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Transverse acoustic impedance of superfluid 3He–B
Journal of Physics and Chemistry of Solids 66 (2005) 1352–1354 www.elsevier.com/locate/jpcs
Transverse acoustic impedance of superfluid 3He–B Yasushi Nagatoa,*, Mikio Yamamotob, Seiji Higashitanib, Katsuhiko Nagaib a Information Media Center, Hiroshima University, Kagamiyama 1-4-2, Higashihiroshima, 739-8511 Japan Faculty of Integrated Arts and Sciences, Hiroshima University, Kagamiyama 1-7-1, Higashihiroshima, Japan
b
Abstract A microscopic theory of transverse acoustic impedance of the B phase of superfluid 3He is presented. We calculate the stress tensor of superfluid 3He–B as a response to the oscillating rough wall. For that purpose, we extend a quasi-classical theory of the rough wall effects based on the random S-matrix model to time dependent problem. We show that the structures in the temperature dependence found in the experiments of transverse acoustic impedance can be related to the excitation of surface bound states which are characteristic surface effects in anisotropic paring Fermi superfluids. q 2005 Elsevier Ltd. All rights reserved. Keywords: D. Acoustical properties
1. Introduction Acoustic technique has been a useful tool to probe the properties of liquid 3He [1]. In particular, the transverse acoustic impedance provides a useful information in situations where damping is strong. Since, the pioneering work by Bekarevich and Khalatnikov, [2] the acoustic impedance has been studied theoretically using the quasi-particle kinetic equation in both the normal and the superfluid states. [3–5] Those theories predict that in the superfluid phase at low temperatures the transverse impedance becomes frequency independent and tends to zero like Zw
3 rv Y ðTÞ 16 F 1
(1)
where Y1Z1/(ebDC1). This is natural because the number of existing Bogoliubov quasi-particles becomes small at low temperatures. In the superfluid state, however, the oscillating wall will create the Bogoliubov quasi-particles by pair breaking mechanism. In a previous report [6], we proposed a theory that can incorporate the pair breaking and showed that * Corresponding author. E-mail address: [email protected] (Y. Nagato).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.05.072
the transverse acoustic impedance is finite at low temperatures and exhibits characteristic frequency dependence in the s-wave pairing system and in the ABM state. In this paper, we present the results of calculations in the B phase of superfluid 3He. At the surface of 3He–B, quasiparticle states are characterized by surface bound states which have lower energy than the bulk energy gap, [8] which is common to anisotropic pairing systems with an order parameter component that changes sign at the reflection by the surface. We show that the excitations of the surface bound states give rise to structures in the frequency and temperature dependence of the acoustic impedance. We extend our quasi-classical Green function theory [7,8] of rough wall effects to time dependent problems using Keldysh technique and calculate the stress tensor as a linear response to the oscillating rough wall.
2. Formulation We consider a system in which liquid 3He fills the zO0 domain and a rough plane wall at zZ0 is oscillating in x-direction like R(t)fe-iUt. We consider a quasi-classical Green function in Keldysh space ! R 0 0 K 0 0 ðK; K ; z; t; t Þ G ðK; K ; z; t; t Þ G ab ab G ab ðk; k 0 ; z; t; t 0 Þ Z ; GAab ðK; K 0 ; z; t; t 0 Þ (2)
Y. Nagato et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1352–1354
1353
Now we consider the effect by the wall motion by perturbation theory. From Eq. (4), the linear response of the Green’s function is given by
s = (x,y)
dG aa ðK; 0Þ Z ðG s C iaÞG dS G ðG s K iaÞ (K,–k)
and from the self energy Eq. (6) one obtains X 2 t0Þ Z P ½ðKiðKx K Qx ÞðRðtÞ dSðt; Q0 1 Q
(K,k) K
K Rðt 0 ÞÞÞG ðQ; t K t 0 Þ C
θ
X Q
–k
k
z
where a,b are direction index and K, K 0 are Fermi momentum component parallel to the wall as depicted in Fig. 1. When one moves to a reference frame moving with the wall, one can use the boundary condition of the static wall for the quasi-classical Green function.[7] Using those boundary conditions and taking the average over the wall roughness, we obtain the Green functions at zZ0 which are diagonal in K space (3)
The right hand side corresponds to the usual quasiclassical Green function associated with the Fermi momentum (K, ak). The explicit form of G aa ð0Þis given by G aa ð0Þ Z G s C ðG s C iaÞG ðG s K iaÞ;
1 K1 G s
K S
;
Q0
1 (8)
S G: dG Z Gd
(9)
The stress tensor at the wall is calculated from the Keldysh part of the Green function X K1 Q Kv xz Z 2 2vK x K K 1 Ki K K ðdGCCðK; t; tÞ K dGKKðK; t; tÞÞ ; ! Tr 2 2
(10)
where vKZvF cos q is the z-component ofQthe Fermi _ velocity. The acoustic impedance is given by xz =R: For the s-wave pairing superfluid, it is straight-forward to obtain an analytical expression for Z.[6] One can show that in the low frequency limit U/0, Z agrees with Eq. (l). At higher frequencies, however, Z becomes frequency dependent and shows singularity at the pair breaking temperature.
