

collisionless limit, in principle C11and C66modes couple to the clapping mode, but it turns out that the effect is almost

impossible to detect. Neglecting the coupling to the clapping mode and working in the limit 4πTc≫ 2Γ ≫ vF|q|,ω,

we obtain the following results near Tc.

δCl

Cl

αl

αn

= −λl1

2

?

2π3

7ζ(3)

1 −

?ω

?

2Γ

1 −T

?2?

1 −

?

4π3

7ζ(3)

T

Γ

?

1 −T

Tc

???

,

l

= 1 −

T

ΓTc

, (6)

where α and αn are the attenuation coefficients in the superconducting and the normal states respectively. The

shift of the sound velocity and attenuation coefficient decrease linearly in (1 − T/Tc) as T → Tc. This result for the

longitudinal sound wave is consistent with the experimental observation reported in [15]

In the case of the transverse sound waves, the leading behaviors of the sound velocity and the attenuation coeffient

can be obtained simply by replacing Cland λlby Ctand λtin the diffusive limit. However, the absolute value of the

transverse sound velocity is much smaller than the longitudinal one and the coupling to the electron system is also

much weaker than the case of the longitudinal sound waves. Thus, it would be hard to observe any change at T = Tc

for the transverse wave. This may explain the experimental finding that the transverse velocity does not show any

change across Tc. We also obtained the attenuation coefficient for all temperatures below Tc. It is given by Eq.(36)

and Fig.2 shows its behavior.

The rest of the paper is organized as follows. In section II, the clapping mode is briefly discussed. In section III,

we provide a brief summary of the formalism used in Ref. [16] to explain how the sound velocity and the attenuation

coefficients are related to the autocorrelation functions of the stress tensor. We present the results of the study on the

sound propagation in the collisionless and diffusive limits in sections IV and V, respectively. We conclude in section

VI. Further details which are not presented in the main text are relegated to the Appendix A and B.

II. COLLECTIVE MODES IN SR2RUO4

As in the s-wave superconductors, the phase and amplitude modes of the order parameter also exist in the p-wave

superconductors. On the other hand, due to the internal structure of the Cooper pair in the p-wave superconductor,

there exist other types of collective mode associated with the order parameter. The nature of these modes is determined

by the structure of the order parameter.

There are collective modes associated with the oscillation of the spin vectorˆd, which we have already discussed

in [11] and [12]. There exists another collective mode associated with the orbital part. Using the notation e±iφ=

(k1±ik2)/|k|, the oscillation of the orbital part e±iφ→ e∓iφgives rise to the clapping mode with ω =√2∆(T). This

mode couples to the sound waves as we will see in the next section. Therefore, the detection of the clapping mode

will provide a unique evidence for the p-wave superconducting order parameter. The derivation of the clapping mode

and the coupling to the sound wave is discussed in Appendix A.

III. DYNAMICS OF SOUND WAVE VIA STRESS TENSOR

In ordinary liquids, the sound wave is a density wave. In superconductors, the density is not only coupled to the

longitudinal component of the normal velocity, but also to the superfluid velocity and to temperature. The role of

these couplings and their consequences in the dynamics of sound wave can be studied by looking at the autocorrelation

function, ?[τij,τij]?, of stress tensor, τij.

?[τij,τij]?(r − r′,t − t′) ≡ −iθ(t − t′)?[τij(r,t),τij(r′,t′)]? ,(7)

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where

τij(r,t) =

?

σ

?

(∇ − ∇

′)i

2i

(∇ − ∇

2im

′)j

ψ†

σ(r,t)ψσ(r′,t)

?

r′=r

. (8)

Here ψ†

The other operators whose correlation functions are needed for the ultrasonic attenuation and the sound velocity

change are the density operator

σis the electron creation operator with spin σ.

n(r,t) =

?

σ

ψ†

σ(r,t)ψσ(r,t), (9)

and the current operator

j(r,t) =

?

σ

?

(∇ − ∇

2im

′)j

ψ†

σ(r,t)ψσ(r′,t)

?

r′=r

. (10)

Assuming that the wave vector of the sound wave is in the ˆ x direction, q = qˆ x, the sound velocity shift, δC, at low

frequencies can be computed from

δCl

Cl

=Cl(ω) − Cl

Cl

????

