Bose spectrum of superfluid solutions 3He4He

Volume 101A, number 3

PHYSICS LETTERS

19 March 1984

BOSE SPECTRUM OF SUPERFLUID SOLUTIONS 3 H e - 4 H e

P.N. BRUSOV Research Institute o f Physics o f Rostov on Don State, M.A. Suslov University, Rostov on Don. USSR and

V.N. POPOV Leningrad Branch o f the V.A. Steklov Mathematical Institute of Academy o f Sciences o f the USSR, Leningrad, USSR

Received 3 May 1983 Revised manusczipt received 18 January 1984

The Bose spectrum of superfluid solutions 3He-4He is investigated in the case of p-pairing of Fermi particles. Both three-dimensional and two-dimensional models are considered. It is shown that the Bose-Fermi interaction changes the sound velocities in both subsystems. The other collective modes (17 in the 3D-model and 11 in the 2D-model) remain almost unchanged.

Solutions of 3He in superfluid 4He have been investigated theoretically and experimentally for many years [1]. A naturally arising question is whether it is possible for the Fermi component of the solution to go into the superfluid state via Cooper pairing [2]. Nowadays there is no experimental evidence for such a transition. Nevertheless, many theoretical papers are devoted to this problem (see ref. [3], and references therein). Many authors supposed that Cooper pairs in the s-state (with zero orbital and spin momenta) may arise in 3-dimensional (3D) systems. According to Bardeen et al. [2] the transition temperature for this case is of the order of 2 X 10 -6 K (at zero tem. perature). Landau et al. [4] gave as an estimate of the transition temperature 10 - 4 K for pressures between 10 and 20 arm. Hoffberg [5] has shown that s-pairing is preferable at low concentrations of 3He (0 ~6.6%. According to Hofberg the transition temperature increases from 10 - 4 K at x = 7.6% to 10 -3 K at x = 8% and has a maximum value of 1.5 × 10 -2 K at x = 8.85%. Here we investigate some model Fermi-Bose systems with p-pairing of Fermi particles. Both the 3D-model of the 3He-4He solution and the 2D-model of 3He-4He films are considered. We study two possibilities, namely the A and B superfluid phases of the Fermi component which arise in the 3D-system, and in the 2D-system we consider the two superfluid phases a and b of the Fermi component. Such phases in 3He-films were predicted by the authors earlier [11]. We use the functional integral formalism. For a clean He 3 system see ref. [12]. The starting point is the action functional S=S~0 +S x + S ~ x . 154

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Volume 101A, number 3

PHYSICS LETTERS

19 March 1984

For the 3D model we have

s~ = fd4x[~a,Cs

- (2mB)-Iv~v~k +XB~]-

~-

fd3x d3y dr UB(X -y)~k(x)~(y)~(y)~k(x),

Sx = fd4x ~$ [X~aiXs-(2mF)-IVxsV× s + XFXsXs] - ~

f d3x d3y dr ~

UF(X -y)~(x)xs,(Y)Xs,(Y)X.s(X),

$,$

sq,x

- f d3x d3y dr UBF(X -y)~(x)~O(x)~ Xs(Y)Xs(Y) "

=

$

Here S~0 , S x describe Bose particles and Fermi particles respectively, S~0x gives the Bose-Fermi interaction x = (x, r). Using the methods developed in refs. [11,12], we can integrate over Fermi fields and come to the effective (hydrodynamieal) action functional S h. Such a functional for the 3D model of the 3 H e - 4 H e solution has the form

S h =~eB(p)b+(p)b(P) - (tB/2~V) P

+g-1 ~

p,i,a

C+ i a ( P ) C i a ( P ) + i1I n

~

Pl +P2 =P3 +P4

det M[cia,

b+(Pl)b+(P2)b(P3)b(P4)

c + , b, b+]/M[O,O,O,

O] .

(2)

Here b ( p ) is the Fourier transform of the Bose field ~b(x, r), Cia(p ) is the Fourier-transform of the tensorial Bose field cia(x , z) describing collective excitations of the Fermi component, t a is the scattering amplitude of Bose particles,g is a negative constant proportional to the pair scattering amplitude of Fermi quasiparticles, c a ( p ) = i6o - ( 2 m B ) - i k 2 + XB , M is the following operator

M=(Z-I(i6O - ~)Sp, p2 - tBF(/3V)-I ~,pb + (p)b(p + P l - P2), (flv)-l/2(nli - n2i)°aCia(Pl +P2) \ -(~v)-l/2(nli - nEi)°aC~(Pl +P2 ) ' Z - l ( - i 6 o + ~)SplP2 + tBF(/3V)-I ~p b+(p)b(P +P2 - P l ) ]'

