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On the hydrodynamics of Fermi superfluids
ANNALS
OF PHYSICS
109, 349-364 (1977)
On the Hydrodynamics
of Fermi Superfluids*+
R. A. BROGLIA* Stare
Unicrrsity of New and Brookhacen
York at Stony Brook, National Laboratory,
Stony Brook, Upton, New
New York York 11973
11794
A. MOLIP\~ARI lstituto
di Fisica
dell’Unicersita
di Ferrara,
Ferraro,
Italy
T. REGGE Institute
for Advanced
Study,
Princeton,
New Jersey
08540
Received April 25, 1977
The hydrodynamic of a neutral Fermi fluid is derived from the microscopic pairing model in the framework of a time dependent BCS formalism. A discussion of the superfluidity and coherence properties of these fluids and a comparison with the known He II hydrodynamics is attempted. Applications to finite nuclear systems are discussed.
I. INTRODUCTION
In dealing with quantum fluids Landau [I] states that in He ZZthe motion of the superfluid component is characterized by the existence of a velocity potential: v(x) = - Vcp(x)
(1.1)
and by a density field p(x), canonically conjugated to y(x). For long wavelength excitations the motion of the superfluid is essentially controlled by the Euler-Bernoulli equations and of course by the continuity equations. Higher energy excitations arise from vortices, where F(X) is no longer single valued and at the roton wavelength. where Feynman’s backflow (cf., e.g., [2]) contribution can be essentially interpreted as a breakdown of Landau’s criterion. In both casesthe concept of velocity potential * The United States copyright covering this + Supported, in part, E(ll-l)-3001. * On leave from the
Government’s right to retain a nonexclusive royalty-free license in and to paper is acknowledged. by Energy Research and Development Administration, under Contract Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark.
349 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-49
I6
350
BROGLIA, MOLINARI,
AND REGGE
becomeslessdefined and useful. The aim of this paper is to show that a similar but richer structure arisesfor Fermi superfluids containing Bose superfluids as a limiting case in the strong coupling limit. In particular we argue for the suitability of a two and possibly multifluid theory in dealing with the excitation spectrum of a Fermi superfluid where the intrinsic degreesof freedom of the nucleon in the Cooper pairs are no longer negligible.
2. BOSE HYDRODYNAMICS:
A SHORT DERIVATION
We assume in what follows that the ground state wavefunction @(x1,..., x2) IS given by a real nonnegative symmetric function of the boson coordinates .yi . Qualitatively @can be characterized asfollows: (‘1) If the bosons are evenly spaced apart on the average then @ is essentially constant and depends only on the macroscopic fluid density. (2) @ drops quickly to zero if two bosons approach each other becauseof the strong repulsive core at short distances. (3)
@has no nodes for this would increasethe energy of the state.
(4)
@is rigorously invariant under translations and rotations.
It is moreover approximately invariant under volume preserving differentiable maps of R3 into R3 for these maps do not violate (1) (2) (3)). A map of this sort, for instance, is given by: 5’ = Ax,
1” == /.Lj’,
z = vz
hpv = 1,
(2.1)
in which case a cube is transformed into a solid of equal volume. This last property really characterizes @ as the wavefunction of a fluid, as contrasted with that of a solid which certainly violates it. It should be understood, however, that the stretching generated by (2.1) should be sufficiently gentle for the invariance is only approximate. According to Feynman [2] once the wavefunction for the ground state is known, we obtain the wavefunction QF for the fluid in (coherent) motion from: QF = exp i f ~(x~)(i?i/fi) x @ = eio@, (2.2) 1 ( i=l where m is the boson mass. The insertion of this function into the time dependent Schrodinger equation leads to disappointing results in that it is difficult to seehow all the arising terms compensate in the equation. An improved version of the Feynman Ansatz has been proposed in [3]. It reads CD,= exp i g fp(xJ(m/fi) ( i=l ) x @ml
3 t), 5x% 30,-v &&a 3 t)) x (a(()/a(x))liz
(2.3) = eiO 1C& 1.
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351
SUPERFLUIDS
Here 6(x, t) is defined through the equation of flow:
with the boundary
dx’x(5, t)/dt = v(x(E, t), t)
(2.4)
xc& 0) = 5.
(2.5)
conditions:
From x(t: t) we define the inverse function &x, t) which is then used in (2.3). The square root of the Jacobian in (2.3) keeps the Ansatz properly normalized. Our definition is equivalent to the differential equation:
with I @D, I*=f) = CD.
(2.6a)
Although t = 0 plays a special role here we could have selected any time and in particular t = -co as starting point. Both definitions amount to choosing a time evolution where the ground state, besides acquiring the Feynman-Onsager phase [2], also drifts downstream along the velocity. Both (2.3) and (2.6) define a particular time evolution of the wavefunction. The macroscopic density p(x, t) of the fluid is determined completely by the Ansatz (2.3) either through the continuity equation (iip/dt) + div pv = 0 and the boundary
condition 0) = 1 I @p. 12d?Y,
p(x,
or equivalently
(2.7a)
.. .
d3x,
(2.7b)
d?Y.\,
by p(x, I) = yy
The connection the identity
with
the Schrodinger
equation
p(x, 0). is established
12.8) as follows.
We state
0.9) and then use GE as a trial function 6 [
dt
d3.Y, .‘.
in the variational d?YN
($*(ifi(a/at)
principle - H) $J) = 0.
(2.10)
We thus obtain: 6 j” dr c i
d3sj
IJ~(.Y~)
[w
-t -1
+
/” dt (c&r, i%DF)
J
= 0.
(2.11)
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BROGLIA,
MOLINARI,
AND
REGGE
In this variational principle y(x) is considered as the independent variational parameter and p(x) as a dependent one which is determined through (2.8). We now use condition 4. Note that the action of the flow on GF can be reduced locally to a translation and a rotation (of no effect) superimposed on a stretching similar to one determined by (2.1). and finally on a local change of density. We assume, therefore, that QF is essentially a “modulated” ground state where the local density P(X) varies slowly with respect to a typical interboson distance. For a conventional ground state at constant density p we can write: H@ = c e(p) CD= Ne(p) @,. For the modulated
(2.12)
ground state we propose the obvious generalization: (2.13)
H@, = C e(p(si , t)) OF . lnserting
(2.13) into (2.1 I) we find finally:
which leads to the Bernoulli
equation: HI+ + $IHZ~~+ (d(pe)/dp)
which appears now as a variational more details cf. [3, 41). If the velocity of F(X) and its derivative v(x). We write the result in terms of the relations C(x) CQ’) -
= 0
(2.15)
approximation to the Schrodinger equation (for potential is small, we can expand QF in powers boson field C(X) fulfilling the usual commutation C’(i)
C(x) = S3(x - x’)
(2.16)
and the associated density p(x) = c+(s)
C(x).