ω=Cl|q|

????

hl(r,t) =q

= −

ω

mionCl|q|Re?[hl,hl]?(q,ω)

ω

mionCt|q|Re?[ht,ht]?(q,ω)

????

ω=Cl|q|

????

,

δCt

Ct

=Ct(ω) − Ct

Ct

ω=Ct|q|

= −

ω=Ct|q|

, (11)

where

ωτxx(q,t) −ωm

q

n(r,t),

ht(r,t) =q

ωτxy(q,t) − mjy(r,t),(12)

Here Cland Ctrepresent the longitudinal and transverse sound velocities in the normal state respectively. mionand

m are the mass of ions and the mass of electron, respectively. On the other hand, the attenuation coefficient, α, at

low frequencies is obtained from

αl=

ω

mionClIm?[hl,hl]?(q,ω)

????

ω=Cl|q|

,αt=

ω

mionCtIm?[ht,ht]?(q,ω)

????

ω=Ct|q|

. (13)

These relations are extensively discussed in the work of Kadanoff and Falko. [16]

IV. SOUND PROPAGATION IN THE COLLISIONLESS LIMIT

As discussed in the introduction, in a metal like Sr2RuO4 the collisionless limit is somewhat difficult to reach

because we need the sound wave with the frequency ω ∼ O(1) GHz. However, we will also see that this is the regime

where one has the best chance to observe the effect of the collective mode.

The sound velocity shift and the attenuation coefficients can be calculated by looking at the autocorrelation func-

tions, ?[τij,τij]?, of stress tensor, τij, as discussed in the previous section. We will use the finite temperature Green’s

function technique [17] to compute these correlation functions. The single particle Green’s function, G(iωn,k), in the

Nambu space is given by

G−1(iωn,k) = iωn− ξkρ3− ∆(ˆk · ˆ ρ)σ1,(14)

where ρi and σi are Pauli matrices acting on the particle-hole and spin space respectively, ωn= (2n + 1)πT is the

fermionic Matsubara frequency, and ξk= k2/2m − µ. Then, for example, the irreducible correlation function can be

computed from

?[τij,τij]?00(iων,q) = T

?

n

?

p

Tr[

?pipj

m

?2

ρ3G(p,ωn)ρ3G(p − q,iωn− iων)] ,(15)

where ων= 2νπT is the bosonic Matsubara frequency.

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A. Longitudinal sound wave

Let us consider the longitudinal wave with u ? q ? x, which corresponds to the C11mode. Since the stress tensor

couples to the density, the autocorrelation function ?[hl,hl]? is renormalized by the Coulomb interaction. In the long

wavelength limit and for s ≡ ω/vF|q| ≪ 1, the renormalized correlation function ?[hl,hl]?0can be reduced to

?q

where

Re?[hl,hl]?0≈

ω

?2

Rep4

F

4m2?cos(2φ),cos(2φ)? =

p4

4m2N(0)

F

?1

2− 2s2(1 − f)

?

, (16)

?A,B? = T

?

n

?

p

Tr[Aρ3G(p,ωn)Bρ3G(p − q,iωn− iων)] ,(17)

with A and B being some functions of φ or operators. Here φ is the angle between p and q, N(0) = m/2π is the

density of states at the Fermi level and f is the superfluid density in the static limit (ω ≪ vF|q|) given by

∞

?

The derivation of the result in Eq.(16) is given in Appendix B 2. Therefore, the sound velocity shift, δCl, is given by

f = 2πT∆2

n=0

1

ω2

n+ ∆2

1

?ω2

n+ ∆2+ (vFq)2/4

.(18)

δCl

Cl

= −λl[1

2− 2s2(1 − f)], (19)

where λl= p2

the longitudinal sound velocity. In Sr2RuO4, s = 10−2≪ 1 which is very different from s ≫ 1 of3He.

Now let us consider the correction due to the collective modes. The additional renormalization of ?[hl,hl]? (in the

C11mode) due to the collective mode is computed in Appendix B 3 and the result is given by

F/(8πmmionC2

l) is the longitudinal coupling constant. Here we set s = ω/vF|q| = Cl/vF, where Clis

?[hl,hl]? =

?q

ω

?2p4

F

4m2

?1

2− 2s2

?

1 − f −

f(vF|q|)2

4{(vF|q|)2+ 4∆(T)2− 2ω2}

??

