Quasiparticle interaction in liquid 3He

ANNALS

OF PHYSICS

78, 1-38 (1973)

Quasiparticle

Interaction

in Liquid

3He

S. BABU* State University of New York, Stony Brook, N. Y. AND

G. E. BROWN Nordita, Copenhagen, and State University of New York, Stony Brook, N. Y.+ Received April 3, 1972

The quasiparticle interaction in liquid 3He is studied. It is shown that part of the interaction can be considered induced through the exchange of density and spin-density excitations. The coupling of these excitations to quasiparticles is shown to be given rigorously by the quasiparticle interaction itself in the long wavelength limit. The induced part of the interaction is then calculated by determining the density-density and spin-density-spin-density response functions assuming that the irreducible particle-hole vertex functions are frequency independent. The remainder of the interaction is shown to be given by a vertex function which does not have certain singularities and it is argued that it can be approximated by the Brueckner G-matrix. From the net interaction Landau parameters are extracted and compared with the experimentally known ones. Since the agreement was not satisfactory, the consequences of making a theory in terms of “paramagnons” is explored. This leads to much better numerical results, but the theory is less satisfactory. Consideration of the density-density response function arrived at here shows it to be similar to one proposed by Pines and Nozieres [3,24], who assumed a local relation between a density fluctuation and the potential it produces. We come to the conclusion that the numerical discrepancies between our calculated Landau parameters and the experimental ones probably arise from neglect of the exchange of two and more spin fluctuations. In the paper following this one, 0. Sjoberg makes application of our theory to calculate the Landau parameters of nuclear matter. No “soft” excitations such as the spin fluctuations of liquid SHe exist in nuclear matter, and the theory appears to be quantitatively much better there, although some of the crucial Landau parameters are not directly measurable, and a direct check is not possible.

* Present address: Indian Institute of Technology, Kanpur, India. + Research supported by U. S. Atomic Energy Commission Contract No. AT(30-1)-4032. 1 Copyright All rights

Q 1973 by Academic Press, Inc. of reproduction in any form reserved.

BABU AND BROWN I. I~R~DuCTI~N

Any interpretation of the properties of liquid 3He generally starts from the ideas and concepts developed by Landau in the late fifties. He used the notion of a quasiparticle, which was later made precise in terms of formal many-body perturbation theory, to construct a theory of normal Fermi liquids valid for long wavelength excitations. Description of the dynamics of these quasiparticles in terms of a Boltzmann type transport equation and the need to conserve quasiparticle momentum led Landau to a natural definition for the energies Ek of his quasiparticles. Let nk” denote the ground state distribution of the particles and nk the distribution of the quasiparticles. Then the excitation of the system is described by &, = nk - nk”.

(1.1)

Furthermore, if E is the energy of the system, then it is a complicated functional E[n,] of the quasiparticle distribution. The definition Landau was led to is Ek = 6E[n,]/6n,

.

(1.2)

Thus Ek is the excitation energy of the system when only one quasiparticle is present. The important point to note now is that when more than one quasiparticle is present, Ek becomes a functional of the quasiparticle distribution again. This dependence is central to Landau’s theory. Thus, if Ekeis the quasiparticle energy at equilibrium, we are led to write

(1.3) when there are other quasiparticles, indicated by the 6~ , differing from zero, present. The last term in Eq. (1.3) stands for the interaction energy of the quasiparticles and&< is called the effective interaction between quasiparticles, or simply Landau’s*function. Equation (1.3) can be thought of as defining the f-function, or equivalently we can redefine

fkkt = 62E/&?k6nkf.

(1.4)

It is assumed that fkk9 is continuous as k or k’ crosses the Fermi SUrfaCe. This again limits us to normal Fermi systems. Usually, in view of the damping of the quasiparticles, only values off when k and k’ are on the Fermi surface where = p, the chemical potential, are needed. Ek O = & So far, spin indices have been suppressed. Introducing them explicitly, in the absence of a magnetic field, time reversal invariance implies

f kr.k’a’

=f-k-a,-k'-a'

*

(1.5)

QUASIPARTICLE

3

INTERACTION

Further, if the Fermi surface is invariant under reflection k ---f -k, becomes f ka,k’.’ = fk--o.k’--a’ . Thus only two independent components, corresponding parallel spins, remain. It is customary to write these as

Eq. (1.5) (1.6)

to parallel and anti-

parts of the where f ik, and f Ekeare the spin-symmetric and spin-antisymmetric quasiparticle interaction. We are concerned only with isotropic systems. Then for k and k’ on the Fermi surface, fik, and fzk, depend only on the angle 5 between k and k’. Expansion in terms of Legendre polynomials gives

f kks(q)= L$ f ;(“‘P,(cos 5). It is then possible to define a dimensionless interaction in terms of F:(a)

=

m*pF --ffl a3fi3

da)

(1.9)

where m* is the effective mass of the quasiparticles at the Fermi surface, pF is the Fermi momentum, and m*p&i+i3 is the density of states per unit volume at the Fermi surface. The Fz are usually referred to as Landau parameters in the literature. Landau went ahead and constructed a theory of normal liquids leading to the prediction about the existence of zero sound in liquid 3He, which has been subsequently experimentally verified. He further related thef-function to a certain limit of the two-particle-two-hole vertex function. All this and the justification of the assumptions made by Landau starting from many-body perturbation theory are both elegant and powerful and can be found in the book by Nozieres [2] and the references quoted there. We are interested here in calculating the Landau parameters; in particular, how to obtain numerical values for them starting from a microscopic theory. In practice, at least four Landau parameters F,“, Fls, F,,a, and F,” can be determined experimentally. This is possible since within the framework of Landau theory the isothermal compressibility, specific heat, magnetic spin susceptibility and spin echoes of 3He can be expressed in terms of these four parameters [3,4]. From this information, one then predicts transport properties, the velocity of zero sound, etc., assuming

4

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AND

BROWN

that F,S@ = 0 for 1 > 2. Thus Landau theory provides a closed, self-consistent framework within which some of the available experimental data on liquid 3He can be interpreted. It is clear, therefore, that a detailed understanding of Landau parameters, i.e., the effective interaction is irrelevant to Landau theory and it is entirely adequate to think of them as phenomenological parameters of the theory which have to be determined from experimental data. But the unsatisfying feature of such a viewpoint is that h priori it is unknown how many such parameters are needed. In particular it would be desirable to demonstrate the accuracy of the customarily made assumption that Ft = 0 for I 3 2. The fact that F," = 10.8 and Fls = 6.3 casts some doubt on the validity of such an assumption. The question we address ourselves here is how the effective interaction can be obtained from a knowledge of the potential-say a hard core of some radius r plus a Van der Waals attraction or a Lennard-Jones type potential between two bare SHe atoms ? In other words how can the presence of other particles be taken into account? We partially answer this question in the following by deriving an integral equation (Eq. (4.2)) for the effective interaction in the long wavelength limit. This is then used as a model for short wavelengths also and by assuming that the vertex functions behave smoothly at short wavelengths, the experimentally known values for the spin-symmetric Landau parameters are shown to be consistent with the model. We make a theory in terms of paramagnons for the spin-antisymmetric Landau parameters. Our model, as will be pointed out later, also obeys the forward scattering sum rule at all wavelengths and the similarity between the present approach and that employed by Pines in predicting the existence of zero sound at short wavelengths and high temperatures will also be indicated. Before we proceed further it is useful to tabulate the known values of the parameters (Table I) at two different pressures [23]. TABLE

I

P

atm 0.28 27

10.8 75.6

6.3 14.4

-0.67 -0.72

-0.7

The paper is organized as follows. In the next section a brief discussion of the earlier attempts to calculate the Landau parameters is presented. In Section III, an alternate approach to the question of quasiparticle effective interaction is presented where it is argued that part of the effective interaction may be considered induced

QUASIPARTICLE

INTERACTION

5

through the exchange of density and spin-density virtual excitations. In Section IV, using the results derived in Appendix A, the explicit form of the induced interaction, exact in the limit of the momentum of excitations tending to zero, is exhibited (Eqs. (4.21) and (4.23)). Then we propose, as a model, that the induced part of the interaction has the same form for larger momentum also. The evaluation of the response functions, necessary to determine the induced interaction is carried out in Section V. The remainder of the interaction is studied in Section VI. The results and some remarks are included in Section VII where the consequences, for the effective interaction, of exchanging spin fluctuations constructed out of bare particles is explored. It is shown that such a treatment gives much better numerical results, although the treatment is not conceptually satisfying. Some final comments are included in Section VIII.