(2.17)
We find cDF Y [ 1 + (i/n//i) [ y(x) p(s) ~1”s +- (h//i) which can be compactly written
1 r/t d”s z+(s, t) j,(s, t)] @
(2.18)
as
In the treatment of Fermi fluids (2.19) is the starting step in the variational (cf. Section 4).
treatment
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353
SUPERFLUIDS
Assuming (2.20)
where ya = ~JT, , we obtain for (2.19) Q% = (1 + $1
u
‘pl,eP@ C Cl+;,Ck (wa - (k + 4 , a) $jj h
If li-l is much larger than the interparticle
@.
(2.21)
distance then
In this case the p-term predominates over thej-term (current). Quantities of hydrodynamical interest are usually expectation values of products of operators for low values of k (Iong-wave limit). We thus note that the generic case is adequately represented by g)h,fi , of the form utilized in (2.20), i.e., (m/h) y&~at.
3.
BEYOND
THE
BOSE
GAS
In Landau’s quantum hydrodynamics [l] only the degrees of freedom of the fluid which can be expressed through y and p are kept, these two fields being canonically conjugated
[&4 PWI = 6(x - a. The Hamiltonian
(3.1)
is then H =
.r
mp[((V#/2)
+ E(p)] d3x.
(3.2)
If the coupling in a Fermi superfluid is taken very large then Cooper pairs will acquire a very large binding energy and behave more and more like bosons. Therefore as long as we do not probe the fluid with an energy comparable to the energy gap we expect a behavior similar to that of a conventional Bose fluid. In particular there should be a phonon excitation branch with spectrum w = c(hk) where c is the velocity of sound. Clearly this branch has meaning only if c(tik) < d, else the phonon decays into quasi-particles. The question of the unfreezing of the relative degrees of freedom of the fermions inside the Cooper pairs is the one we have to deal with. Since breaking a Cooper pair costs energy, it is not expected that any excitation not much higher than 24 should break all Cooper pairs and see the fluid as essentially a normal Fermi fluid at T = 0, where all hydrodynamical motions become strongly dissipative. Rather we expect the onset of new degrees of freedom as we gradually increase the energy of the excitations which finally destroy the usefulness of the one fluid theory.
354
BROGLIA, MOLINARI,
AND REGGE
We believe that a satisfactory physical theory of these extra degreesof freedom can be acquired through a two- and perhaps multifluid theory with particular regard to the pairing vibrations [5-71, which play an important role in nuclei. A multifluid theory can be carried out either as a phenomenological enlargement of the conventional Landau scheme[l-4] or on a BCS model Hamiltonian [8, 91. We believe that both approaches are mutually related in a fashion similar to the boson case.
4. TIME DEPENDENT
BCS THEORY
In this section we present a hydrodynamical picture similar to the one discussed for the Bose fluid in Section 1. Our starting point is the effective pairing Hamiltonian: H=H,,+H,
(4.1)
where H,, = c (et - p) l1fi.s LS
(4.la)
and H, = c V,,b,‘b, kP
.
(4.lb)
The single-particle energies cJiare measured from the Fermi energy p. The creation and annihilation operators for Fermions of linear momentum k and the z-component of the spin s are denoted c;~ and cks, respectively. They satisfy the usual anticommutation relations (ck,, , c;,,,> = 6(k, k’) 6(s, s’).
(4.2)
The operator trkscounts the number of particles in the state (ks) and is given by nkn
=
+
CknCks
(4.3)
.
The pair creation and annihilation operators are bk+ = c;&
,
(4.4)
and bl, = c-,~c,~,
(4.5)
where et,, = TC&.T-l, T being the time reversal operator. The quantities V,, are the pairing matrix elements. The conventional static BCS treatment of (4.1) is based on the trial wavefunction / BCS) = n (u, + Vkeim’bk+)lo), k
(4.6)
where ) 0) is the vacuum of the operator c;Z,. Becauseof the violation of the particle number in (4.6), each Cooper pair is referred to an intrinsic system, eiQkbeing the
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355
SUPERFLUIDS
two-dimensional rotation in gauge space which relates this system to the laboratory system. The quantities U, and V, are the BCS occupation parameters. The variational principle for the ground state, S(BCSjHjBCS)
=0
is satisfied provided U, and V,: display the following functional single-particle energy and on the pairing matrix elements
(4.7) dependence on the
(4.8) (4.9) where the quasi-particle
energy is defined by Ek = ((‘cl, -
p)2 + dk2)l/”
(4.10)
and where the energy-gap parameter d, satisfies the integral equation A, = - C V,,(A./2E,)
= - 1 vk,Upv,.
7
Y
(4.11)
We can choose 91c= q~, where g, is a constant. This arbitrariness merely reflects the degeneracy of the BCS ground state. A convenient gauge is that which makes (4.1 I) real. We note that a wavefunction similar to (4.6) can also be defined for a Bose fluid. In fact
has the same degeneracy as the BCS wavefunction and is also nondiagonal in the number of particles. In the case of the Bose fluid of Section 1, if motion takes place the phase factor exp{iN(m@)} is replaced by exp{im xr=, y(xJ/fi}. It is now natural to ask what corresponding changes we must implement in the BCS wavefunction in order to represent slow fluid motions. The starting point for this generalization is Eq. (2.19). Thus, we are led to consider generators of the form (4.13) and a trial wavefunction of the kind @ = eie 1 BCSj,
(4.14)
356
BROGLIA,
MOLINARI,
AND
REGGE
with (4.15)
@ = 1 FhkQhk . 1l.P
The quantities 9),& are time dependent and in general not diagonal in the indices h and k. We assume moreover that the parameters Oain (4.8) and (4.9) are also subject to variation and are time dependent. It is understood that all 19, and yhk are small because we are dealing with states of quasi-equilibrium corresponding to small variations around the ground state. The operator Q hfGadd the momentum (11 - k) to the Cooper pair. In analogy to the boson case we expect that the motion of the fluid is generated by the nondiagonal terms in vhli. Note that the diagonal term LJhh merely redefines the partial phases occurring in (4.6). There is no lack of generality in carrying out the different steps of the variational calculation utilizing a phase defined as qJhk = Ah, +- y/a 3
(4.16)
h,, = W, 4 v/;
(4.17)
with and Yhli
=
(1/2i)@(k
k
+
a)
-
@/I
+
4
k))
&(h+k)/21
(4.18)
,
where a = 27r/L is the smallest wavenumber compatible with periodic boundary conditions. We are thus in a way selecting the most gentle possible motion of the fluid. Note that (4.16)-(4.18) define a hermitian operator. In contrast to the boson case we retain an essential dependence of #I(~+~),~, and vrz on their respective indices. As we discuss below, to delete this dependence would imply to freeze the internal degrees of freedom of the Cooper pairs, and thus the physical system under discussion would basically reduce to the Bose fluid of Section 1. We write 0=fl+r, (4.19) with (4.20) A = c hh,nhk > hk
and r
The trial wavefunction
=
c YhkQhk h Ii
is then introduced 6 f(O*(ifi(+?t)
(4.21)
.
into the variational
equation
- H) @) dr = 0
and the integral is expanded up to quadratic terms in the variations.