(4) 3. Density of states of 3He–B at the surface
where G s is the Green function for the specular wall, G ðK; t; t 0 Þ Z
2 P
!½dGðQ; t; t 0 Þ;
Fig. 1. Fermi momenta with the parallel component K fixed. The z-components are denoted by ak (aZG1).
G aa ðK; K 0 ; 0; t; t 0 Þ Z G aa ðK; 0; t; t 0 ÞdKK 0 :
(7)
(5)
In Fig. 2, we show the total density of states of the BW state at the surface. [8] In the BW state, there exist surface
and the surface self energy S is given by X ð2Þ 0 0 iðKKQÞRðt 0 Þ ÞZ h ðK KQÞ!eKiðKKQÞRðtÞ GðQ;t;t Þe : SðK;t;t
3 2.5
(6)
Since we are treating Keldysh Green functions, the products Ðin the above equations should read tÞBð t 0 ÞZ dtAðt; t; t 0 Þ: A Bðt; (2) In Eq. (6), h is a quantity that represents the correlation of the roughness of the wall. Following P Ref. [7], we consider a simplest model h(2)(K-Q)Z2W/ Q 1. One can show that WZ1 corresponds to the diffuse surface boundary condition and WZ0 corresponds to the specular surface boundary condition. In what follows, we consider only the case with diffusive surface (WZ1), because it is obvious that in case of the specular surface the wall motion will not give any effect to the liquid.
n(ω,z=0) / N(0)
Q
2 1.5 1 0.5 0
0
∆*
∆ ω
Fig. 2. Quasi-particle density of states at the diffusive surface of the BW state at TZ0.2Tc. D is the bulk energy gap and D* is the upper edge of the surface bound state band.
Y. Nagato et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1352–1354
bound states whose energy depends on the polar angle q of the Fermi momentum. The bound state energy is zero when qZ0 and increases as a function of q. In case of the specular surface (WZ0), the surface bound states fill up the bulk energy gap. In case of the diffusive surface (WZ1), however, the bound state is broadened and its energy saturates when q increases. As a result, there appears an upper energy edge D* of the bound state band as can be seen in Fig.2. Although a similar result has been obtained by Zhang [9], the origin of this new gap between D* and the bulk energy gap D is still an open, question.
2.5 2
Re(Z ) 1 0.5 0
We calculate numerically the stress tensor for the diffusive wall given by Eq. (10) using the self-consistent order parameter and the quasi-classical Green’s functions obtained in Ref. [8]. We first evaluate the self-energy fluctuation dS given by Eqs. (8) and (9). Although dS is dominated by the first line in Eq. (8), the second line in Eq. (8) yields a finite contribution. Thus, Eqs.(8) and (9) should be solved self-consistently. In Fig. 3, we show our result for the transverse acoustic impedance Z for the BW state with diffusive wall. We first find that there is no jump at UZ2D which was found in s-wave paring superfluid [6] but a slight change in the slope. Quite interesting is the cusp behavior at UZDCD* found both in the real part and the imaginary part. This is a singularity due to the pair excitation of a surface bound state and a propagating Bogoliubov quasi-particle. It indicates that the surface bound states characteristic to anisotropic pairing Fermi superfluids can be observed by acoustic experiments.
Im(Z )
–0.5 –1
4. Transverse acoustic impedance of 3He–B
Acoustic Impedance Z at T = 0.2Tc
1.5
Z / ZN
1354
0
∆*
∆
Ω
2∆* ∆*+∆ 2∆
Fig. 3. Frequency dependence of the transverse acoustic impedance Z of superfluid 3He–B at TZ0.2Tc. Z is scaled by its normal state value. Upper curve is the real part of Z and lower curve is the imaginary part of Z.
been interpreted [11] to be caused by the order parameter collective mode in 3He–B. Although the present analysis has not taken into account the collective modes, it indicates that the surface bound states yield sizable contribution. Further study of Fermi liquid effect as well as coupling with the order parameter collective modes is necessary for quantitative comparison with experiments.