II. BRUECKNER-BETHE

THEORY

Several attempts have been made to calculate the Landau parameters [5, 6, 71. It is clear that any microscopic attempt, to be successful, must handle the highly singular repulsive part of the interatomic potential at short distances. This has been done in Brueckner-Bethe theory-primarily designed for nuclear matter, but applied also to 3He by Brueckner and Gammel[8], Ostgaard [7], and Bertsch [6]and in the method of correlated basis functions by Feenberg and his collaborators [9] who applied it to SHe and to 3He-4He mixtures. The Brueckner-Bethe theory as applied to 3He shall be presently discussed as this really provides the motivation for our approach and also a starting point. We shall, at the end of this section, briefly comment on the work of Feenberg and collaborators also. Brown [lo] discussed the connection between Landau and Brueckner-Bethe theories already and the application of Brueckner-Bethe theory in the calculation of bulk and transport properties has been extensively discussed in a series of papers by 0stgaard [7]. Consequently, the following discussion will be brief. One starts by defining a G-matrix as G=Y-V+G,

(2.1)

where Y is the interatomic potential, hard core and all, Q is the projection operator which excludes all occupied states from being considered as intermediate states and e is an energy denominator and is the difference between the particle and hole energies in the intermediate state. The usefulness of such an object stems from the fact that it is the result of summing repeated scatterings of two particles outside the Fermi sea, the effect of the other particles being taken care of by the Q-operator,

6

BABU

AND

BROWN

and therefore is finite. Then the linked Rayleigh-Schrodinger perturbation expansion for the energy in terms of the large matrix elements of the potential is rearranged in terms of G-matrix elements. The hope for convergence of such a rearranged expansion is then generally proportional to what has been called the size of the correlation hole or the wound in the wave function as explained by Brandow [Ill. Unfortunately this parameter, while of the order of 0.1-0.2 for nuclear matter, is much larger for liquid 3He and is of the order 0.5-0.6 and therefore the convergence of the G-matrix expansion for the ground-state energy is much more in doubt in case of liquid 3He. This, in fact, is demonstrated by the calculations of 0stgaard on the bulk properties of liquid 3He. He chose, by putting the intermediate state potential equal to zero, i.e., the effective mass equal to one, in the evaluation of the G-matrix elements, to calculate the two-body and threebody energies separately and then add them. This leads to a gap in the energy spectrum at the Fermi surface. He found, using the Frost-Musulin potential [12] the contribution of the two-body clusters to the energy to be -3.05”K per particle and that of the three-body clusters to be --1.95”K, a value comparable to the former. He proceeded then to calculate the Landau parameters by functionally differentiating the ground state energy twice with respect to the occupation numbers. Such a procedure requires the evaluation of the dependence of the derivatives of the G-matrix elements on the relative momentum which, however, is complicated to calculate or estimate. But as will be pointed out later, such terms, generally referred to as the rearrangement terms since they represent how the manybody medium adjusts itself when an extra particle is added or removed, contribute dominantly to the effective interaction and consequently it is essential that they be treated accurately. The inability of the theory starting from Brueckner-Bethe type formalism to do this, we believe, is a serious handicap. Further, long wavelength virtual excitations near the Fermi surface, though their contributions to the total energy is small, significantly contribute to the effective interaction. But the gap in the energy spectrum at the Fermi surface, introduced for the sake of calculational convenience, as pointed out already by both Bertsch and Ostgaard, artificially damps such excitations. In view of this, it is safe to conclude that while the Brueckner-Bethe formalism may be capable of determining the bulk properties like isothermal compressibility, spin susceptibility, etc. it is ill-suited, at least at present, for a calculation of the Landau parameters, as has already been done by Bertsch and 0stgaard. In the following we show how low momentum excitations can be treated exactly. The task of extending such a treatment for momenta up to 2pF is nontrivial and we must use a model to extrapolate from low momenta to higher momenta also. This is discussed in Section IV. Now we briefly comment on the work of Feenberg and collaborators [5,9]. Since it has been discussed by Feenberg [9] in his book “Theory of Quantum Fluids,” the discussion below is short. Liquid 3He is related to a boson system of

QUASIPARTICLE

INTERACTION

7

particles with the same mass and from the properties of the boson system a convergent approximation for the fermion system is developed. In this way a theory of the ground state of 3He is constructed. An orthonormal basis set from “correlated basis functions” is constructed and used for a limited diagonalization of the Hamiltonian in terms of quasiparticles. Hing-Tat Tan and Feenberg [5] use this formalism to determine the quasiparticle interaction, transport properties, etc. of liquid 3He. It appears to us that this formalism is well suited to treat the ground state properties and not for excited states. Further, from the point of view of someone raised in conventional diagramatic many-body perturbation theory the question of what kind of diagrams are included is hard to answer.

III.

EXCITATION

OF THE MEDIUM

It seemed to us worthwhile to develop an alternate approach to the problem of effective interaction in liquid 3He. The G-matrix will form a good building block to start from, since any consideration of the effective interaction must include the short-range correlations between the atoms due to the strong repulsion at short distances. But while F,,s, Fls are large and repulsive, the spin-averaged G-matrix at the Fermi surface is relatively small, attractive and, for instance, gives a negative binding energy. The many-body medium therefore somehow transforms this small attractive G-matrix into a large repulsivef-function. This change is so accomplished that FiT and Fi” are almost equal, F,,a being negative and very small. In addition, when the pressure is increased, both Fi’ and FiL increase by an almost identical amount leaving Foa almost unaffected. An effort to understand how the many-body medium acts can be made by looking upon Fklkz as the change in the energy of a quasiparticle with momentum k, , when one with momentum k, is added. The additional particle disturbs the medium which gets excited, and these disturbances travel as excitations of the system and interact with the other quasiparticles in it. In other words, an induced interaction exists between the quasiparticles, just as in the case of dilute mixtures of 3He-4He where 4He phonons-real and virtual, depending on the temperatureare the excitations [13]. Since only quasiparticles on the Fermi surface are under consideration, only virtual excitations are relevant and they can carry momentum up to twice the Fermi momentum. The corresponding wavelength is smaller than the average interparticle spacing in liquid 3He. In that sense these are not long wavelength excitations. These excitations can have total spin equal to one or zero. The latter correspond to density fluctuations, almost exhausted by zero sound for long wavelengths, and contribute only to FTT while the former depending on the z-component of the spin contribute to either FTT or FTL. Let us digress here briefly to discuss the sign of the interaction resulting from

8

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AND

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the exchange of these virtual excitations. Generally the exchange of virtual quanta between particles at frequencies low compared to those characteristic of the particles themselves result in an attractive interaction between the particles as is the case with the exchange of 4He phonons between 3He atoms in 3He-4He mixtures and with phonon exchange between electrons at the Fermi surface in metals. But here the resulting interaction will be repulsive sincefis a particle-hole interaction. The latter point can be seen by the following argument [IO], where a problem involving a weak interaction V, which need be handled only to first order, is considered. The particle-hole interaction connecting particle-hole states of total momentum q can be thought of as

@= c (k+ q,k I VIk’,k’ + cr)(d* -w+J(a+*at+&

(3.1)

k’k

that is, as the matrix element of CDbetween the particle-hole states ak+‘+& I 0) and &+,,ak 10), where 10) is the vacuum. Such matrix elements enter into the description of collective excitations that are coherent linear combinations of particle-hole states. The momentum of the excitation is q so that in the long-wavelength limit, q + 0, the Operator pair ak+&+, becomes nk = &+& and similarly @!&‘+4 becomes nk’ . Hence the expectation values of CDbecome

Referring back the definition offas a functional derivative one can see the relation between f and the particle-hole interaction. Returning now to the view of the interaction proposed above, the feasibility of utilizing it clearly depends on being able to calculate the coupling between the excitations and the quasiparticles themselves and the appropriate propagators for the excitations. Let us note that since only quasiparticles at the Fermi surface create and absorb these excitations in our considerations, only the zero-energy propagators of these excitations are needed. Since the time-ordered propagators and their retarded counterparts differ only in their imaginary parts, which vanish for zero energy, the response functions of the medium describe the propagation of these excitations and have to be calculated. This is done in detail in the following. Also it turns out that the coupling between the quasiparticles and the excitations of the medium can be calculated exactly in the long wavelength limit within the framework of Landau theory and this is done in Appendix A. Finally, one should add to this induced interaction a term corresponding to what we shall call the direct interaction. In the 3He-4He mixtures also a direct interaction had to be added and this almost cancelled the induced interaction. But the direct interaction here is of a different origin since we are considering a homo-

QUASIPARTICLE

9

INTERACTION

geneous medium. We will take the G-matrix to represent such a direct interaction in our case, arguing that the effects of the hard core will dominate any such term, and that it is not included in the induced interaction. Our numerical results will show that it is necessary to include more than just the G-matrix in the direct interaction, and we shall argue that one should include, also, exchange of two and more spin fluctuations.