(3)

where ~ = CF(k - kF) , Oa(a = 1, 2, 3) are the Pauli matrices, Z is a renormalization constant, V is the volume of the system, p = (k, 6o), 6o = (2n + 1)lrT,/3 = T -1 , tBF is the Fermi-Bose scattering amplitude, c F is a Fermi velocity, k F is a Fermi momentum. Suppose that both Bose and Fermi components of the system are in the superfluid state. This means that there exist a condensate of the tensorial field cia. Let us perform the shift transformations b(p) -~ b(p) + b(O)(p) = b(p) + ot([3v)l/28pO , Cia(P ) "-+Cia(P) + c~O~p), where b(0)(p), c~O)(p) are the condensate wave functions o f b and Cia fields. If we know the quadratic form o f S h (after the shift), we can f'md the Bose spectrum from the equation det Q = 0, where Q is the matrix of the quadratic form. For the case of the B-phase of the Fermi component the quadratic part of S h is [i6o - ( 2 m B ) - I k 2 - (~2t B - C ( p ) ] b+(p)b (p) - ½ ~_1 [a2t B + C ( p ) ] [b (p)b (-p) + b+(p)b+(-p)] p

P _

1

P

+

+

{Ai/(P)C+ (P)Cja(P)+ ~Bi/ab(P)[Cia(P)C/b(--p) + Cia(P)Cjb(--p)]}

-- ~ [b ( - p ) + b+(P)l[cia(P) P

- c+(-p)]Dia(p),

(4)

where 155

Volume 101A, number 3

PHYSICS LETTERS

A i / ( P ) = -Si]g-1 - 4Z2(3V) -1

Bi/ab(P) = 4Z2A2(/JV) -1

C(P)=Z2a2t2F(f3V)-I

19 March 1984

~ MllM221nlinl](i~°l + ~1)(ic°2 + ~2), Pl +P2 =P

~ M-{1M21nlinl](2nla nlb - 6ab), " P 1+P2 =P

~

Ml-11M2 -1 [06°1 + ~1)(--i6°2 +~2) _ A 2 ] ,

Pl +P2 =P

Dia(P) = - D i a ( - P ) = 4Z2~tBFA({3V) -1

~ M-flM21nlinla(i6°l + ~1)" (5) Pl +P2 =P Here M i = co2 + ~2 + A2(i = 1,2) (A is a gap in the Fermi spectrum), ~2 = P0 is the density of the Bose condensate of the field b ( p ) . The last term in (4) corresponds to the interaction between b ( p ) and Cia(P ) fields. We have Dia(p ) ~ ~.~Sia for small p. This means that the only mode which interacts with b and b + is cii(P ) - c+(-p). In 3 He B this mode is the sound mode. We see that the interaction between Bose and Fermi components implies the interaction between sound modes of both components. In a first approximation there is no interaction between the Bose component and the other 17 collective modes of the Fermi component. The equation for sound velocities has the form u 4 - u2(~c 2 + ot2tBm~1) + c 2 ~ 2 ( 3 m B ) - l ( t B - Z 2 t 2 F k 2 n - 2 c ~ l )

= O.

(6)

It shows that the Bose-Fermi interaction does not change Ul2 + u2,2 but diminishes u21u2 as compared to their values for noninteraeting systems (tBF = 0). We have considered also the case of the A-phase of the Fermi component. All conclusions including eq. (6) remain unchanged for this case. The only collective mode of the Fermi component interacting with the sound mode of the Bose component corresponds to the variable Cll(P) - ic21 (p) - [C~l( - p ) + ic~l ( - p ) ] . This is the sound mode in 3 He A. We have considered also 2D-models of the films 3 He a - 4 He and 3 He b - 4 He. In ref. [ 11 ] we called the superfield phases with the order parameter proportional to respectively

(100) (100) and

i 00

(7)

010

the a and b phases of the 3He film. For both a and b phases of the Fermi component of the 3 H e - 4 H e film one has to replace eq. (6) by

u 4 - U2(~C2F + ot2tBm~1) + c2o~2(2mB) -1 [t B -- Z 2t2FkF(TrcF)-I ] = O. All other conclusions concerning the Bose spectrum turn out to be valid for both 3D and 2D models. Namely, only the sound mode of the Fermi component interacts with the sound mode of the Bose component. The full number of collective modes in the 2D model is equal to 13 = 12 + 1 instead of 19 = 18 + 1 in the 3D case.

References [1 ] I.M. Khalatnikov, Theory of superfluidity (Nauka, Moscow, 1971). [2] J. Bardeen, G. Baym and D. Pines, Phys. Roy. Lett. 17 (1967) 372. [3] G. Baym and Pethick, The physics of liquid and solid helium (New York, 1978) p. 123. 156

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PHYSICS LETTERS

19 March 1984

J. Landau et al., Phys. Rev. A2 (1970) 2472. M.B. Hoffberg, Phys. Rev. A5 (1972) 1963. G.E. Volovik, V.P. Mineyev and I.M. Khalatnikov, Zh. Eksp. Teor. Fiz. 69 (1975) 675. V.A. Andrianov and V.N. Popov, Vestn. Leningr. Univ. 22 (1976) 11. E.P. Bashkin, Zh. Eksp. Teor. Fiz. 78 (1980) 360. L.F. Andrianov, Zh. Eksp. Teor. Fiz. 50 (1966) 1415. K.N. Zinovieva and S.T. Boldurev, Zh. Eksp. Teor. Fiz. 56 (1969) 1089. P.N. Brusov and V.N. Popov, Phys. Lett. 87A (1982) 472. V. Alonso and V.N. Popov, Zh. Eksp. Teor. Fiz. 73 (1977) 1445. V.N. Popov and P.N. Brusov, Preprint LOMI P-4-82, Leningrad (1982).

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