+ imionC11

l

ω

αl(ω) .(20)

As one can see from the above equation, there is no resonance because ω ≪ vF|q|. However, we will be able to see a

shadow of the collective mode in the sound velocity change, which we discuss in the following.

In the limit s ≪ 1 and setting s = C11

δC11

l

C11

l

vF

l/vF, the above equation leads to the sound velocity shift given by

= −λ11

l

?

1

2− 2

?C11

l

?2?

1 − f −

f

4{1 + (2∆/vFq)2}

??

. (21)

Note that the sound wave gets soften more by the collective mode. In Fig. 1, we show I = 1 −f −

for vF|q|/∆(0) = 0,1,2,3 for 0.7 < t < 1.0 where t = T/Tc. Note that the coupling to the collective mode can be

observed for vF|q| ∼ 2 − 3∆(0) which corresponds to ω ∼ O(1) GHz.

The attenuation coefficient, αl, is given by

f

4[1+(2∆(T)/vF|q|)2]

αl(ω)

αn

l(ω)=1ω

?∞

∆

dω′

ω′(ω′+ ω) − ∆2

√ω′2− ∆2?(ω′+ ω)2− ∆2

− θ(ω − 2∆)1

ω

∆

?

tanhω + ω′

2T

− tanhω′

2T

?

?ω−∆

dω′

ω′(ω′− ω) − ∆2

√ω′2− ∆2?(ω′− ω)2− ∆2

?

tanhω′

2T

?

,(22)

where αn

ductors.

We can carry out a parallel analysis for the longitudinal wave with u ? q ? z which corresponds to the C33mode.

Unfortunately, this sound wave does not couple to the clapping mode. Therefore, the velocity shift of this sound wave

is simply given by

lis the attenuation coefficient in the normal state. This form is the same as the one in the s-wave supercon-

δC33

C33

l

l

= −λ33

l

?

1

2− 2

?C33

l

vF

?2

(1 − f)

?

. (23)

.

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B. Transverse sound wave

Here we consider first the C66 mode that has u ? y and q ? x. In this case, the sound velocity change can be

obtained from the evaluation of ?[ht,ht]?. Assuming that the current contribution is negligible at low frequencies and

following the same procedure used in the case of the longitudinal sound wave, we obtain

δC66

C66

t

t

= −λ66

t

?

1

2+ 2

?C66

t

vF

?2?

1 − f +

f

4{1 + (2∆/vFq)2}

??

, (24)

where the λt= p2

upon entering the superconducting state. However, due to the fact that the transverse velocity is rather small and the

coupling to the electron system is also weak compared to the longitudinal case, it will be hard to observe the change

of the transverse sound velocity at T = Tc.

Another transverse sound mode, C44, that has u ? z and q ? x, does not couple to the clapping mode. Thus the

sound velocity change in this case is given by

F/(8πmmionC2

t) is the transverse coupling constant. Note that the transverse sound velocity increases

δC44

C44

t

t

= −λ44

t

?

1

2+ 2

?C44

t

vF

?2

(1 − f)

?

. (25)

V. THE DIFFUSIVE LIMIT

In the frequency range kHz ∼ MHz, the diffusive limit is more realistic. In this limit, the incorporation of the

quasi-particle damping is very important. Here we assume for simplicity that the quasi-particle scattering is due to

impurities. Unlike the case of s-wave superconductors, we treat the impurity scattering in the unitary limit. Then

the effect of the impurity is incorporated by changing ωnto ˜ ωn(renormalized Matsubara frequnecy) in Eq.(14) [18].

The impurity renormalized complex frequency, ˜ ωnis determined from

˜ ωn= ω + Γ

?˜ ω2

n+ ∆2

˜ ωn

,(26)

where Γ is the quasi-particle scattering rate and the quasi-particle mean free path is given by l = vF/(2Γ).

In order to compare the results in the normal state and those in the superconducting state, let us first work out

the correlation functions in the normal state, where ∆ = 0.

A. Normal state

We can use Eq.(B5) to compute ?[hl,hl]?0. In the limit of ω,vFq ≪ 2Γ, we get

?cos(2φ),cos(2φ)? = ?cos2(2φ)

?

1 −

ω

ω + 2iΓ − ζ

?

? ≈1

2

?

1 − (ω

2Γ)2+ iω

2Γ

?