IV.

INDUCED

INTERACTION

Let I’+, , k, ; 1) be the fully reducible particle-hole vertex and k, , k, and 1 be four vectors. We shall use the notation of Nozieres [2] as far as possible. As l---f 0, as pointed out by Landau [l], r does not have a unique limit and depends on the ratio ( I l/Z, . Then Landau showed that (4.1)

where Zk is the residue of the single particle propagator at the quasiparticle pole. We refer the reader to appendix A, Eq. (A6), where it was shown that (4.1) can be rewritten, for k, w k, , as

Here R”(P,

4

= -a&

-po)

%p -

4.

(4.3)

In order to relate this to our feeling of how the many-body system responds to an additional particle and to generalize Eq. (4.2) for short wavelengths, an external potential SU,(q, w), which acts only on quasiparticles of momentum p, is assumed to be present. The dynamical evolution of the quasiparticle distribution is characterized in Landau theory by a transport equation which has the form [3], in the presence of the potential SU,(q, w), of Eq. (4.5) (w - Q . VP) hhl> w) + q . VDE

C&’ p P’

6n,,(q, w) + SU,(q, w) = r,[&l,f]

(4.5)

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AND

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where V, is the velocity of a quasiparticle of momentum p and ID is a collision term, a functional of the quasiparticle distribution, and in our case is equal to zero since we are dealing with a system at absolute zero temperature. A few comments on the nature of this equation are probably relevant. First using a procedure well-known in kinetic theory a Boltzmann equation is written for the distribution function of quasiparticles assuming the latter to be independent described by a classical Hamiltonian Q , Eq. (1.3). Such an equation will refer to the total distribution function for all quasiparticles. As such it is not very useful since nk is defined only in the vicinity of the Fermi surface and further, only excited quasiparticles in the immediate neighborhood of the Fermi surface can be really considered independent. Hence a more useful transport equation describing the flow of only excited quasiparticles must be extracted from it. This can be done by rewriting nk using &k defined in Eq. (1.1) and retaining only terms linear in &k . Fourier transformation of this linearized transport equation results in Eq. (4.5). A further point about the external potential should also be made. It is that one knows from first principles only the force exerted on a bare particle and to deduce that exerted on a quasiparticle a detailed knowledge of the structure of the self-energy cloud, i.e., the various constituents of the quasiparticle, is needed. But, fortunately, we are interested in the response of the system to a scalar field, which produces a force proportional to the system density. Since one quasiparticle “contains” one bare particle, their numbers are equal and the force felt by the quasiparticle is equal to that felt by a bare particle. This makes it possible to write the transport equation in the presence of such an external potential. The resulting equation, in Fourier space, since we assume our SU,(q, w) to be such an external potential, is the one written above. Now Eq. (4.5) can be rewritten, suppressing the arguments, as

an, Q*VD ( SUD, )cl.0+ CJJ- q * vp ccat-,

[IUD” p”

(g)),

w + s,,,]

0

(4.6)

Let

ano _ -a(P q) ’ .

aB,-

(4.7)

Then iteration of equation (4.6) yields =

dp,

q)

sDD’

+

a(P,

q)hD’a(P’,

q)

(4.8)

QUASIPARTICLE

INTERACTION

11

Hence,

Now, if the sums over the frequencies, which can be done trivially because of the S-functions in R*, are carried out in Eq. (4.2), one can immediately write that

(4.10) valid for k, = kz . It should be remarked here that q is not arbitrary but is fixed, given k, and k, , being equal to (k, - k,) and although our Eq. (4.10) somewhat resembles Eq. (IV.19) of C.J. Pethick [14], it has a different structure. This result is important. It can be interpreted as follows. Let us ignore the first term on the right hand side for a moment. The second term has the form that characterizes an induced interaction-a coupling constant, a propagator, and then a further coupling constant. The important fact is that in the long wave length limit, i.e., k, w k, , where the above formula applies, the coupling constant is the Landau?function itself. This is because we are making a theory in terms of quasiparticles and not bare particles and this, for example, is in sharp contrast with paramagnon theory [15] where the coupling constant is a parameter which should be fixed in principle by the static susceptibility. The latter is a point which is ambiguous [16] and needs to be explored. It is convenient to represent the induced interaction diagramatically as in Fig. 1.

FIG.

1. Shaded part stands for (&z~~,,!~U&~

12

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Then it is evident why the coupling constant is the f-function &, in the limit q -+ 0. The intermediate excitations are represented by the response function:

As stated earlier, since only zero frequency propagators enter, there is no difference between time-ordered propagators and the response function here. Since I k, - k, I can be as large as 2p, where pF is the Fermi momentum, this picture of the induced interaction needs to be generalized. The vertex will then be given by a generalizedf-function depending on all the four momenta k, , k, , p1 , and p1 + q whose description lies beyond the scope of Landau theory. Further the notion of quasiparticle in this region is extremely fuzzy and characterization of the intermediate states in terms of the change in Sn, , a quasiparticle distribution, is questionable. In addition to this, there is another problem. This arises from the fact that since even at long wavelengths the coupling constants are f-functions themselves, Eq. (4.10) is really an integral equation for the f-function. As such, its solution depends upon the knowledge of the first term on the right hand side. This is considered in Section VI. It turns out that its detailed structure is difficult to obtain. Consequently, in view of this and the constraints imposed by the earlier mentioned problem, the following attitude, perhaps a little pragmatic, will be adopted. Since Landau parameters for 1 < 1 are well known experimentally, these same values will be assumed as coupling constants. Since the response function for small wavelengths will also be determined in terms of Landau parameters, these become our input and in a sense replace the interatomic potential from which any calculation should start. Thus, at least for the induced part of the effective interaction, no direct reference to the interatomic potential need be made. Then the difference between the resulting Landau parameters and the input values can be taken as an indication of the accuracy of the techniques and picture employed here. Beginning from this philosophy, the problem mentioned earlier can be attacked with more ease. We propose that the contribution of long wavelength excitations to the effective interaction is dominant since the response function tends to cut off the higher momentum contributions. This can be seen in a simple calculation, for example, where only one F. is nonzero. Then the static response function is given

by

-m%

x(qPkF) %-2 1 + &x(d=d

’

(4.11)

where x(x) = 1 - -$ In -x+1 X-l

.

(4.12)

QUASIPARTICLE

INTERACTION

13

Then it is clear that the cut-off is steeper for the case F,,a f 0 than that for I;o” # 0. The former corresponds to propagation of spin density excitations while the latter to density excitations. Such a cut-off will continue to occur when Fl , for 1 < 2, are different from zero, assuming no harmful behavior at the vertices. Consequently, it will be assumed that the vertex function is still given by the Landau f-function itself even for nonzero q. More generally, Eq. (4. IO) itself in the form written down will be used for j q 1 up to 2pF . Let us note immediately that in such a form the induced part of the interaction is trivially obtained if the response functions obtained from Landau theory are used. Since these response functions so determined would be accurate only for q + 0, we shall calculate them in the next section in a somewhat better fashion, though still approximately. For the present, we simply note that a symmetric treatment1 of the vertex suggests that the coupling for large q. f klpl , valid for small q, be replaced by f(kl+k2)/2,P1 Now the induced part of the interactlon shall be rewritten in a more convenient form. So far the spin indices have been ignored. It now becomes necessary to restore them. To prevent confusion let us confine attention to the unsymmetrized coupling constants. Symmetrization will be carried out later, trivially. Denoting the induced part of the interaction by

we have

ufk",pl)

($f!-).

o (f&k2

+

P ' %f&k,).

c4*13)

For convenience the two cases oIz = &toZz will be considered separately. First, for ulz = gsz = 7,

1 We are grateful to C. J. Pethick for pointing out the advantage of doing this.

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It is expedient now to assume that the external field depends only on the Z-component of the spin and that SU,,, = SU,, = 6 U, in the first term while 6 U,, = -AU,, = SU,’ in the second. Such a choice is possible since the two terms are independent and allows the sum over p to be evaluated trivially resulting in

Thus a decoupling of the density and spin density excitation is accomplished. Further understanding of the induced interaction can be achieved by writing f in terms of Landau parameters, as follows. As is customary, only the Landau parameters Fi*“, I < 2 shall be assumed to be different from zero. Then, letting k stand for the unit vector in the direction of k,

+ 2 c UY4 k -i-w3 ( snDITs;‘snplL P, ), o(foa+ k2*fijcp”) PlP2 (4.16) We now approximate Then

k, * fil by fr, * pl/pF and kz * & by f, * p2/pr .