(4.22) In particular
(@, H@) = (BCS j H 1BCS:) + 1’(BCS j [@, H] 1BCS) - &(BCS [ [@, H] 1BCS) (4.23) up to second order in 0.
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SUPERFLUIDS
Several of the first order terms vanish as implied by the BCS variational condition (4.7). The remaining first order terms vanish because the operator I’ does not conserve the momentum and thus (BCS 1 [I’, H] 1BCS) = 0. By the same token there are no bilinear terms /lr in the variation of (4.22). The remaining quadratic terms can be classified as follows: (a)
Terms quadratic
in Ll and 68, . An explicit calculation
SH, = ~2E,(S%,,.)” 1.
- G c cos2%,cos2%,S0,68,~ k‘y
~- (G/8) 2 sin2Q,sin28,(6y,L I.,a (b)
gives
- 6~~)~ ,
(4.24)
Terms quadratic in r and arising from H,,, . They are equal to 6H, = 1 ykrL jz (c,; - E/J cos 28,; . k f2
(4.25)
(c) Terms quadratic in N arising from Hint. In the Bose fluid such terms are not present because of a trivial reason; namely, that the potential terms are strictly local in the boson coordinates and the Nterms correspond essentially to the FeynmanOnsager (cf. [3. 31) phase which is also local and therefore cancels out. Also the model BCS Hamiltonian is not complete, in that it represents only condensation in Cooper pairs of zero momentum. For this and similar reasons, the contribution of the terms under discussion is difficult to evaluate and is anyway expected to be small. We decided not to include them in the present calculation. (d)
Terms arising from the time derivative,
= -Re
(+BCS
i.e.,
1@eio 1 BCS) + Re (e’“BCS
/ e’% k j SC%>. (4.26)
Note that r
0 = $1
= 6 (BCS ! BCS‘, - Re (BCS ; 3 ] KS,<..
But (BCS ; %/?t / BCS) is real and thus equal to zero. which term in (4.26) is also zero. Thus Ret@. #‘h/Et)
@) = -17 c q& sin2 8, l2
(4.27)
implies that the last (4.28)
The terms (4.24), (4.25), and (4.28) are all the second order contributions to (4.22). The terms proportional to (1” do not contribute to (4.25). This is natural since the terms of type (b) are related to the kinetic energy of the moving fluid and this motion can only be achieved by nondiagonal terms of the type l? Both the fl and the r terms
358
BROGLIA,
MOLINARI,
AND
REGGE
contain an additional k-dependence which needs some clarification before we proceed further. The physical meaning of this dependence must depend on those features of the Fermi fluid which are not present in the Bose fluid and these can only be the relative degrees of freedom of the fermions in each Cooper pair. These manifest themselves in the surviving internal index k which labels the relative momentum of the fermions in the pairs. Therefore, pairs are distinguished by the label k which indicates they possess an internal k-degree of freedom besides the ordinary translational degrees of freedom. Alternatively we may consider the fluid as the mixture of infinitely many fluids, labeled by k, the dynamics of the system allowing the transmutation of a fluid into the other. The distinction between the two pictures is clearly only one of semantics. The first one implies that just as the velocity potential should depend on x, then it should also depend on k. In the second picture it is natural to assume that the different fluids may perform different motions according to the k-dependence on yJi . In either case a k dependence in the velocity potential appears natural. A strong k-dependence really implies that the multifluid picture is not very useful in that it shows that the Fermi structure is still very imporant and that the grouping into pairs is not a significant reduction of the complexity of the problem. Similarly, yk is meaningful near the Fermi surface. Outside it, the product U,.V, is small and any change in ~~ merely redefines the wavefunction @ by a global phase factor. As an intermediate step in the derivation of the equations of motion for the Fermi fluid, we shall make more explicit the contribution of each Cooper pair to the kinetic energy (b). In particular we discuss the role of the variables x, (h - k), and (/I + k)/2. The first two variables were already encountered in the treatment of the Bose fluid, where the argument x in the velocity potential y(x) is conjugated to (/I - k). We assume that in the fermion case, the conjugate relation between the two variables is preserved. This assumption is easily shown to be in accordance with (4.16)-(4.18) when #I(~~+,~)?I = #. Thus y(s)
= C (A,,, --t yhk) e’(“-‘)~” = 9 t 1‘.k
$ sin(ax)
(4.29)
where a = I h - k j. Thus x and (11 - k) refer to the same degree of freedom, namely, to the center of mass of each Cooper pair. On the other hand, the variable ((h + k)/2) IS associated with the internal degrees of freedom of the Cooper pair, and can be used to label the set of fluids corresponding to each species of these pairs. Thus, a velocity potential which takes both external and internal degrees of freedom of the Fermi condensate into account should depend on both ((II + k)/2) and ((II - k)/2). A simple potential which fulfills such requirements is (4.30) d-4 = 1 ~r,+(l,z),~--cl,,)e’z”.
1
Utilizing
the definition (4.16H4.18) ydX)
for P)?,~,~we obtain =
yk
+
$12 sin@x).
(4.31)
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SUPERFLUIDS
359
This velocity potential can give rise to a richer variety of flows, albeit less coherent, than (4.29). The k-fluid moves with velocity vJx) == V,,(X) I- -V(#, sin ax) and 6H, is given by
where Q is the volume of the physical system under consideration. In deriving the above expression we have made use of the smallness of a to expand (Ed, ~ cl) and cos 28,< appearing in (4.25), in powers of (h - k). We can now collect all the variations in the form of a quadratic effective Lagrangian in the fields 9J.y) and SO,;(x) as follows:
- (G/8) 1 sin 28, sin 28,,(6~, - SF,)”
(4.33 j
h 7~
This Lagrangian is then assumed as a starting point for the investigation which follows in Section 5. It should be noted, however, that the set of all x dependent fields vk(x), SO,(x), is more general than the present derivation from the BCS model Hamiltonian allows for. Indeed we go beyond the type of variation represented by (4.32) and admit the validity of (4.33) for a generic x-dependence of the fields. Note, however, that the x-dependence should not be too strong and that we should only consider modes for which a < k, . As it stands the Lagrangian (4.32) bears a strong similarity to the one characteristic of the Landau single fluid approach. The term sin” I!&&, shows that sin2 &.(x)/Q is the conjugate variable to @L(x) and that it plays the role of partial density of the fluid of pairs with momentum k. Indeed the number equation
where N is the number of fermions in the system, merely expresses the total density of pairs p(x) as the sum of partial densities of k-pairs. The term -(G/8)
1 sin 20, sin 20, x (6~~ - Sq$
(4.35)
7:p
plays the same role in the k space as the kinetic energy does in x space and is essential in maintaining some degree of coherence in the fluid for it increases the energy of those excitations in which vDkhas a strong dependence on k. Finally the terms quadratic in 68, correspond to the free energy of a fluid, which in Landau’s theory depend only on the density of the fluid.