. (27)

One can show that ?cos(2φ),1? is of higher order in ω/2Γ and vF|q|/2Γ while ?1,1? ≈ 2?cos(2φ),cos(2φ)? to the lowest

order. Thus, as in the previous section, ?[hl,hl]? is well approximated by ?cos(2φ),cos(2φ)? times a multiplicative

factor. This gives us

δCn

Cl

l

≈ −λl1

2

?

1 − (ω

2Γ)2?

,αn

l≈ λl|q|(ω

2Γ) , (28)

where ω is set to Cl|q|.

It is not difficult to see that the results of the transverse sound wave is essentially the same as the longitudinal case

up to the lowest order with a simple replacement of λl and Cl by λt and Ct. Therefore, in the diffusive limit, the

longitudinal and transverse sound velocities have the same form with different coupling constants.

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B. Superconducting state near Tc

Now we turn to the case of the superconducting state near Tc, where the correlation functions can be computed

from Eq.(B6) after replacing ωnby ˜ ωn. In this section, we will assume 4πTc≫ 2Γ ≫ vF|q| and use

As in the previous sections, the leading contribution in ?[hl,hl]? can be computed from ?cos(2φ),cos(2φ)?.

After some algebra, we finally obtain

∆

2πT≪ 1 near Tc.

?cos(2φ),cos(2φ)? ≈1

2−1

+i1

2(ω

2(ω

2Γ)2

?

?

1 −

∆2

πΓT

∆2

2πΓT

?

ψ(1)(1

ψ(1)(1

2+

Γ

2πT) −

Γ

2πT) −

Γ

8πTψ(2)(1

Γ

4πTψ(2)(1

2+

Γ

2πT)

Γ

2πT)

??

2Γ)

1 −

?

2+2+

??

, (29)

where

ψ(n)(z) =

?d

dz

?n

ψ(z) = (−1)n+1n!

∞

?

k=0

1

(z + k)n+1. (30)

Here ψ(n)(z) is the poly-Gamma function and ψ(z) is the di-Gamma function. This leads to

δCl,t

Cl,t

αl,t

αn

= −λl,t1

2

∆2

2πΓTψ(1)(1

?

1 −

?ω

2Γ

?2?

2+

1 −

Γ

2πT) .

∆2

πΓTψ(1)(1

2+

Γ

2πT)

??

,

l,t

= 1 −(31)

where αn is the ultrasonic attenuation coefficient in the normal state. Here we combined the subscripts l and t,

because the above analysis applies to the case of the transverse sound wave as well. Only the coupling constants λl,t

are different. In particular, when Γ/2πT ≈ Γ/2πTc≪ 1, the above equations can be further reduced to

δCl,t

Cl,t

2 2Γ

αl,t

αn

l,t

7ζ(3)ΓTc

= −λl,t1

?

1 −

T

?ω

?

?2?

1 −

?

4π3

7ζ(3)

T

Γ

?

1 −T

Tc

???

,

= 1 −

2π3

1 −T

. (32)

Note that the sound velocity change and the attenuation coefficients decrease linearly in (1 − T/Tc) as T → Tc.

This result, Eq. (32), for the longitudinal sound wave is consistent with the experimental observation reported in [15]

However, in the experiment, the transverse sound velocity does not show any change across Tc. The absolute value

of the transverse sound velocity is much smaller than the longitudinal one and the coupling to the electron system is

also much weaker than that of the longitudinal sound waves. Therefore, it is difficult to observe any change at T = Tc

for the transverse wave, which may explain the experimental results.

Here we neglect the coupling to the collective mode. Indeed, even in the diffusive limit, C11 and C66 modes do

couple to the collective mode. However, our investigation showed that the coupling to the collective mode in these

cases is almost impossible to detect although we do not present the details of the analysis here.

C. Ultrasonic attenuation for all temperature regimes

The general expression of the sound attenuation coefficient for T < Tccan be obtained by following the procedure of

Kadanoff & Falko, and Tsuneto [16]. To obtain the ultrasonic attenuation coefficient, αt, we compute the imaginary

part of the correlation function ?[τxy,τxy]?. We finally arrive at

Im?[τxy,τxy]? =p4

−∞

F

m2N(0)ω

?∞

dω

?