(4.17)

QUASIPARTICLE

15

INTERACTION

Here m is the bare particle mass, J, and J,’ are the currents defined as

and (4.18) J, is the usual current density that is related to the density by the continuity equations; J,’ can be analogously related to the spin-density C,, (6nDT - an,,&) through the continuity equation

The response functions appearing in Eq. (4.17) can easily be identified.

is the density-density

response function while

is the current-density

response function,

is the density-current

response function and

Thus

is the current-current response tensor. Thus the cross terms in the Landau parameters couple the density to the current or the velocity field while the direct terms yield the coupling between density and density and between current and current. Since the current can be expressed in terms of 6nl(q, w), the 1 = 1 component in 59517811-2

16

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the expansion of Sn(q, w), it is possible to guess, qualitatively, the effect of including higher order Landau parameters in the f-function. Thus inclusion of f2 will add the result of coupling the I = 2 component of the distortion of the Fermi surface to I = 0, 1, and 2 components, the strength of the coupling being given byfofi , Fiji , andfZ2 respectively. Assuming that thefi form a decreasing sequence in value with Z, then the coupling between density and the higher harmonics, unless it vanishes, is the most significant for each additional fi included. If some of them should be zero, then the first nonvanishing one is dominant. Thus, e.g., in the case of Fl* in the long wavelength limit the static density-current and currentdensity response functions, which are equal, are zero, as shown for example by Nozibres [2]. This can also be seen more simply from the continuity equation which can be used even at small wavelengths. Thus the current-current coupling is important and is calculated later on. No such exact result is known for the higher components but the contribution from their coupling may be expected to be small. The current-current coupling obtained above is similar to that derived by Bardeen, Baym and Pines in the case of SHe-4He mixtures [13]. This can be seen by looking at one of the vertices where the coupling can be written as fld-$k~J=f,S~k.v

fi”

=-...--

3

mPF ,,akmv

(4.20) where m* is the effective mass and am* = m* - m. So far only the terms in which fis enter were considered but the case of the remaining terms is similar. The cross terms will again vanish and only the direct terms involving foa and flu remain. Symmetrization of the vertex between the incoming and outgoing quasiparticle momenta can be made now by simply replacing k1 and k, by their mean (kl + k,)/2. Since both k, and k, are on the Fermi surface this vector is perpendicular to their difference k, - k, , which is also equal to q, the momentum transfer in the crossed channel. The longitudinal part of the currents J,, and J,’ is along the direction of q. Consequently looking at Eq. (4.17) it is clear that the longitudinal part of the current does not couple to the quasiparticles and only the transverse part of the current-current response tensor contributes to the effective interaction. Let us point out that such is also the case, before symmetrization, in the limit q + 0.

QUASIPARTICLE

The induced part of the interaction -fgy

17

INTERACTION

for ulz = uzz = T, finally takes the form

= foSXPP(%O)f,” + (1 - q2/4PF2) f$

xi&,

0) f$ (1 - q2/4pF2).

(4.21)

Here [l - (q2/4pF2] is the square of the length of the vector (k, + k2)/2p, ; xPDis the density-density response function; xg is the transverse component, along any direction perpendicular to q, of the current-current response tensor; and x00 and xFJP refer to analogous quantities for spin density and “spin current” J’ respectively. Now the case of vT1== -(TV= shall be considered. Such a situation is depicted in Fig. 2 where only the spins of the particles are indicated.

FIG.

2.

The intermediate states in the crossed channel have a total spin Z-component, m, , equal to fl. These states cannot be simply characterized in terms of a an,,, , where 01 is either up or down. The coupling can be determined by observing that the vertex, in Nozieres notation, is lim lo~o,lll+o l~(k , P; 0 in the long wavelength limit, q -+ 0. (Consult the appendix.) The superscript on r refers to the total Z component of spin. As was shown by Nozieres, lT(k, ) p; I) = or(kp, Immediately,

f r:* -fL$.

pa; E) - or&v,

p - a; I).

(4.22)

one deduces that the vertex is given, for small q, by 2f&

=

Further, since the liquid is isotropic the response of the m, = 0, spin antisymmetric mode is equal to the response of the m, = 1 mode. Consequently, we shall write the induced interaction for the case glz = -ozz as

18

BABU

AND

BROWN

The additional factors of two come from those in the coupling, 2f O. Thus the induced interaction is completely determined once the response functions, to whose evaluation we shall turn in the next section, are known. Let us point out that the induced interaction now depends only on 1q ] = 1kl - kB 1 and hence can be expanded in terms of Legendre polynomials yielding the “induced part” of the Landau parameters. The remainder of the interaction, i.e., the first term on the right hand side of Eq. (4.10), which will be called the “direct interaction” term, for the lack of a better name, will be considered in Section VI.

V. RESPONSE FUNCTIONS In Landau theory, the response functions are determined starting from, for example, Eq. (4.6); in fact, its solution, which can be easily evaluated for o = 0, is written out earlier [see Eq. (4.8)]. We now intend to generalize this for larger q. This follows from the observation that ~u(p,q) is simply the long wavelength limit of the function

where n90 is the quasiparticle occupation number. This leads us to propose that the density-density and the spin density-spin density response functions for large q be calculated from the following equation:

Before indicating under what assumptions such an equation can be derived it should be remarked that Eq. (5.1) is the same as the result of generalized RPA [3] except that the Hartree-Fock values for the effective interaction and particle-hole energies are replaced by the f-function and the quasiparticle-hole energies. The validity of such an equation depends on the frequency dependence of the vertex functions, defined by Eq. (5.3) below. To see this, following Nozieres [2], we can write

Sn,(q, 4 j-m $9.r &; q)G(P- 4) --03 whl,w>= ZD

G (p + ;),

where G(p) is the quasiparticle

propagator,

Z, is the usual renormalization

(5.2)

at the

QUASIPARTICLE

quasiparticle pole, p0 is the fourth component vertex operator defined as &G

4) = 1 + c T(P, P’; q) G (p’ P’ L

19

INTERACTION

1 + c Z(P> I”; 4) G (P’ -

of the four-vector and (l(p; q) is the

- f) G (p’

+ ;)

(5.3) ;) G (pf -I- 4) A($;

q),

P’

where ZQ, p’; q) and Z(p, p’; q) are the usual fully reducible and irreducible vertex functions respectively. Our (1 corresponds to A4 of Nozieres. In obtaining (4.6) from (5.3) only the limiting behavior of (l(p; q) as q - 0 is of interest and the procedure employed to study it is standard in Fermi liquid theory [2]. To find A(p, q) for nonzero values of q, the above integral equation has to be solved. This is complicated by the fact that the irreducible vertex function, which is unknown, is frequency dependent. This frequency dependence is especially important for small q where its neglect would result in errors in fl(p, q) by a factor of 2, which differs considerably from unity in the case of liquid 3He. But then the exact result for the response function for small q is known and is given by Eq. (4.6) and hence it is necessary that any result, to be valid at all, for the density-density and spin-density-spin-density response functions obtained by assuming some frequency dependence, say, for the irreducible vertex function must coincide with the result of Eq. (4.6) in the limit q --f 0. Such a result is obtained in the simplest way by assuming Z(p, p’; q) to be independent of p,, , which we do. This is our main assumption, and we have no justification for it, other than it gives the correct q + 0 limit and seems to give reasonable extrapolation properties. It has been convenient when the dielectric function, which is related to the vertex operator A(p, q), of an electron gas at metallic densities is considered, to replace the irreducible vertex function by a static screened interaction [17]. The equation that results for A(p, q) is considerably simpler to handle and in fact has been solved variationally by Langreth [18], who found at least for the ground state energy a result that compares favorably with the result of the interpolation scheme developed by Pines and Nozibres [19]. In summary, since we can determine the behavior of our response functions only for q + 0, we must assume some extrapolation to nonzero q. Our assumption that Z(p, p’; q) is independent of p. allows us to make this extrapolation. We end up with something similar to the polarization potential theory of Pines and Nozieres [3,24], which they arrived at by assuming a local relation between a density fluctuation and the potential it produces. We shall return to a comparison of our theory with theirs later. Although Z(p, p’; q) is assumed independent of p,, , since Z(p,p’; q) = Z(p’,p; -q) this is equivalent to a static interaction only in the q - 0 limit. Such a behavior on the part of Z(p, p’; q) makes A(p; q) independent