360
BROGLIA,
MOLINARI,
AND REGGE
5. THE EQUATIONS
OF MOTION
It is now a simple matter to carry out the variation of the Lagrangian (4.33) for generic fields. The second variation in 6~~ gives -sin 28&O, 7m(G/2) sin 20, x sin 20,(S~~ ~ SF,)
while the one in ek corresponds to sin 2f$&~, + 4E&$
- ~GCOS~&..COS Y
2%,68, = 0.
(5.2)
Making the Ansatz
(5.3) we obtain the set of equations wQk - (d - m2(+,
Tk = - 7,
2(k*p)2 $1
(5.4)
and
-4EkQk
+ w sin2i&T, = -2Gcos2
t&Y,
(5.5)
where Y = 2 cos 2e,p, ) P
(5.6)
A’= 1 sin 2&T,.
(5.7)
Dk = co2sin 28, - 4&(d - ~~~(h/nr)~2(k . p)“(X?,/&,)),
(5.8)
and Defining
the amplitudes Qk and Tk are given by
--d + (Gj’
2m2(k * p)” 2 k
Qk=-s
!
w
sin 8,
(5.9)
and
(5.10)
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Utilizing
-;K
361
SUPERFLUIDS
(5.4)-(5.10) the dispersion relation fixing the frequency w is obtained: 2Ek sin20.--
’
sin 28, cos 28,
-UCk
-/------ sin 28, cos 219, Dk
G
(d - 2tn2 (4)’
-2x
Dk
(,k - p)’ $)
cos2 20,
1 G
Dk
ii
--
= 0.
i
(5.11)
We now study the long wavelength modes of the pairing spectrum. The energy denominator Dk is given by Dk = -sin2 &(4&
+ u;~),
(5.12)
where (5.13) In (5.13) the spatial averaging (k . p)” = k2p2/3has been utilized. Expanding the different elements of (5.11) in terms of p and w and retaining up to quadratic terms one obtains a11
=
-;DI;2Ek
w - $;
sin 26 _-L “G
1[T
(5.14) (i-)”
k2p2-- co?]&-I
)
and aI2 = a21=
sin 28, cos 28, (5.15)
The relation (4.22) can now be written as -+$‘[~(+)‘k2p2-m2]& x
Gco;s
cos2 2ek
Gzr-1 k
= 0,
(5.16)
h
where only constant terms were retained in a22, as a,, is proportional quantities. We are now in position to calculate the sound velocity.
to second order
362
BROGLIA, MOLINARI,
AND REGGE
Assuming us1 = 0, i.e., no coupling between the gauge and gap modes, we obtain, from the condition a,, = 0, (5.17) In general one obtains ti * i-1m sin22f9,, 4E,;f
(5.18)
for the velocity of normal sound in a superconductor, when the coupling to the pairing degreesof freedom is taken into account. The modes associated with variations 60, , namely, the pairing vibrations, have a finite mass,which is in all caseslarger than or equal to 24.
6. FINITE SYSTEMS Settingp = 0 and utilizing the relations Ui2 - Vk2 = cos 20, and 2U,V, = sin 20, we can rewrite (5.11) as w c (UiZ - V,2) ,; 4Ek2 - w2 2Ek(Uk2 - Vk2) 2c 4Ek2 - w2 k
= 1 -1 0.
(6.1)
G
This relation coincides with [6, Eq. (2.42)], which was obtained diagonalizing the pairing Hamiltonian in the RPA. In this case(4.1) was expressedin terms of the quasiparticle operators cq+ = u,c,+ - V#,q . (6.2) and afiG’= u,c,-i- + V&k )
(6.3)
and retaining all terms quadrilinear in the al’sbut those which give rise to scattering processes. In finite systems, k is to be interpreted as labeling the set of quantum numbers of a particle moving in a shell model potential well. The dispersion relation (6.1) gives rise to two types of collective modes. This can best be seenby neglecting the off-diagonal elements. Thus (6.4)
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has a solution w = 0 (i.e., & (l/2&) = l/G, known as the gap equation in the BCS theory), its wavefunction is proportional to the operator number of particles acting on the BCS ground state. This is the spurious state associated with the nonconservation of the number of particles and has its counterpart, in the case of extended systems, in the normal sound at rest (k = 0). On the other hand the dispersion relation
I = &- ’ which
(6.5)
is equivalent to (6.6)
has its lowest root at w = ?A. These collective modes are associated with the intrinsic (gap) degree of freedom of the Cooper pairs and are known as pairing vibrations [5, IO]. They correspond in extended matter to massive phonons associated with both isotropic and quadrupole oscillations of the Fermi surface. The quadrupole oscillations are generated by the terms proportional to (k . P)~ in a22 (cf. Eq. (5.11)). While the field associated with normal sound is constant (Uk2 + Vk2 = l), the one associated with pairing vibrations changes sign at the Fermi surface implying that particles with Ed > p move in one direction while those with ek < p move in the opposite direction, and there is no net flow of mass generated by pairing vibrations. This can occur through hybridization with the normal sound, controlled by the magnitude of the off-diagonal element in (5.11) or (6. I). The conditions for the appearance of a roton minimum from this hybridization as argued in Ref. [II] are yet unclear, since our results are essentially long wavelength results. The fact that the result (6.1) is obtained utilizing the RPA, in which case statistics plays an essential role because of the quasi-boson commutators, and through the Lagrangian formalism discussed above, which heavily relies on the existence of a velocity potential, indicates that both solutions of the pairing problem are equivalent and their viewpoints complementary.
REFERENCES 1. L. LANDAU, J. Exp. Theor. P&x 11, (1941), 592. 2. R. P. FEYNMAN, Phys. Rev. 94 (1954), 262. 3. M. RASETTI AND T. REGGE, J. Low Tevzp. Phys. 13 (1973), 249. 4. M. RASETTI AND T. REGGE, “Superfluid Motion,” preprint IAS, 5. D. R. BES AND R. A. BROGLIA, Nucl. Phys. 80 (1966), 289. 6. R. A. BROGLIA, “Proceedings of the Eleventh Summer Meeting
October 1976.
of Nuclear Physicists” (L. Sips, Ed.), Vol. 1, p. 201, Herceg Novi, August 1966, Federal Nuclear Energy Commission of Yugoslavia. 7. A. BOHR AND B. R. MOTTELSON, “Nuclear Structure,” Vol. II, Benjamin, New York, 1971. 8. J. BARDEEN, L. N. COOPER, AND J. R. SCHRIEFFER, Phys. Rev. 108 (1957), 1175.
364
BROGLIA,
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AND
REGGE
9. J. R. SCHRIEFFER, Theory of superconductivity, in “Frontiers in Physics,” Benjamin, New York, 1964. 10. The pairing vibrations are one of the low-lying collective modes of nuclei. They have been systematically observed in two-nucleon transfer processes and their main properties are well described by the solutions of (6.1). For more detail we refer to R. A. BROGLIA, 0. HANSEN, AND C. RIEDEL, Adv. Nucl. Phys. 6 (1973). 11. R. A. BROGLIA, A. MOLINARI, AND T. REGGE, Ann. Phys. (N. Y.) 97 (1976), 289.