−∂nF

∂ω

?

g(˜ ω)

Im√˜ ω2− ∆2

1 + y2/2 −?1 + y2

y4

, (33)

where y =

given by

vFq

2Im√˜ ω2−∆2and nF(ω) = 1/(eω/T+ 1) is the Fermi distribution function. The coherence factor g(˜ ω) is

7

Page 8

g(˜ ω) =1

2

?

1 +|˜ x|2− 1

|˜ x2− 1|

?

, (34)

where ˜ x = ˜ ω/∆ is determined from

˜ x =ω

∆+ iΓ

∆

√˜ x2− 1

˜ x

. (35)

Similar analysis can be also done for αl.

In the limit of |q|l ≪ 1, the above result leads to the following ratio between the attenuation coefficients in the

superconducting state, αl,t, and the normal state, αn

l,t.

αl,t

αn

l,t

=

Γ

8∆

?∞

0

dω

Tsech2(ω

2T)

g(˜ ω)

Im√˜ x2− 1

.(36)

Notice that the Eq. (36) applies for both of the transverse and longitudinal sound waves. This result is evaluated

numerically and shown in Fig.2 for several Γ/Γcwhere Γc= ∆(0)/2 is the critical scattering rate which drives Tcto

zero.

VI. CONCLUSION

We have identified a unique collective mode called the clapping mode in a p-wave superconductor with the order

parameter given by Eq.(1). This collective mode couples to the sound wave and affects its dynamics.

The effect of the clapping mode on the sound waves was calculated in the collisionless limit. However, unlike the

case of3He, the detection of the collective mode appears to be rather difficult. One needs, at least, the high frequency

experiment with ω ∼ O(1) GHz.

In the diffusive limit, we worked out the sound velocity change near T = Tcand found that it decreases linearly

in 1 − T/Tcwhich is consistent with the experiment reported by Matsui et al [15]. We also obtained the ultrasonic

attenuation coefficient for the whole temperature range, which can be tested experimentally. On the other hand, the

coupling of the collective mode is almost invisible in the diffusive limit.

ACKNOWLEDGMENTS

We thank T. Ishiguro, K. Nagai, Y. Maeno, and M. Sigrist for helpful discussion and especially E. Puchkaryov for

drawing Fig.2. The work of H.-Y. Kee was conducted under the auspices of the Department of Energy, supported (in

part) by funds provided by the University of California for the conduct of discretionary research by Los Alamos Na-

tional Laboratory. This work was also supported by Alfred P. Sloan Foundation Fellowship (Y.B.K.), NSF CAREER

Award No. DMR-9983731 (Y.B.K.), and CREST (K.M.).

APPENDIX A:

1. Clapping mode and its coupling to the stress tensor

The fluctuation of the order parameter corresponding to the clapping mode can be written as δ∆ρ3∼ e±2iφσ1ρ3.

The relevant correlation functions for the couplings are

?δ∆,cos(2φ)?(iων,q) = T

?

?

n

?

?

p

Tr[δ∆ρ3G(p,ωn)cos(2φ)ρ3G(p − q,iωn− iων)] ,

?δ∆,δ∆?(iων,q) = T

n

p

Tr[δ∆ρ3G(p,ωn)δ∆ρ3G(p − q,iωn− iων)] . (A1)

After summing over p, we get

8

Page 9

?δ∆,cos(2φ)? =

?

πTN(0)

?

n

−iων∆

2?ω2

1 +

n+ ∆2?

ω2

n+ν+ ∆2

?ω2

n+ ∆2+

n+ ∆2+

?

ω2

ω2

n+ν+ ∆2

??ω2

?

n+ν+ ∆2?2

?

?

+ ζ2

?

,

?δ∆,δ∆? =

?

πTN(0)

?

n

ωnωn+ν

?ω2

n+ ∆2?

ω2

n+ν+ ∆2

?ω2

n+ ∆2+

n+ ∆2+

ω2

n+ν+ ∆2

??ω2

ω2

n+ν+ ∆2?2

+ ζ2

?