20

BABU

AND

BROWN

of p,, . Since &I; 4) corresponds to the probability that a quasiparticle p is scattered, in the presence of a unit external potential, to the state p + q, the assumption made earlier about the behaviour of I(p, p’; q) corresponds to the frequency independence of such a probability. A posteriori such an assumption then seems reasonable at least for large q, which we are particularly interested in. To proceed now, Eq. (5.2) can be rewritten as h&k ~1 = 2 fwd% fJJ>

n(P; 4)

w -

nD+wz) - %-(a/z) ‘D+h/Z) + ‘P-h/Z) + ir] ’

(5.4)

Further Eq. (5.3) can be rewritten as hl~, q) = 1 + c I(P, P’; q) G2(p’) + 2+Zi*R(p’; 8’

q) A@‘; q),

(5.5)

where

W; 4) =

%‘+(,/2) w

-

Ep'+(a/2)

-

nP'-(,/2) XPO' +

W-(0/2)

+

-

PI-

i7

(5.6)

Defining (5.7) renormalization

standard to Fermi liquid theory can be carried out resulting in 4~;

q) = A’(P) + c r”(P, P’) 2riZ?R(p’, P’

4) &‘;

s),

(5.8)

or z,n(P;

4) = 1 + CAP, P’) P’

h,+(,/,)

w -

%i’+(q/z)

w-h/z))

+

&A(p’; 4) + i ’

%I’-G/2)

(5.9)

where the Ward identity Z+‘lO(p) = 1 has been used. Eliminating A(p; q) from this in favour of Sn,(q, CO) using Eq. (5.4) results in Eq. (5.1) proposed earlier and which will now be used to calculate the short wavelength density-density and spin-density-spin-density response functions. Such a calculation is straightforward and is described below for the sake of completeness. The spin-symmetric and spin-antisymmetric parts of the distribution function satisfy two independent equations, which are %I+, - n, ~4+l, snp%l, w) + w - s+a + cp + iv C.cpp pr

w) + SU,(q, 0) = 0

(5.10)

QUASIPARTICLE

21

INTERACTION

and c .f;,, w+-l,

w> + suahl, w) = 0,

(5.1 I)

P’

where

and (5.12) Thus for SUP, = [ SUP, I only one of the potentials is nonzero. From the similarity between the two equations it is clear that both density-density and spin densityspin density response functions have the same form except thatfs in the former is replaced by f a in the latter. Let us consider Eq. (5.10). Letting (5.13)

eJ>

21+1 = 2 x

/

l nP - n,+a -1 0~ - cp+q + ep + iq Nx%

[pfPr

WI +

~UP(%

41 Pl@ * $)46

* 9)

(5.14)

Let us note that al(w) depends on 1p Jonly. Letting, as usual, fis = 0 for 1 > 2 and using 6 .$’ = (fi . a)($’ * lj) + sin 19sin 8’ cos($ - 4’) (5.15) Eq. (5.14) can be rewritten 21+1 al(w) = --7y-

j

-1 nu - HP+, -I w - 5+q + ep + iv [f&4

+ 5 Ii . 4YWj (5.16)

where (5.17)

22

BABU

AND

BROWN

Writing out Eq. (5.16) for I = 0, 1 and summing over p yields x(w) = fox(w) + c ~Ql9 P

0) %(cr, w) + (h/3) Y(W) dq,

w)

and (5.18) where

Solving Eqs. (5.18) for x(q, w) and observing that

we get

%h 411 - fiS~2(%011+ fiS~12h,WI = 11-fiS~2h, QJ)lU- foS%h,o)l - .h%8%2(q,WI * (5*20) The result of Landau theory can be easily derived by evaluating oI(q, w), IyI(q, w) and a2(q, w) in the limit q -+ 0. This can be done trivially to yield 010= -a(s),

012= -(Q + s2a2(s)),

al = -m(s),

(5.21)

where s = w/qv*

and 01(s)= ($ In -s+l S-l Then it is easy to show by substitution Landau

X00

-_ -(m*pF/v”>

(+I/(1

-

9

m*PFJ7r2.

(5.22)

that + F,, + 1 +$,3j

s2) 4~)).

(5.23)

To proceed further, oI,(q, w) defined by Eq. (5.19) must be evaluated for n = 0, 1,2. This is done in the appendix B for the case of interest to us, w = 0.

QUASIPARTICLE

23

INTERACTION

The evaluation of the current-current response functions, to which we shall turn our attention now, is made easier by recognizing that only the static functions are needed. This is fortunate since no practical theory, other than machine integration-in the case of classical fluids-of Newton’s equations, exists for evaluating them. It should be pointed out that it has become possible recently to evaluate the second and fourth frequency moments of the correlation function, again only for classical fluids [20], in terms of the interatomic potential and the particle distribution functions. Application of a similar analysis to quantum fluids has not been carried out. As pointed out earlier only the transverse part of the current-current response functions enter in the induced interaction and this can be obtained most directly by simply appealing to its definition [3],

where (Jcl&, perpendicular frequency of infinitesimal

is the matrix element of an arbitrary component of the current J, to q between the ground state 10) and an excited state 1n), wno is the the excited state measured with respect to the ground state and 7 is an real positive number. So

xh

0) = - c (2

KJdLo

(5.25)

12ho>.

n

Since J,, = C&(q, 0) * fiL where j,(q, 0) is the current carried by an individual particle with momentum p and $, is a unit vector J- to q. The state 1n) coupled to the ground state / 0) through the current J,,(q, 0) can be thought of as the set of states / p) consisting of only one current-carrying particle with momentum p, the current being $I *j,(q, 0), x&L

0) = - c (2 KP I ML 0) . $I / Oil%J(P, P

4)).

(5.26)

Using Galilean invariance arguments we find 4P + q) =

(P + d2 - P2 _-- P . q

Since

2m

m

q2 ‘2%.

(5.27) (5.28)

- d+,)(P . $3’ x%l,0)= - ; c* %10(1 P . q + (q2/2)

= -: ,.g,

(P .41)” P . q + (q2/2) *

(5.29)

24

BABU

AND

BROWN

The last step follows from a rearrangement angular integrations we obtain

xh

of the indices. Carrying out the

(q’2) . lDp d.pP[P~- W/4)1 In pp +- cq,2j

0) = - & + &

(5.30)

Let us note that lim,,, x&q, 0) = -(N/m), where m is the bare mass, is a result that can be calculated starting from Landau theory. The last integral is straightforward to carry out, but not particularly illuminating. It will be evaluated numerically. Our calculation of the induced interaction depends mainly upon our model for the density-density response function. We finish this section by noting that the theory resulting from our assumption that I@, p’; q) does not depend upon p,,-which we can check only in the limit of w = 0, q -+ 0 where it gives the correct result-gives us results similar to those of Nozieres and Pines [3] and of Pines [24] in their theory of the polarization potential. The analogy is most clearly seen from ref. [24]. Here, a local relation between the density fluctuation and the potential it produces is assumed; namely,

$b31(%0) = .hJ
(5.31)

which can serve as the definition of the polarization potential r&i . If one defines xsc(q, w) as a measure of the response induced by the sum of the external field, $, and the polarization field, &-,Ol , then (P(%

WI>

=

Xsd%

4(!N%

w>

+

$4xJlol~

WI>

(5.32)

Since x(q, w), which we call xO,(q, w), can be defined as Ml,

WI> = xh, w> WY WI,

(5.33)

one easily finds

X(%w> = x&9 w)/[l -fox*cGl, w>l.