OF PHYSICS
109, 349-364 (1977)
On the Hydrodynamics
of Fermi Superfluids*+
R. A. BROGLIA* Stare
Unicrrsity of New and Brookhacen
York at Stony Brook, National Laboratory,
Stony Brook, Upton, New
New York York 11973
11794
A. MOLIP\~ARI lstituto
di Fisica
dell’Unicersita
di Ferrara,
Ferraro,
Italy
T. REGGE Institute
for Advanced
Study,
Princeton,
New Jersey
08540
Received April 25, 1977
The hydrodynamic of a neutral Fermi fluid is derived from the microscopic pairing model in the framework of a time dependent BCS formalism. A discussion of the superfluidity and coherence properties of these fluids and a comparison with the known He II hydrodynamics is attempted. Applications to finite nuclear systems are discussed.
I. INTRODUCTION
In dealing with quantum fluids Landau [I] states that in He ZZthe motion of the superfluid component is characterized by the existence of a velocity potential: v(x) = - Vcp(x)
(1.1)
and by a density field p(x), canonically conjugated to y(x). For long wavelength excitations the motion of the superfluid is essentially controlled by the Euler-Bernoulli equations and of course by the continuity equations. Higher energy excitations arise from vortices, where F(X) is no longer single valued and at the roton wavelength. where Feynman’s backflow (cf., e.g., [2]) contribution can be essentially interpreted as a breakdown of Landau’s criterion. In both casesthe concept of velocity potential * The United States copyright covering this + Supported, in part, E(ll-l)-3001. * On leave from the
Government’s right to retain a nonexclusive royalty-free license in and to paper is acknowledged. by Energy Research and Development Administration, under Contract Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark.
349 Copyright All rights
0 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSN
0003-49
I6
350
BROGLIA, MOLINARI,
AND REGGE
becomeslessdefined and useful. The aim of this paper is to show that a similar but richer structure arisesfor Fermi superfluids containing Bose superfluids as a limiting case in the strong coupling limit. In particular we argue for the suitability of a two and possibly multifluid theory in dealing with the excitation spectrum of a Fermi superfluid where the intrinsic degreesof freedom of the nucleon in the Cooper pairs are no longer negligible.
2. BOSE HYDRODYNAMICS:
A SHORT DERIVATION
We assume in what follows that the ground state wavefunction @(x1,..., x2) IS given by a real nonnegative symmetric function of the boson coordinates .yi . Qualitatively @can be characterized asfollows: (‘1) If the bosons are evenly spaced apart on the average then @ is essentially constant and depends only on the macroscopic fluid density. (2) @ drops quickly to zero if two bosons approach each other becauseof the strong repulsive core at short distances. (3)
@has no nodes for this would increasethe energy of the state.
(4)
@is rigorously invariant under translations and rotations.
It is moreover approximately invariant under volume preserving differentiable maps of R3 into R3 for these maps do not violate (1) (2) (3)). A map of this sort, for instance, is given by: 5’ = Ax,
1” == /.Lj’,
z = vz
hpv = 1,
(2.1)
in which case a cube is transformed into a solid of equal volume. This last property really characterizes @ as the wavefunction of a fluid, as contrasted with that of a solid which certainly violates it. It should be understood, however, that the stretching generated by (2.1) should be sufficiently gentle for the invariance is only approximate. According to Feynman [2] once the wavefunction for the ground state is known, we obtain the wavefunction QF for the fluid in (coherent) motion from: QF = exp i f ~(x~)(i?i/fi) x @ = eio@, (2.2) 1 ( i=l where m is the boson mass. The insertion of this function into the time dependent Schrodinger equation leads to disappointing results in that it is difficult to seehow all the arising terms compensate in the equation. An improved version of the Feynman Ansatz has been proposed in [3]. It reads CD,= exp i g fp(xJ(m/fi) ( i=l ) x @ml
3 t), 5x% 30,-v &&a 3 t)) x (a(()/a(x))liz
(2.3) = eiO 1C& 1.
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351
SUPERFLUIDS
Here 6(x, t) is defined through the equation of flow:
with the boundary
dx’x(5, t)/dt = v(x(E, t), t)
(2.4)
xc& 0) = 5.
(2.5)
conditions:
From x(t: t) we define the inverse function &x, t) which is then used in (2.3). The square root of the Jacobian in (2.3) keeps the Ansatz properly normalized. Our definition is equivalent to the differential equation:
with I @D, I*=f) = CD.
(2.6a)
Although t = 0 plays a special role here we could have selected any time and in particular t = -co as starting point. Both definitions amount to choosing a time evolution where the ground state, besides acquiring the Feynman-Onsager phase [2], also drifts downstream along the velocity. Both (2.3) and (2.6) define a particular time evolution of the wavefunction. The macroscopic density p(x, t) of the fluid is determined completely by the Ansatz (2.3) either through the continuity equation (iip/dt) + div pv = 0 and the boundary
condition 0) = 1 I @p. 12d?Y,
p(x,
or equivalently
(2.7a)
.. .
d3x,
(2.7b)
d?Y.\,
by p(x, I) = yy
The connection the identity
with
the Schrodinger
equation
p(x, 0). is established
12.8) as follows.
We state
0.9) and then use GE as a trial function 6 [
dt
d3.Y, .‘.
in the variational d?YN
($*(ifi(a/at)
principle - H) $J) = 0.
(2.10)
We thus obtain: 6 j” dr c i
d3sj
IJ~(.Y~)
[w
-t -1
+
/” dt (c&r, i%DF)
J
= 0.
(2.11)
352
BROGLIA,
MOLINARI,
AND
REGGE
In this variational principle y(x) is considered as the independent variational parameter and p(x) as a dependent one which is determined through (2.8). We now use condition 4. Note that the action of the flow on GF can be reduced locally to a translation and a rotation (of no effect) superimposed on a stretching similar to one determined by (2.1). and finally on a local change of density. We assume, therefore, that QF is essentially a “modulated” ground state where the local density P(X) varies slowly with respect to a typical interboson distance. For a conventional ground state at constant density p we can write: H@ = c e(p) CD= Ne(p) @,. For the modulated
(2.12)
ground state we propose the obvious generalization: (2.13)
H@, = C e(p(si , t)) OF . lnserting
(2.13) into (2.1 I) we find finally:
which leads to the Bernoulli
equation: HI+ + $IHZ~~+ (d(pe)/dp)
which appears now as a variational more details cf. [3, 41). If the velocity of F(X) and its derivative v(x). We write the result in terms of the relations C(x) CQ’) -
= 0
(2.15)
approximation to the Schrodinger equation (for potential is small, we can expand QF in powers boson field C(X) fulfilling the usual commutation C’(i)
C(x) = S3(x - x’)
(2.16)
and the associated density p(x) = c+(s)
C(x).