,(A2)

where ?···? on the right hand side of the equations represents the angle average. Now summing over ωnand analytic

continuation iων→ ω + iδ lead to

?δ∆,cos(2φ)? = N(0)?ω

?δ∆,δ∆? = g−1− N(0)?ζ2+ 2∆2− ω2

4∆F?,

4∆2

F? ,(A3)

where F is given by

F(ω,ζ) = 4∆2(ζ2− ω2)

?∞

∆

dEtanh(E/2T)

√E2− ∆2

(ζ2− ω2)2− 4E2(ω2+ ζ2) + 4ζ2∆2

[(ζ2− ω2)2+ 4E2(ω2− ζ2) + 4ζ2∆2]2− 16ω2E2(ζ2− ω2)2. (A4)

In the limit of ω ≪ vF|q|, the contribution (in ?[hl,hl]?; see Appendix B 3) due to the coupling with the clapping

mode becomes

?δ∆,cos(2φ)?2

g−1− ?δ∆,δ∆?= N(0)

ω2?F?2

4?(ζ2− 2∆2− ω2)F?

s2f

4[1

≈ N(0)

2− (2∆/vFq)2− s2]. (A5)

where f = limq→0limω→0?F? is the superfluid density and given by Eq.(18). We can see that the frequency of the

clapping mode is given by√2∆ from ?δ∆,δ∆?.

APPENDIX B:

1. Longitudinal sound wave in the collisionless limit

The longitual sound velocity shift is given by the real part of ?[hl,hl]?. The irreducible correlation function for the

stress tensor can be obtained from

?[τxx,τxx]?00= T

?

n

?

p

Tr[p4

F

m2(cos φ)4ρ3G(p,ωn)ρ3G(p − q,iωn− iων)] ,(B1)

where φ is the angle between p and q. Since the stress tensor couples to the density, the correlation function is

renormalized as

?[hl,hl]?0= ?[hl,hl]?00+V (q)?[hl,n]??[n,hl]?

1 − V (q)?[n,n]?

, (B2)

where V (q) = 2πe2/|q| is the Coulomb interation. This equation can be simplified in the long wave length limit

(|q| → 0) as

?q

It is useful to define the following quantity for notational convenience.

?[hl,hl]?0≈

ω

?2

[?[τxx,τxx]?00−?[τxx,n]??[n,τxx]?

?[n,n]?

] .(B3)

?A,B? = T

?

n

?

p

Tr[Aρ3G(p,ωn)Bρ3G(p − q,iωn− iων)] ,(B4)

9

Page 10

where A and B can be some functions of φ or operators. Using this notation, Eq.(B3) can be rewritten as

?[hl,hl]?0≈

?q

?q

ω

?2p4

?2 p4

F

m2

?

?cos2φ,cos2φ? −?cos2φ,1??1,cos2φ?

?

?1,1?

?

=

ω

F

4m2

?cos(2φ),cos(2φ)? −?cos(2φ),1??1,cos(2φ)?

?1,1?

?

. (B5)

Then, each correlation function can be computed from

?1,1?(iων,q) = T

?

?

?

n

?

?

?

p

Tr[ρ3G(p,ωn)ρ3G(p − q,iωn− iων)] ,

?1,cos(2φ)?(iων,q) = T

n

p

Tr[cos(2φ)ρ3G(p,ωn)ρ3G(p − q,iωn− iων)] ,

?cos(2φ),cos(2φ)?(iων,q) = T

n

p

Tr[cos2(2φ)ρ3σ1G(p,ωn)ρ3G(p − q,iωn− iων)] . (B6)

Summing over p leads to

?1,1? =

?

πTN(0)

?

n

1 −

ωnωn+ν+ ∆2

n+ ∆2?

?ω2

ω2

n+ν+ ∆2

?ω2

n+ ∆2+

n+ ∆2+

?

ω2

ω2

n+ν+ ∆2

??ω2

?

n+ν+ ∆2?2

+ ζ2

?

, (B7)

where ζ = vF · q and N(0) = m/2π is the two dimensional density of states. Similar equations can obtained for

?cos(2φ),1? and ?cos(2φ),cos(2φ)? with additional angle factors, cos(2φ) and cos2(2φ) respectively. After summing

over ωnand analytic continuation iων→ ω + iδ, we get the following results in the limit of ω ≪ vF|q|.

ζ2− ω2f

ζ2− (ω + iδ)2? ≈ N(0)

?cos(2φ),1? = N(0)?cos(2φ)

ζ2− ω2f

ζ2− (ω + iδ)2? ≈ N(0)

We find that the second term in the last line of Eq.(B5) is of higher order in ω/vF|q|(≡ s) so that we can ignore

it. In Sr2RuO4 or metals, s ≪ 1. Thus the effect of the coupling to the density is merely to change the vertex

associated with τxxfrom

?cos(2φ),cos(2φ)? leads to

?q

where f is the superfluid density.