Acknowledgements This work is supported in part by a Grant-in-Aid for COE Research (No. 13CE2002) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
References 5. Discussion We have proposed a quasi-classical theory of acoustic impedance of Fermi superfluids and applied it to superfluid 3 He–B. Our formulation can incorporate the pair breaking mechanism. We have shown that the pair excitation of a surface bound state and a propagating Bogoliubov quasiparticle gives rise to cusps in the frequency dependence of acoustic impedance. In our preliminary calculations, we have found that the temperature dependence of Z shows a similar behavior to the frequency dependence of Fig. 3 and is in qualitative agreement with experiments.[10–12] So far the structures in the temperature dependence of the imaginary part of Z have
[1] W.P. Halperin, E. Varoquaux, in: W.P. Halperin, L.P. Pitaevskii (Eds.), Helium Three, Elsevier, Amsterdam, 1990, p. 353. [2] I.L. Bekarevich, I.M. Khalatonikov, Zh. Eksp. Teor. Fiz. 39 1699 (Sov. Phys.-JETP 12 (1961) 1187). [3] E.G. Flowers, R.W. Richardson, Phys. Rev. B 17 (1978) 1238. [4] H. Højgaard Jensen, et al., J. Low Temp. Phys. 41 (1980) 473. [5] D. Einzel, et al., J. Low Temp. Phys. 53 (1983) 695. [6] M. Yamamoto, et al., J. Low Temp. Phys. 138 (2005) 349. [7] Y. Nagato, et al., J. Low Temp. Phys. 103 (1996) 1. [8] Y. Nagato, M. Yamamoto, K. Nagai, J. Low Temp. Phys. 110 (1998) 1135. [9] W. Zhang, Phys. Lett. 130 (1988) 314. [10] P.R. Roach, J.B. Ketterson, J. Low Temp. Phys. 25 (1976) 637. [11] S. Kalbfeld, et al., Phys. Rev. B 48 (1993) 4160 (J. Low Temp. Phys. 98 (1995) 549). [12] Y. Aoki, et al., J. Low Temp. Phys. 138 (2005) 783.
Transverse acoustic impedance of superfluid 3He–B Yasushi Nagatoa,*, Mikio Yamamotob, Seiji Higashitanib, Katsuhiko Nagaib a Information Media Center, Hiroshima University, Kagamiyama 1-4-2, Higashihiroshima, 739-8511 Japan Faculty of Integrated Arts and Sciences, Hiroshima University, Kagamiyama 1-7-1, Higashihiroshima, Japan
b
Abstract A microscopic theory of transverse acoustic impedance of the B phase of superfluid 3He is presented. We calculate the stress tensor of superfluid 3He–B as a response to the oscillating rough wall. For that purpose, we extend a quasi-classical theory of the rough wall effects based on the random S-matrix model to time dependent problem. We show that the structures in the temperature dependence found in the experiments of transverse acoustic impedance can be related to the excitation of surface bound states which are characteristic surface effects in anisotropic paring Fermi superfluids. q 2005 Elsevier Ltd. All rights reserved. Keywords: D. Acoustical properties
1. Introduction Acoustic technique has been a useful tool to probe the properties of liquid 3He [1]. In particular, the transverse acoustic impedance provides a useful information in situations where damping is strong. Since, the pioneering work by Bekarevich and Khalatnikov, [2] the acoustic impedance has been studied theoretically using the quasi-particle kinetic equation in both the normal and the superfluid states. [3–5] Those theories predict that in the superfluid phase at low temperatures the transverse impedance becomes frequency independent and tends to zero like Zw
3 rv Y ðTÞ 16 F 1
(1)
where Y1Z1/(ebDC1). This is natural because the number of existing Bogoliubov quasi-particles becomes small at low temperatures. In the superfluid state, however, the oscillating wall will create the Bogoliubov quasi-particles by pair breaking mechanism. In a previous report [6], we proposed a theory that can incorporate the pair breaking and showed that * Corresponding author. E-mail address: [email protected] (Y. Nagato).