(5.34)

Note the similarity of this equation to our Eq. (5.20), provided we dropf,” (which was not included in the considerations of Pines), and identify a,,(q, w) with his x&q, w). Our choice of aO(q, w) was dictated by the need to obtain the known answer for the induced interaction in the limit w = 0, q -+ 0. Since we use the empirical& , we automatically obtain the correct compressibility and, consequently, zero sound velocity from our density-density response function. Thus, we have essentially carried out the program outlined by Pines [24], although we sidestepped the question of the compressibility by building in the empirical value, and our assumption that I(p, p’; q) is independent of p. is known to give accurate results

QUASIPARTICLE

25

INTERACTION

only in the w = 0, q -+ 0 limit; otherwise, it is just an assumption about the extrapolation. In addition to the sum rules which Pines [24] desired to build in, our approximation automatically satisfies the Landau condition that the forward particle-particle scattering amplitude Atk, for particles of like spin goes to zero for k = k’, not only for w = 0, but for all s, so long as w < Ed, q < kF . The Landau condition gives a sum rule involving thefts andf,“. If we define F = $(FS + Fa)

(5.32)

z = $(FS - F”),

this sum rule is

c1 W[l + NV + l>l+ -&Al+ Z1/(21 +

1111= 0

(5.33)

and follows directly from antisymmetry. If one calculates the F’s and Z’s in perturbation theory to given order, one will often find this sum rule violated. However, this sum rule is automatically built in by our approximations. This can be seen by noting that the phonon-induced interaction, Fig. 1, built into F, is just the exchange term corresponding to the phononinduced interaction built into the particle-particle scattering amplitude A. Since the Pauli principle is then satisfied, the sum rule Eq. (5.33) follows automatically. We will not expand upon this point further, since it will be discussed in detail in a forthcoming paper by Sjiiberg who makes application of our method to calculation of the Landau parameters in nuclear matter. Finally, we should mention that we end up with a density-density correlation function very similar in form to that of Singwi et al. [25].

VI. DIRECT INTERACTION So far our attention has been focussed on the induced interaction. Turning now to the first term on the right hand side of equation (4.10) we shall refer to it as the direct term-that part of the interaction that cannot be considered as induced. We shall write it as, using (A4)

f&, = -2niZi 1 1,-o, lim1/1*0I (kl + k , k, + f ; k, - kl) + c I (kl + f , k; k, - kl) G2(k) I (k; k, - ; ; k, - kl) + ... . k

(6.1)

26

BABU AND BROWN

As pointed out in the appendix, since I is not irreducible in the channel in which the particle and hole momenta differ by I, it is nonanalytic as I + 0. A rigorous calculation of the direct term should therefore include a study of the analytic properties of I. The complexity of such a study is compounded by the absence of any practical scheme for the calculation of I and the renormalization factors 2, . In fact, one only knows that I is the four-point vertex function irreducible in the crossed-channel as explained in the appendix A and as such one can only enumerate what diagrams should and should not be included, which is hardly of any help for either an analytic study or for a numerical evaluation except in some simple cases. Consequently we are forced to make some approximations in the present case. It is useful to point out, at this stage, the intimate relationship that exists between the contribution of the incoherent background to&‘& for [ k, - k, [ --t 0, as given by the first and higher interactions on the right hand side of Eq. (6.1),-the reason for calling these terms incoherent background contribution will become apparent in the following-and the renormalization factor Z, , at the quasiparticle pole. To do this, let us consider two diagrams that enter in the self-energy, one shown in Fig. 3a, is a skeleton diagram and the other shown in Fig. 3b, has a self-energy insertion in one of the internal lines.

j--& ).---+% 4 FIG. 3a.

FIG. 3b.

Now opening the lines labeled kz, which corresponds to functional differentiation with respect to 8nk, , and writing the diagrams as particle-hole interactions results in Figs. 4 and 5 respectively.

FIG. 4.

FIG. 5.

It is clear that while Fig. 4 belongs to the irreducible interaction Z(k, , k, ; k, -k,), Fig. 5 belongs to the 2nd term in the right hand side of Eq. (6.1). This property is more general and opening the internal lines of more complicated self energy inser-

QUASIPARTICLE

INTERACTION

27

tions results in the iterated terms on the right hand side of Eq. (6.1). But the self energy insertions really correspond to clothing the quasiparticle and contribute to the incoherent part of the spectral density of the one particle propagator. Therefore we argue that we can ignore all the terms on the right hand side of equation (6.1) except the first if we set Zkl = 1. The question remains as to what to choose for the first term. We approximate it here by the G-matrix, that is, we assume that

fit& = G(k, 7kd

(6.2)

Our justification for this is that we should have taken most of the collective effects into account in our treatment of the phonon-induced interaction in the last section. Whereas this seemed reasonable to us at first, we then saw that processes involving the exchange of two and more phonons enter into f d, the division into vertex interactions reducible (induced interaction) and irreducible (f,“,,,) in the crossed channel as made in Appendix A separating off only the one-phononexchange terms into the former. We believe the numerical inadequacies in our treatment in terms of Landau parameters to result mainly from the neglect of multiple phonon exchange, and are currently trying to build this in.

VII.

RFWJLTS AND THE EFFECT OF PARAMAGNONS

In the previous sections formulae have been derived forf [see Eqs. (4.10), (4.21) (4.23), (6.7)] and these depend on q = 1k, - k, j = 2PF sin 5/2 where 5 is the angle between the vectors k, , kz on the Fermi surface. It is straightforward to extract Landau parameters from this. Since it was convenient earlier to treat flT fl” and separately we shall continue to talk about them here also. It is a simple matter to translate these to the more conventionalf,s andf,“. We use TT(TS) FC =

m*pp TT(T.~ -ffl 7rw

-rn*p,21+l div -f 2

(7.1) 1 “(“)(2pF

1-1

sin 5/2) P,(cos 5) d(cos 5).

The results for 1 = 0, 1 are presented in Tables II, III for the two pressures 0.28 atm and 27.0 atm, where the contribution of individual excitations is listed separately. Table III is incomplete since the G-matrix at 27 atm is unknown. But two points are evident. First, the interaction in the spin up-down channel is grossly underestimated and the reasons for this are discussed below. Secondly, in the spin up-up

28

BABU

AND

BROWN

TABLE

II

P = 0.28 atm Density Fluctuations Spin Fluctuations with m, = 0 Spin Fluctuations with m, = il Current Current Coupled through F18 Current Current Coupled through Fla G-Matrix (Zero Pressure) Total Experimental

8.07 0.27 0 1.50 0 -4.16

1.11 0 3.29 0 -2.12

5.78 10.13

TABLE

Density Fluctuations Spin Fluctuations with m, = 0 Spin Fluctuations with m, = i 1 Current Current Coupled through F," Current Current Coupled through 4" G-Matrix

71.44 0.32 0 7.85 0

Experimental

74.88

0 0 0.54 0

4

0 0 4

4 -1.5

0 4 -1.3

2.28 5.6

-0.96 11.47

-1.3 7.0

2.51

0 0 0.64 0

III

-0 0 10.88 0 unknown 13.7

4

0 0 4 0 4

76.32

15.1

channel there is clearly a sort of “boot-strapping” at both pressures; namely, exchange of density and spin excitations and current-current correlations in the crossed channel builds up the effective interaction in the direct channel to almost its proper value, though some of it is cancelled by the G-matrix. We expect this to hold even when all the vertex corrections are included since these should be comparatively small. The reason for the unsatisfactory state in the spin up-down channel probably results from the G-matrix being a rather poor approximation for the direct interactionfd, into which multiple phonon exchange processes enter, as discussed in the last section. On the other hand, the paramagnon model [l&21] for liquid 3He handles, in some phenomenological and poorly understood way, effects of multiple phonon

QUASIPARTICLE

29

INTERACTION

exchange, etc., by incorporating everything into an effective zero-range interaction, which has nothing directly to do with the Landau parameters. The coupling constant in such a case is given not byfa but by a parameter I adjusted to fit the spin susceptibility. The consequences of such a theory in terms of bare particles for the Landau interaction is explored below, where as usual in paramagnon theory a contact interaction is assumed:

Hint = z 1 d3XLPX’n?(x)n,(x’) 6(x - x’). Contributions to the effective mass m* from paramagnons [15,21] shown in Fig. 6.