(2.17)
We find cDF Y [ 1 + (i/n//i) [ y(x) p(s) ~1”s +- (h//i) which can be compactly written
1 r/t d”s z+(s, t) j,(s, t)] @
(2.18)
as
In the treatment of Fermi fluids (2.19) is the starting step in the variational (cf. Section 4).
treatment
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353
SUPERFLUIDS
Assuming (2.20)
where ya = ~JT, , we obtain for (2.19) Q% = (1 + $1
u
‘pl,eP@ C Cl+;,Ck (wa - (k + 4 , a) $jj h
If li-l is much larger than the interparticle
@.
(2.21)
distance then
In this case the p-term predominates over thej-term (current). Quantities of hydrodynamical interest are usually expectation values of products of operators for low values of k (Iong-wave limit). We thus note that the generic case is adequately represented by g)h,fi , of the form utilized in (2.20), i.e., (m/h) y&~at.
3.
BEYOND
THE
BOSE
GAS
In Landau’s quantum hydrodynamics [l] only the degrees of freedom of the fluid which can be expressed through y and p are kept, these two fields being canonically conjugated
[&4 PWI = 6(x - a. The Hamiltonian
(3.1)
is then H =
.r
mp[((V#/2)
+ E(p)] d3x.
(3.2)
If the coupling in a Fermi superfluid is taken very large then Cooper pairs will acquire a very large binding energy and behave more and more like bosons. Therefore as long as we do not probe the fluid with an energy comparable to the energy gap we expect a behavior similar to that of a conventional Bose fluid. In particular there should be a phonon excitation branch with spectrum w = c(hk) where c is the velocity of sound. Clearly this branch has meaning only if c(tik) < d, else the phonon decays into quasi-particles. The question of the unfreezing of the relative degrees of freedom of the fermions inside the Cooper pairs is the one we have to deal with. Since breaking a Cooper pair costs energy, it is not expected that any excitation not much higher than 24 should break all Cooper pairs and see the fluid as essentially a normal Fermi fluid at T = 0, where all hydrodynamical motions become strongly dissipative. Rather we expect the onset of new degrees of freedom as we gradually increase the energy of the excitations which finally destroy the usefulness of the one fluid theory.
354
BROGLIA, MOLINARI,
AND REGGE
We believe that a satisfactory physical theory of these extra degreesof freedom can be acquired through a two- and perhaps multifluid theory with particular regard to the pairing vibrations [5-71, which play an important role in nuclei. A multifluid theory can be carried out either as a phenomenological enlargement of the conventional Landau scheme[l-4] or on a BCS model Hamiltonian [8, 91. We believe that both approaches are mutually related in a fashion similar to the boson case.
4. TIME DEPENDENT
BCS THEORY
In this section we present a hydrodynamical picture similar to the one discussed for the Bose fluid in Section 1. Our starting point is the effective pairing Hamiltonian: H=H,,+H,
(4.1)
where H,, = c (et - p) l1fi.s LS
(4.la)
and H, = c V,,b,‘b, kP
.
(4.lb)
The single-particle energies cJiare measured from the Fermi energy p. The creation and annihilation operators for Fermions of linear momentum k and the z-component of the spin s are denoted c;~ and cks, respectively. They satisfy the usual anticommutation relations (ck,, , c;,,,> = 6(k, k’) 6(s, s’).
(4.2)
The operator trkscounts the number of particles in the state (ks) and is given by nkn
=
+
CknCks
(4.3)
.
The pair creation and annihilation operators are bk+ = c;&
,
(4.4)
and bl, = c-,~c,~,
(4.5)
where et,, = TC&.T-l, T being the time reversal operator. The quantities V,, are the pairing matrix elements. The conventional static BCS treatment of (4.1) is based on the trial wavefunction / BCS) = n (u, + Vkeim’bk+)lo), k
(4.6)
where ) 0) is the vacuum of the operator c;Z,. Becauseof the violation of the particle number in (4.6), each Cooper pair is referred to an intrinsic system, eiQkbeing the
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355
SUPERFLUIDS
two-dimensional rotation in gauge space which relates this system to the laboratory system. The quantities U, and V, are the BCS occupation parameters. The variational principle for the ground state, S(BCSjHjBCS)
=0
is satisfied provided U, and V,: display the following functional single-particle energy and on the pairing matrix elements
(4.7) dependence on the
(4.8) (4.9) where the quasi-particle
energy is defined by Ek = ((‘cl, -
p)2 + dk2)l/”
(4.10)
and where the energy-gap parameter d, satisfies the integral equation A, = - C V,,(A./2E,)
= - 1 vk,Upv,.
7
Y
(4.11)
We can choose 91c= q~, where g, is a constant. This arbitrariness merely reflects the degeneracy of the BCS ground state. A convenient gauge is that which makes (4.1 I) real. We note that a wavefunction similar to (4.6) can also be defined for a Bose fluid. In fact
has the same degeneracy as the BCS wavefunction and is also nondiagonal in the number of particles. In the case of the Bose fluid of Section 1, if motion takes place the phase factor exp{iN(m@)} is replaced by exp{im xr=, y(xJ/fi}. It is now natural to ask what corresponding changes we must implement in the BCS wavefunction in order to represent slow fluid motions. The starting point for this generalization is Eq. (2.19). Thus, we are led to consider generators of the form (4.13) and a trial wavefunction of the kind @ = eie 1 BCSj,
(4.14)
356
BROGLIA,
MOLINARI,
AND
REGGE
with (4.15)
@ = 1 FhkQhk . 1l.P
The quantities 9),& are time dependent and in general not diagonal in the indices h and k. We assume moreover that the parameters Oain (4.8) and (4.9) are also subject to variation and are time dependent. It is understood that all 19, and yhk are small because we are dealing with states of quasi-equilibrium corresponding to small variations around the ground state. The operator Q hfGadd the momentum (11 - k) to the Cooper pair. In analogy to the boson case we expect that the motion of the fluid is generated by the nondiagonal terms in vhli. Note that the diagonal term LJhh merely redefines the partial phases occurring in (4.6). There is no lack of generality in carrying out the different steps of the variational calculation utilizing a phase defined as qJhk = Ah, +- y/a 3
(4.16)
h,, = W, 4 v/;
(4.17)
with and Yhli
=
(1/2i)@(k
k
+
a)
-
@/I
+
4
k))
&(h+k)/21
(4.18)
,
where a = 27r/L is the smallest wavenumber compatible with periodic boundary conditions. We are thus in a way selecting the most gentle possible motion of the fluid. Note that (4.16)-(4.18) define a hermitian operator. In contrast to the boson case we retain an essential dependence of #I(~+~),~, and vrz on their respective indices. As we discuss below, to delete this dependence would imply to freeze the internal degrees of freedom of the Cooper pairs, and thus the physical system under discussion would basically reduce to the Bose fluid of Section 1. We write 0=fl+r, (4.19) with (4.20) A = c hh,nhk > hk
and r
The trial wavefunction
=
c YhkQhk h Ii
is then introduced 6 f(O*(ifi(+?t)
(4.21)
.
into the variational
equation
- H) @) dr = 0
and the integral is expanded up to quadratic terms in the variations.