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jpcs.2005.05.072
the transverse acoustic impedance is finite at low temperatures and exhibits characteristic frequency dependence in the s-wave pairing system and in the ABM state. In this paper, we present the results of calculations in the B phase of superfluid 3He. At the surface of 3He–B, quasiparticle states are characterized by surface bound states which have lower energy than the bulk energy gap, [8] which is common to anisotropic pairing systems with an order parameter component that changes sign at the reflection by the surface. We show that the excitations of the surface bound states give rise to structures in the frequency and temperature dependence of the acoustic impedance. We extend our quasi-classical Green function theory [7,8] of rough wall effects to time dependent problems using Keldysh technique and calculate the stress tensor as a linear response to the oscillating rough wall.
2. Formulation We consider a system in which liquid 3He fills the zO0 domain and a rough plane wall at zZ0 is oscillating in x-direction like R(t)fe-iUt. We consider a quasi-classical Green function in Keldysh space ! R 0 0 K 0 0 ðK; K ; z; t; t Þ G ðK; K ; z; t; t Þ G ab ab G ab ðk; k 0 ; z; t; t 0 Þ Z ; GAab ðK; K 0 ; z; t; t 0 Þ (2)
Y. Nagato et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1352–1354
1353
Now we consider the effect by the wall motion by perturbation theory. From Eq. (4), the linear response of the Green’s function is given by
s = (x,y)
dG aa ðK; 0Þ Z ðG s C iaÞG dS G ðG s K iaÞ (K,–k)
and from the self energy Eq. (6) one obtains X 2 t0Þ Z P ½ðKiðKx K Qx ÞðRðtÞ dSðt; Q0 1 Q
(K,k) K
K Rðt 0 ÞÞÞG ðQ; t K t 0 Þ C
θ
X Q
–k
k
z
where a,b are direction index and K, K 0 are Fermi momentum component parallel to the wall as depicted in Fig. 1. When one moves to a reference frame moving with the wall, one can use the boundary condition of the static wall for the quasi-classical Green function.[7] Using those boundary conditions and taking the average over the wall roughness, we obtain the Green functions at zZ0 which are diagonal in K space (3)
The right hand side corresponds to the usual quasiclassical Green function associated with the Fermi momentum (K, ak). The explicit form of G aa ð0Þis given by G aa ð0Þ Z G s C ðG s C iaÞG ðG s K iaÞ;
1 K1 G s
K S
;
Q0
1 (8)
S G: dG Z Gd
(9)
The stress tensor at the wall is calculated from the Keldysh part of the Green function X K1 Q Kv xz Z 2 2vK x K K 1 Ki K K ðdGCCðK; t; tÞ K dGKKðK; t; tÞÞ ; ! Tr 2 2
(10)
where vKZvF cos q is the z-component ofQthe Fermi _ velocity. The acoustic impedance is given by xz =R: For the s-wave pairing superfluid, it is straight-forward to obtain an analytical expression for Z.[6] One can show that in the low frequency limit U/0, Z agrees with Eq. (l). At higher frequencies, however, Z becomes frequency dependent and shows singularity at the pair breaking temperature.
(4) 3. Density of states of 3He–B at the surface
where G s is the Green function for the specular wall, G ðK; t; t 0 Þ Z
2 P
!½dGðQ; t; t 0 Þ;
Fig. 1. Fermi momenta with the parallel component K fixed. The z-components are denoted by ak (aZG1).
G aa ðK; K 0 ; 0; t; t 0 Þ Z G aa ðK; 0; t; t 0 ÞdKK 0 :
(7)
(5)
In Fig. 2, we show the total density of states of the BW state at the surface. [8] In the BW state, there exist surface
and the surface self energy S is given by X ð2Þ 0 0 iðKKQÞRðt 0 Þ ÞZ h ðK KQÞ!eKiðKKQÞRðtÞ GðQ;t;t Þe : SðK;t;t
3 2.5
(6)
Since we are treating Keldysh Green functions, the products Ðin the above equations should read tÞBð t 0 ÞZ dtAðt; t; t 0 Þ: A Bðt; (2) In Eq. (6), h is a quantity that represents the correlation of the roughness of the wall. Following P Ref. [7], we consider a simplest model h(2)(K-Q)Z2W/ Q 1. One can show that WZ1 corresponds to the diffuse surface boundary condition and WZ0 corresponds to the specular surface boundary condition. In what follows, we consider only the case with diffusive surface (WZ1), because it is obvious that in case of the specular surface the wall motion will not give any effect to the liquid.
n(ω,z=0) / N(0)
Q
2 1.5 1 0.5 0
0
∆*
∆ ω
Fig. 2. Quasi-particle density of states at the diffusive surface of the BW state at TZ0.2Tc. D is the bulk energy gap and D* is the upper edge of the surface bound state band.