.\ 0I 4I ,0 I

(7.2)

are of the two types

W-B --m-m e-w m-w

---c

1L

FIG. 6. Contributions of paramagnons to the effective mass. The dashed lines represent the i-function interaction. The marks // indicate the lines we are going to break in order to obtain the effective interaction.

The spin susceptibility calculated in the RPA plays the role of the propagator for the intermediate particle-hole pair; indeed, if we drop the lines with marks // through them in Fig. 6 we have just the graphs for the susceptibility. The contribution of the m = fl paramagnon modes to the susceptibility is [15,211 X0(%w - Z%(O)WJ, 91, (7.3) where x0 is the Pauli (free-gas) susceptibility,

NT(O) = pFm/2d

is the density of

30

BABU

AND

BROWN

states of one spin in the free gas, U(q, 0) is the Lindhard function. The contribution of the mZ = 0 mode is2

whdq, 0)Wq,w - m, om

(7.4)

IN,(O) = 1.

(7.5)

where we now write

The m, = &l and m, = 0 paramagnons contribute in Fig. 7. The mZ = 0 mode contributes only to Err.

to Fp’ and PJ as shown

--w-m --w-s -m-m--m-- /w m-v--

i FIG.

7. Contributions

to FTT and FTJ with a S-function interaction (by paramagnon exchange).

One easily finds

(F;rpar-agnons= g (w) (F;Ly--magnons

= 2$

(w)

i j’ &OS x) PROSx) 1 yp;(: -1

o> 9

(7.6)

i 1’ &OS x) P(COS x) 1 _ i:(q o>. -1

3

From these expressions, evaluated numerically, we obtained the results shown in Table IV as a function of i. These results are in striking contrast to the earlier ones. The spin up-down channel interactions have been built up through the exchange of paramagnons in 2 Because of the d-function interaction and Fermi statistics, only odd numbers of bubbles can enter into the graph, Fig. 5a, or the corresponding graph for the susceptibility. Such a sum of an odd number of bubbles is expressed as $JwI, a)

1 [ 1 - mfl,

WI +

1 1 + mq,

0) I *

The first term in brackets represents the paramagnon; the second term, the density fluctuation, which was, however, already treated in Section IV in a better way.

QUASIPARTICLE

Contribution

0.87 0.88 0.89 0.90 0.95

Exp’tl

31

INTERACTION

TABLE IV from Paramagnon Exchange to Landau Parameters

3.66 3.91 4.18 4.50 6.95

3.4 3.75 4.15 4.62 8.91

9.92 10.45 11.03 11.69 16.75

6.79 7.49 8.30 9.23 17.81

10.13

5.6

11.47

7.0

the crossed-channel to the requisite level for P = 0.27 atm if 1 is assumed to be 0.90. The static susceptibility for this 1 in the paramagnon model is equal to (1 - 1)--l = 10 which compares with the experimental value of 9.2. Thus somehow by making the rather simple approximations of the paramagnon model we have succeeded in accounting for the interactions in the spin up-down channel. We believe the essential point to be that the ladder graphs in the paramagnon theory are handled in just the same way in calculating the effective mass m*, which follows from Fl* = &(FiPT+ q$). Thus, one is using the theory in making relations between empirical quantities, and the only significance of the a-function interaction is as an artifice in doing this. None the less, the paramagnon theory has been immensely suggestive with respect to a wide range of phenomena. For the present we will propose that density fluctuations be treated as in Section IV and V, but the spin terms be evaluated using the paramagnon model. The following results, listed in Table V, can be extracted from earlier tables. TABLE P = 0.28 atm Density fluctuations (from Table 2) Current-current coupled through F,’ G-matrix Paramagnons (I = 0.9) Total Exp’tl

Fi’ 8.07 1.5

-4.16

V FTT 1

FT”0

FTL 1 _.__

1.11 3.29 -2.12

0 0 -1.5

0 0

4.50

4.62

11.69

9.89

4.9 5.6

10.19 11.47

10.13

-1.3 9.23

7.93 7.0

The last two lines in Table V compare favorably with each other. Let us note that F,” = 0.33 in contrast to the experimental value of -0.667, but then one is 595/78/1-3

32

BABUANDBROWN

subtracting two numbers FJT and FJL which are quite close to each other. Since we are interested in the values of Fl for I 3 2, we list below some values (Table VI). TABLE

P = 0.28atm Density fluctuations Current-current Coupled through Fls Paramagnons (1 = 0.9)

VI

FT' a

FT' 3

-0.01

-0.03

FTJ 2

FTA 3

0

0

0.64

0.05

0

0

1.85

0.77

3.69

1.54

Unfortunately, the G-matrix contributions to these are not known. Two comments should be made. Firstf(k, , k,) is given by [GZ(k,)jGn(k,)] where C(k,) is the self-energy. Since in Fig. 6 any of the internal lines can have the label k, , to get the effective interaction one should sum diagrams that are obtained by cutting any one of the internal lines and not just the lines marked ‘1 as we did, but then the renormalization factors 2, should be included and this again gets very complicated. VIII.

CONCLUSION

We have shown how the exchange of density and spin density excitations contribute to the quasiparticle interaction. The failure of the treatment of these excitations on the same footing to lead to satisfactory results led us to consider the paramagnon model. This introduced a new parameter I whose interpretation in terms of physical quantities is unclear. The aesthetic satisfaction derived from the equal treatment of density and spin-density excitations has been lost, but the results stand favourably in comparison with experiment. We believe the numerical inadequacies in our treatment of the induced interaction via Landau parameters to arise from the neglect of multiple spin-fluctuation exchange. The treatment of the interaction induced through zero-sound exchange should be better, and this is evidenced by our good results for F”M‘, which comes mainly from this mechanism. The velocity of zero-sound is high compared with quasiparticle velocity, so the probability of multiple exchange is relatively small. It will be shown later (Sjoberg) that in application to nuclear matter, where spin fluctuations are relatively unimportant, our method appears to give rather accurate results. As noted, we believe the paramagnon model to include effects of multiple spin-fluctuation exchange in a rough phenomenological way. Finally, we note that even with our relatively poor numerical results, our treat-

QUASIPARTICLE

INTERACTION

33

ment has shown how the phonon-induced reaction in liquid 3He can provide the highly repulsive particle-hole interaction evidenced in the large and positive Landau parameters, whereas the interaction entering into calculations of the binding energy must necessarily be attractive. An application of our paper to nuclear matter is made by 0. Sjiiberg in the following paper. As noted in our abstract, the theory should be quantitatively much better there, because there are no “soft” excitations; i.e., no excitations with phase velocity small compared with the Fermi velocity of the quasiparticles, so the neglect of the two-phonon exchange should be less serious there. It gives us great pleasure to acknowledge stimulating discussions with Professor A. Sjiilander, who offered many suggestions, used in this work. We are also very grateful to Professor C. Pethick for criticism and key suggestions.

APPENDIX

A3

Here we show that in the long wave length limit the vertices coupling 3He quasiparticles to density and spin fluctuations become precisely the spin symmetric and spin asymmetric Landausfunctions. While one can apply the procedure used by Saam [22] in the case of 3He-4He mixtures to determine the coupling between long wave length 4He phonons and 3He quasiparticles we shall follow an alternate path to bring out clearly the interplay between the singularities in the momentum transfer in the crossed channel and the coupling to collective modes. Let r(p, p’; I), where p, p’ and I are four vectors, denote the reducible four point vertex function of Fig. 8 without the external lines. This vertex function can be considered as a particle-hole interaction where the incoming particle-hole pair energy-momenta are p + l/2 and p - l/2, while that of the outgoing pair are given by p’ + l/2 andp’ - I/2. One can then define an irreducible interaction Z(p, p’; Z), which has no “1 cuts”, as in the first line of Fig. 8. As is standard in Fermi liquid theory Z(p, p’; I) can be normalized and one easily shows that the Landau function f(p,p’) is obtained from F(p, p’; I) by letting 1 ---f 0 such that / I i/Z, -+ 0 and restricting p and p’ to lie on the Fermi surface. In the limit I -+ 0, r(p, p’; I) depends on I only through the ratio cy= 11 l/Z,, . This is usually indicated by writing P(p, p’; I), meaningful only for small I. Hencef(p, p’) corresponds to 01= 0. Alternately we can consider T(p, p’; 1) as the reducible particle-hole interaction where the incoming particle-hole pair is represented by p + l/2 and p’ -+ l/2 as is done in the second line of Fig. 8, where the irreducible function 1(p + Z/2, F - Z/2; p -p’) is defined which has no p -p’ cuts. Here p = (p + p’)/2. 3 The arguments below were developed in collaboration

with Professor A. SjGlander.