(4.22) In particular
(@, H@) = (BCS j H 1BCS:) + 1’(BCS j [@, H] 1BCS) - &(BCS [ [@, H] 1BCS) (4.23) up to second order in 0.
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357
SUPERFLUIDS
Several of the first order terms vanish as implied by the BCS variational condition (4.7). The remaining first order terms vanish because the operator I’ does not conserve the momentum and thus (BCS 1 [I’, H] 1BCS) = 0. By the same token there are no bilinear terms /lr in the variation of (4.22). The remaining quadratic terms can be classified as follows: (a)
Terms quadratic
in Ll and 68, . An explicit calculation
SH, = ~2E,(S%,,.)” 1.
- G c cos2%,cos2%,S0,68,~ k‘y
~- (G/8) 2 sin2Q,sin28,(6y,L I.,a (b)
gives
- 6~~)~ ,
(4.24)
Terms quadratic in r and arising from H,,, . They are equal to 6H, = 1 ykrL jz (c,; - E/J cos 28,; . k f2
(4.25)
(c) Terms quadratic in N arising from Hint. In the Bose fluid such terms are not present because of a trivial reason; namely, that the potential terms are strictly local in the boson coordinates and the Nterms correspond essentially to the FeynmanOnsager (cf. [3. 31) phase which is also local and therefore cancels out. Also the model BCS Hamiltonian is not complete, in that it represents only condensation in Cooper pairs of zero momentum. For this and similar reasons, the contribution of the terms under discussion is difficult to evaluate and is anyway expected to be small. We decided not to include them in the present calculation. (d)
Terms arising from the time derivative,
= -Re
(+BCS
i.e.,
1@eio 1 BCS) + Re (e’“BCS
/ e’% k j SC%>. (4.26)
Note that r
0 = $1
= 6 (BCS ! BCS‘, - Re (BCS ; 3 ] KS,<..
But (BCS ; %/?t / BCS) is real and thus equal to zero. which term in (4.26) is also zero. Thus Ret@. #‘h/Et)
@) = -17 c q& sin2 8, l2
(4.27)
implies that the last (4.28)
The terms (4.24), (4.25), and (4.28) are all the second order contributions to (4.22). The terms proportional to (1” do not contribute to (4.25). This is natural since the terms of type (b) are related to the kinetic energy of the moving fluid and this motion can only be achieved by nondiagonal terms of the type l? Both the fl and the r terms
358
BROGLIA,
MOLINARI,
AND
REGGE
contain an additional k-dependence which needs some clarification before we proceed further. The physical meaning of this dependence must depend on those features of the Fermi fluid which are not present in the Bose fluid and these can only be the relative degrees of freedom of the fermions in each Cooper pair. These manifest themselves in the surviving internal index k which labels the relative momentum of the fermions in the pairs. Therefore, pairs are distinguished by the label k which indicates they possess an internal k-degree of freedom besides the ordinary translational degrees of freedom. Alternatively we may consider the fluid as the mixture of infinitely many fluids, labeled by k, the dynamics of the system allowing the transmutation of a fluid into the other. The distinction between the two pictures is clearly only one of semantics. The first one implies that just as the velocity potential should depend on x, then it should also depend on k. In the second picture it is natural to assume that the different fluids may perform different motions according to the k-dependence on yJi . In either case a k dependence in the velocity potential appears natural. A strong k-dependence really implies that the multifluid picture is not very useful in that it shows that the Fermi structure is still very imporant and that the grouping into pairs is not a significant reduction of the complexity of the problem. Similarly, yk is meaningful near the Fermi surface. Outside it, the product U,.V, is small and any change in ~~ merely redefines the wavefunction @ by a global phase factor. As an intermediate step in the derivation of the equations of motion for the Fermi fluid, we shall make more explicit the contribution of each Cooper pair to the kinetic energy (b). In particular we discuss the role of the variables x, (h - k), and (/I + k)/2. The first two variables were already encountered in the treatment of the Bose fluid, where the argument x in the velocity potential y(x) is conjugated to (/I - k). We assume that in the fermion case, the conjugate relation between the two variables is preserved. This assumption is easily shown to be in accordance with (4.16)-(4.18) when #I(~~+,~)?I = #. Thus y(s)
= C (A,,, --t yhk) e’(“-‘)~” = 9 t 1‘.k
$ sin(ax)
(4.29)
where a = I h - k j. Thus x and (11 - k) refer to the same degree of freedom, namely, to the center of mass of each Cooper pair. On the other hand, the variable ((h + k)/2) IS associated with the internal degrees of freedom of the Cooper pair, and can be used to label the set of fluids corresponding to each species of these pairs. Thus, a velocity potential which takes both external and internal degrees of freedom of the Fermi condensate into account should depend on both ((II + k)/2) and ((II - k)/2). A simple potential which fulfills such requirements is (4.30) d-4 = 1 ~r,+(l,z),~--cl,,)e’z”.
1
Utilizing
the definition (4.16H4.18) ydX)
for P)?,~,~we obtain =
yk
+
$12 sin@x).
(4.31)
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SUPERFLUIDS
359
This velocity potential can give rise to a richer variety of flows, albeit less coherent, than (4.29). The k-fluid moves with velocity vJx) == V,,(X) I- -V(#, sin ax) and 6H, is given by
where Q is the volume of the physical system under consideration. In deriving the above expression we have made use of the smallness of a to expand (Ed, ~ cl) and cos 28,< appearing in (4.25), in powers of (h - k). We can now collect all the variations in the form of a quadratic effective Lagrangian in the fields 9J.y) and SO,;(x) as follows:
- (G/8) 1 sin 28, sin 28,,(6~, - SF,)”
(4.33 j
h 7~
This Lagrangian is then assumed as a starting point for the investigation which follows in Section 5. It should be noted, however, that the set of all x dependent fields vk(x), SO,(x), is more general than the present derivation from the BCS model Hamiltonian allows for. Indeed we go beyond the type of variation represented by (4.32) and admit the validity of (4.33) for a generic x-dependence of the fields. Note, however, that the x-dependence should not be too strong and that we should only consider modes for which a < k, . As it stands the Lagrangian (4.32) bears a strong similarity to the one characteristic of the Landau single fluid approach. The term sin” I!&&, shows that sin2 &.(x)/Q is the conjugate variable to @L(x) and that it plays the role of partial density of the fluid of pairs with momentum k. Indeed the number equation
where N is the number of fermions in the system, merely expresses the total density of pairs p(x) as the sum of partial densities of k-pairs. The term -(G/8)
1 sin 20, sin 20, x (6~~ - Sq$
(4.35)
7:p
plays the same role in the k space as the kinetic energy does in x space and is essential in maintaining some degree of coherence in the fluid for it increases the energy of those excitations in which vDkhas a strong dependence on k. Finally the terms quadratic in 68, correspond to the free energy of a fluid, which in Landau’s theory depend only on the density of the fluid.