?1,1? = N(0)?

?

1 − is(1 − f)

?

√1 − s2

2s2(1 − f)(1 + i

?1

?

,

ζ2− ω2f

ζ2− (ω + iδ)2? ≈ N(0)

1 − 2s2

2s√1 − s2)

2√1 − s2

?

,

?cos(2φ),cos(2φ)? = N(0)?cos2(2φ)

2− 2s2(1 − f) − i(1 − f)s

?

. (B8)

p2

mcos2φ to

F

p2

2mcos(2φ) as far as the lowest order contribution is concerned. Evaluation of

F

Re?[hl,hl]?0≈

ω

?2

Rep4

F

4m2?cos(2φ),cos(2φ)? =

?q

ω

?2p4

F

4m2N(0)

?1

2− 2s2(1 − f)

?

, (B9)

2. Contribution coming from the coupling to the clapping mode

The correction due to the collective mode leads to the renormalized correlation function as follows.

?[hl,hl]? = ?[hl,hl]?0+?[hl,δ∆ρ3]??[δ∆ρ3,hl]?

g−1− ?[δ∆ρ3,δ∆ρ3]?

,(B10)

where g is the coupling constant between the stress tensor and the collective mode. δ∆ρ3represents the fluctuation

associated with the clapping mode. Using the fact that ?1,e2iφ? = 0 and δ∆ ∼ e2iφσ1, the above equation can be

further reduced to

?[hl,hl]? =

?q

?q

ω

?2 p4

?2p4

F

4m2

?

?

?cos(2φ),cos(2φ)? +?cos(2φ),δ∆??δ∆,cos(2φ)?

?2?

g−1− ?δ∆,δ∆?

f(vF|q|)2

4{(vF|q|)2+ 4∆(T)2− 2ω2}

?

=

ω

F

4m2

1

2− 2

?Cl

vF

1 − f −

??

+ imionCl

ω

αl(ω) ,(B11)

10

Page 11

where f is the superfluid density and given by Eq.(18).

[1] T. M. Rice and M. Sigrist, J. Phys. Condens. Matter 7, L643 (1995).

[2] M. Sigrist , D. Agterberg, A. Furusaki, C. Honerkamp, K. K. Ng, T. M. Rice, and N. E. Zhitomirsky, Physica C, 317-318,

134 (1999); M. Sigrist and K. Ueda, Rev. Phys. Mod. 63, 239 (1991).

[3] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q. Mao, Y. Mori, and Y. Maeno, Nature, 396, 658 (1998).

[4] The Superfluid Phases of Helium 3, D. Vollhardt and P. W¨ olfle (Tayor & Francis, New York, 1990).

[5] G. M. Luke et al., Nature 394, 558 (1998).

[6] S. Nishizaki, Y. Maeno, S. Farner, S. Ikeda, and T. Fujita, J. Phys. Soc. Jpn. 67 560 (1998).

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[8] M. Sigrist and M. E. Zhitomirsky, J. Phys. Soc. Jpn. 67 3452 (1996).

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[11] H. Y. Kee, Y. B. Kim, and K. Maki, Phys. Rev. B 61, 3584 (2000).

[12] L. Tewordt, Phys. Rev. Lett., 83, 1007 (1999).

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3687 (1998).

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Ambegaokar, Low Temp. Physics LT9A(G. Daunet et al., eds) Plemum Press New York 1964 p524. ; They treat the three

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11

Page 12

0.7 0.80.9

t

0.4

0.5

0.6

0.7

0.8

0.9

I

FIG. 1. The function I representing the reduction in the sound veloc-

ity as a function of the reduced temperature t = T/Tcwith 0.7 < t < 1.0

for vF|q|/∆(0) = 0,1,2,3.

12

Page 13

0 0.20.4 0.60.8

1

t

0

0.2

0.4

0.6

0.8

1

Α

????????

Αn

0 0.1 0.20.3 0.4

FIG. 2. The normalized attenuation coefficient as a function of the

reduced temperature t = T/Tc for Γ/Γc = 0,0.1,0.2,0.3,0.4.

13