34

BABU

AND

BROWN

FIG. 8. Different ways of splitting up the vertex function. The shaded pieces represent the irreducible interaction.

Since we shall be concerned with this representation of the vertex function, it is written out explicitly in eq. (Al) (for simplicity we assume all the spins on the external lines are same and suppress spin indices)

= I(j+;,j+-pl

) +p(,+;,kP-Pp’)

x G k++)G(k-+)r(k,p-&p-p’) (

(Al)

Note that I(p + l/2, p - l/2; p - p’) is nonanalytic as I -+ 0 in exactly the same way as I(p, p’; 1) is for p --+ p’. This means that Qp, p’; I) has to be handled carefully when both I and p - p’ are small. Because of the Pauli principle we have r (p + f ) j-l - f ; p - p)) = --r(p,

p’; Z).

W9

We shall confine our attention here to the case when both p - p’ and I are small. We shall now use the superscript CYto indicate the dependence of r on p-p’; i.e., a = (p - p’)/p,, in the development below. Since we take p -+ p’, we can put p =p in the first two arguments in r. Further [2] G(k++)G(k-+)

m G2(k) + 2riZk2RU(k, p - p’),

(A3)

QUASIPARTICLE

35

INTERACTION

where a* k/m*

(A3.1)

R=(k, p - p’) = 1 _ a: , klm* a(/.‘ - ko) ‘%tJ - %)q

Here the singular behavior is confined to R”. This decomposition is unique for finite values of p’ - p up to a function of p’ - p that vanishes with p’ - p. Then TO(P + U/2), P - (42); P - P’) = Z(P + (l/2), P - U/2); P - P’) + 14~

+ U/2), k; p - P’) GYk)

k

x r”(k,

P -

U/2); P - P’>.

The first term on the right hand side is nonanalytic for I -+ 0 whereas in the last term we can set 2 = 0 since the difficulties in the small region k up + Z/2 disappear, because of vanishing phase space, in the summation over k. We can further replace T”(k, p - l/2; p - p’) by (2&?YpZ&1f(k, p). Using (A4), we can eliminate Z from (Al), to give r*(P

+ W),

P -

W);

P - P’)

= ~O(P + V/2), p -

U/2); P - P’) + c (27Q7,)-1f(~,

k>

k

x 2niZ,R”(k, p - p’) F=(k, p - (Z/2); p - p’).

645)

Again the way of going to the limit I - 0 must be defined in the first term on the right hand side. Letting l-+ 0 in P and using eq. (A2) we get --BP, P’> = l,Jjs~,o 27G2Gir”(p + Cf(p, k) R”k

+ (VS P - U/2); P - P’>

P - ~‘)fOr,

P>

k

+ C f(p, k> R”(k, P - ~‘>fOr,

k’) RmW;

P - p’>fOr’,

P>+ ee.2

kk’

W)

which is the desired result. Here R” enters, because we took p. = p,,’ = eFfirst, in putting the quasiparticles on the Fermi surface, so that p. - p,,’ = 0. Later we went to Landau angle fi * fi’ zero, so as to take the long wavelength limit in the crossed channel. In that limit Rm(k, p - p’) = -8(/..t - k,) 801 - %c)*

(A7)

36

BABU AND BROWN

On substituting this into equation (A6), ignoring the first term, one obtains trivially, when only f, is nonzero, the bubble sum shown in Fig. 9, which can be shown to be dominated by zero sound for largef, .

+

FIG. 9.

The case of spin fluctuations is similar as can be shown by considering of the external lines explicitly, though paramagnons do not dominate Finally in the absence of detailed information about the first term on hand side of equation (A6), one has to make certain assumptions. This is in the text in Section VI.

the spins the sum. the right discussed

APPENDIX B

The lyn(q, w) defined by Eq. (5.19) will be evaluated below for w = 0 and n = 0,l and 2. The integration has to be carried out over the shaded area, as shown in the figure above. In cylindrical coordinates, as defined in Fig. 10, EP

-

%+a

=

-zq/m* PF2 - (z + q/a2

Pl

=

L-32

= PF2 - (z - a2

P t- 2

FIG.

10. Fermi sphere, under translation by q, in cylindrical coordinates.

QUASIPARTICLE

INTERACTION

31

The value of $ * 4 is dependent on which of the two shaded areas 1 is located. If we define F(z, p) as F(z> P) = K--4/2 + 41(p2 + (q/2 - d2P21 then it is easy to see that $ .d = F(z, p) in one shaded area while $ . 4 in the other. Hence

032)

F(-z>P)

033)

For n = 0 it is easy to evaluate the integrals. The result is 034)

For n = 1,2 the second integral is harder to evaluate and has been done numerically. REFERENCES 1. L. D. LANDAU, Sou. Phys. JETP 3 (1956), 920; 5 (1957), 101; 8 (1959), 70. 2. P. NOZ&RES, “Theory of Interacting Fermi Systems,” W. A. Benjamin, New York, 1964. 3. D. PINES AND P. NOZ~RES, “The Theory of Quantum Liquids,” W. A. Benjamin, New York, 1966. 4. A. J. LEGGET AND M. J. RICE, Phys. Rev. Letters 20 (1968), 586. 5. HING-TAT TAN AND E. FEENBERG, Phys. Rev., 176 (1968), 370 and earlier references cited there. 6. G. BERTSCH, Phys. Reo., 184 (1969), 187. 7. E. OSTGAARD, Phys. Rev. 187 (1969), 371 and earlier references cited there. 8. K. A. BRUECKNER AND J. L. GAMMEL, Phys. Reu. 109 (1958), 1040. 9. E. FEENBERG, “Theory of Quantum Fluids,” Academic Press, New York, 1969. 10. G. E. BROWN, Rev. Mod. Phys. 43 (1971), 1. 11. B. BRANDOW, Rev. Mod. Phys., 39 (1967), 721. 12. A. A. FROST AND B. MUSULIN, J. Chem. Phys. 22 (1954), 1017. 13. J. BARDE~N, G. BAYM, AND D. PINES, Phys. Rev. 156 (1967), 207. 14. C. J. PETHICK, in Vol. XI-B of“Lectures in Theoretical Physics,” (K. Mahantappa and W. E. Brittin, Eds.) Gordon and Breach, New York, 1969. 15. S. DONIACH AND S. ENGELSBERG, Phys. Rev. Letters 17 (1966), 750. 16. J. R. SCHRIEFFER AND N. BERK, Phys. Letters 24A (1967), 604. S. MA, M. T. BEAL-MONOD, AM, D. R. FREDKIN, Phys. Rev. 174 (1968), 227. 17. J. HUBBARD, Proc. Roy. Sot. (London) A. 243 (1957), 336.

38

BAEKJ

AND

BROWN

18. D. C. LANGRETH, Phys. Rev. 181 (1968), 753. 19. P. Nozxrbm AND D. PPIES, Phys. Rev. 111 (1958), 442. 20. D. FORSTER, P. C. MARTIN AND S. Yp, i%ys. Rev. 170 (1968), 155. 21. W. BRENIG AND H. J. MIKESKA, Phys. Letters 24A (1967), 332. 22. W. F. SAAM, Ann. Phys. (N.Y.) 53 (1969), 219. 23. J. C. WHEATLEY, in “Quantum Fluids,” (D. F. Brewer, Ed.), Wiley, New York, 1966. 24. D. PINES, 1965 Tokyo Summer Lectures in Theoretical Physics, SyokaM, Tokyo and Benjamin, N. Y. 25. SINGWI, Tosr, LAND AND SJ~~LANDER, Phys. Rat. 176 (1968), 589; SINGWI, SJ~LANLWR, TOSI AND LAND, Phys. Rev. Bl (1970), 1044.