360
BROGLIA,
MOLINARI,
AND REGGE
5. THE EQUATIONS
OF MOTION
It is now a simple matter to carry out the variation of the Lagrangian (4.33) for generic fields. The second variation in 6~~ gives -sin 28&O, 7m(G/2) sin 20, x sin 20,(S~~ ~ SF,)
while the one in ek corresponds to sin 2f$&~, + 4E&$
- ~GCOS~&..COS Y
2%,68, = 0.
(5.2)
Making the Ansatz
(5.3) we obtain the set of equations wQk - (d - m2(+,
Tk = - 7,
2(k*p)2 $1
(5.4)
and
-4EkQk
+ w sin2i&T, = -2Gcos2
t&Y,
(5.5)
where Y = 2 cos 2e,p, ) P
(5.6)
A’= 1 sin 2&T,.
(5.7)
Dk = co2sin 28, - 4&(d - ~~~(h/nr)~2(k . p)“(X?,/&,)),
(5.8)
and Defining
the amplitudes Qk and Tk are given by
--d + (Gj’
2m2(k * p)” 2 k
Qk=-s
!
w
sin 8,
(5.9)
and
(5.10)
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Utilizing
-;K
361
SUPERFLUIDS
(5.4)-(5.10) the dispersion relation fixing the frequency w is obtained: 2Ek sin20.--
’
sin 28, cos 28,
-UCk
-/------ sin 28, cos 219, Dk
G
(d - 2tn2 (4)’
-2x
Dk
(,k - p)’ $)
cos2 20,
1 G
Dk
ii
--
= 0.
i
(5.11)
We now study the long wavelength modes of the pairing spectrum. The energy denominator Dk is given by Dk = -sin2 &(4&
+ u;~),
(5.12)
where (5.13) In (5.13) the spatial averaging (k . p)” = k2p2/3has been utilized. Expanding the different elements of (5.11) in terms of p and w and retaining up to quadratic terms one obtains a11
=
-;DI;2Ek
w - $;
sin 26 _-L “G
1[T
(5.14) (i-)”
k2p2-- co?]&-I
)
and aI2 = a21=
sin 28, cos 28, (5.15)
The relation (4.22) can now be written as -+$‘[~(+)‘k2p2-m2]& x
Gco;s
cos2 2ek
Gzr-1 k
= 0,
(5.16)
h
where only constant terms were retained in a22, as a,, is proportional quantities. We are now in position to calculate the sound velocity.
to second order
362
BROGLIA, MOLINARI,
AND REGGE
Assuming us1 = 0, i.e., no coupling between the gauge and gap modes, we obtain, from the condition a,, = 0, (5.17) In general one obtains ti * i-1m sin22f9,, 4E,;f
(5.18)
for the velocity of normal sound in a superconductor, when the coupling to the pairing degreesof freedom is taken into account. The modes associated with variations 60, , namely, the pairing vibrations, have a finite mass,which is in all caseslarger than or equal to 24.
6. FINITE SYSTEMS Settingp = 0 and utilizing the relations Ui2 - Vk2 = cos 20, and 2U,V, = sin 20, we can rewrite (5.11) as w c (UiZ - V,2) ,; 4Ek2 - w2 2Ek(Uk2 - Vk2) 2c 4Ek2 - w2 k
= 1 -1 0.
(6.1)
G
This relation coincides with [6, Eq. (2.42)], which was obtained diagonalizing the pairing Hamiltonian in the RPA. In this case(4.1) was expressedin terms of the quasiparticle operators cq+ = u,c,+ - V#,q . (6.2) and afiG’= u,c,-i- + V&k )
(6.3)
and retaining all terms quadrilinear in the al’sbut those which give rise to scattering processes. In finite systems, k is to be interpreted as labeling the set of quantum numbers of a particle moving in a shell model potential well. The dispersion relation (6.1) gives rise to two types of collective modes. This can best be seenby neglecting the off-diagonal elements. Thus (6.4)
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363
SUPERFLUIDS
has a solution w = 0 (i.e., & (l/2&) = l/G, known as the gap equation in the BCS theory), its wavefunction is proportional to the operator number of particles acting on the BCS ground state. This is the spurious state associated with the nonconservation of the number of particles and has its counterpart, in the case of extended systems, in the normal sound at rest (k = 0). On the other hand the dispersion relation
I = &- ’ which
(6.5)
is equivalent to (6.6)
has its lowest root at w = ?A. These collective modes are associated with the intrinsic (gap) degree of freedom of the Cooper pairs and are known as pairing vibrations [5, IO]. They correspond in extended matter to massive phonons associated with both isotropic and quadrupole oscillations of the Fermi surface. The quadrupole oscillations are generated by the terms proportional to (k . P)~ in a22 (cf. Eq. (5.11)). While the field associated with normal sound is constant (Uk2 + Vk2 = l), the one associated with pairing vibrations changes sign at the Fermi surface implying that particles with Ed > p move in one direction while those with ek < p move in the opposite direction, and there is no net flow of mass generated by pairing vibrations. This can occur through hybridization with the normal sound, controlled by the magnitude of the off-diagonal element in (5.11) or (6. I). The conditions for the appearance of a roton minimum from this hybridization as argued in Ref. [II] are yet unclear, since our results are essentially long wavelength results. The fact that the result (6.1) is obtained utilizing the RPA, in which case statistics plays an essential role because of the quasi-boson commutators, and through the Lagrangian formalism discussed above, which heavily relies on the existence of a velocity potential, indicates that both solutions of the pairing problem are equivalent and their viewpoints complementary.
REFERENCES 1. L. LANDAU, J. Exp. Theor. P&x 11, (1941), 592. 2. R. P. FEYNMAN, Phys. Rev. 94 (1954), 262. 3. M. RASETTI AND T. REGGE, J. Low Tevzp. Phys. 13 (1973), 249. 4. M. RASETTI AND T. REGGE, “Superfluid Motion,” preprint IAS, 5. D. R. BES AND R. A. BROGLIA, Nucl. Phys. 80 (1966), 289. 6. R. A. BROGLIA, “Proceedings of the Eleventh Summer Meeting
October 1976.
of Nuclear Physicists” (L. Sips, Ed.), Vol. 1, p. 201, Herceg Novi, August 1966, Federal Nuclear Energy Commission of Yugoslavia. 7. A. BOHR AND B. R. MOTTELSON, “Nuclear Structure,” Vol. II, Benjamin, New York, 1971. 8. J. BARDEEN, L. N. COOPER, AND J. R. SCHRIEFFER, Phys. Rev. 108 (1957), 1175.
364
BROGLIA,
MOLINARI,
AND
REGGE
9. J. R. SCHRIEFFER, Theory of superconductivity, in “Frontiers in Physics,” Benjamin, New York, 1964. 10. The pairing vibrations are one of the low-lying collective modes of nuclei. They have been systematically observed in two-nucleon transfer processes and their main properties are well described by the solutions of (6.1). For more detail we refer to R. A. BROGLIA, 0. HANSEN, AND C. RIEDEL, Adv. Nucl. Phys. 6 (1973). 11. R. A. BROGLIA, A. MOLINARI, AND T. REGGE, Ann. Phys. (N. Y.) 97 (1976), 289.