Y. Nagato et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1352–1354
bound states whose energy depends on the polar angle q of the Fermi momentum. The bound state energy is zero when qZ0 and increases as a function of q. In case of the specular surface (WZ0), the surface bound states fill up the bulk energy gap. In case of the diffusive surface (WZ1), however, the bound state is broadened and its energy saturates when q increases. As a result, there appears an upper energy edge D* of the bound state band as can be seen in Fig.2. Although a similar result has been obtained by Zhang [9], the origin of this new gap between D* and the bulk energy gap D is still an open, question.
2.5 2
Re(Z ) 1 0.5 0
We calculate numerically the stress tensor for the diffusive wall given by Eq. (10) using the self-consistent order parameter and the quasi-classical Green’s functions obtained in Ref. [8]. We first evaluate the self-energy fluctuation dS given by Eqs. (8) and (9). Although dS is dominated by the first line in Eq. (8), the second line in Eq. (8) yields a finite contribution. Thus, Eqs.(8) and (9) should be solved self-consistently. In Fig. 3, we show our result for the transverse acoustic impedance Z for the BW state with diffusive wall. We first find that there is no jump at UZ2D which was found in s-wave paring superfluid [6] but a slight change in the slope. Quite interesting is the cusp behavior at UZDCD* found both in the real part and the imaginary part. This is a singularity due to the pair excitation of a surface bound state and a propagating Bogoliubov quasi-particle. It indicates that the surface bound states characteristic to anisotropic pairing Fermi superfluids can be observed by acoustic experiments.
Im(Z )
–0.5 –1
4. Transverse acoustic impedance of 3He–B
Acoustic Impedance Z at T = 0.2Tc
1.5
Z / ZN
1354
0
∆*
∆
Ω
2∆* ∆*+∆ 2∆
Fig. 3. Frequency dependence of the transverse acoustic impedance Z of superfluid 3He–B at TZ0.2Tc. Z is scaled by its normal state value. Upper curve is the real part of Z and lower curve is the imaginary part of Z.
been interpreted [11] to be caused by the order parameter collective mode in 3He–B. Although the present analysis has not taken into account the collective modes, it indicates that the surface bound states yield sizable contribution. Further study of Fermi liquid effect as well as coupling with the order parameter collective modes is necessary for quantitative comparison with experiments.
Acknowledgements This work is supported in part by a Grant-in-Aid for COE Research (No. 13CE2002) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
References 5. Discussion We have proposed a quasi-classical theory of acoustic impedance of Fermi superfluids and applied it to superfluid 3 He–B. Our formulation can incorporate the pair breaking mechanism. We have shown that the pair excitation of a surface bound state and a propagating Bogoliubov quasiparticle gives rise to cusps in the frequency dependence of acoustic impedance. In our preliminary calculations, we have found that the temperature dependence of Z shows a similar behavior to the frequency dependence of Fig. 3 and is in qualitative agreement with experiments.[10–12] So far the structures in the temperature dependence of the imaginary part of Z have
[1] W.P. Halperin, E. Varoquaux, in: W.P. Halperin, L.P. Pitaevskii (Eds.), Helium Three, Elsevier, Amsterdam, 1990, p. 353. [2] I.L. Bekarevich, I.M. Khalatonikov, Zh. Eksp. Teor. Fiz. 39 1699 (Sov. Phys.-JETP 12 (1961) 1187). [3] E.G. Flowers, R.W. Richardson, Phys. Rev. B 17 (1978) 1238. [4] H. Højgaard Jensen, et al., J. Low Temp. Phys. 41 (1980) 473. [5] D. Einzel, et al., J. Low Temp. Phys. 53 (1983) 695. [6] M. Yamamoto, et al., J. Low Temp. Phys. 138 (2005) 349. [7] Y. Nagato, et al., J. Low Temp. Phys. 103 (1996) 1. [8] Y. Nagato, M. Yamamoto, K. Nagai, J. Low Temp. Phys. 110 (1998) 1135. [9] W. Zhang, Phys. Lett. 130 (1988) 314. [10] P.R. Roach, J.B. Ketterson, J. Low Temp. Phys. 25 (1976) 637. [11] S. Kalbfeld, et al., Phys. Rev. B 48 (1993) 4160 (J. Low Temp. Phys. 98 (1995) 549). [12] Y. Aoki, et al., J. Low Temp. Phys. 138 (2005) 783.