IL NUOVO CIMENTO

Vor,. XIX, N. 4

16 Fcbbraio 1961

Flow Properties of Superfluid Systems of Fermions (*). A. E. GLASSGOLD Lawrence Radiatio~t .Laboratory, University o] Cali]ornia - Berkeley, Cal.

A. M. SESSLER (**) Tlte Ohio State University - Columbus, Ohio

(ricevuto il 14 0 t t o b r e 1960)

- - The nonspherically symmetric solutions to the BardeenCooper-Schrieffer theory are given a physical interpretation in terms of an anisotropic fluid model. These solution have been used previously to predict a phase transition in liquid 3He by EMERY and SESS~ER and by BRUECKNER, SODA, ANDERSON and MOREL. An investigation of the flow properties of such systems is made t h a t involves the calculation of the effective mass for flow in a straight channel and the moment of inertia of a cylindrical container of the liquid. The angular dependent energy gap characteristic of this type of theory leads to an effective mass for flow t h a t depends on the angle between the axis of symmetry of the fluid and the direction of flow. The effective mass for flow vanishes as the absolute temperature tends to zero, although not as rapidly as for a sphericaiiy symmetric gap. The moment of inertia, when the symmetry direction for the fluid and the rotation axis are the same, is simply related to the mass for flow. Summary.

1. - I n t r o d u c t i o n .

T h e work of BARDEEN, COOPEI~ a n d SCn•IEFFER (BCS) p r o v i d e s a r e m a r k a b l y successful s o l u t i o n to t h e p r o b l e m of s u p e r c o n d u c t i v i t y (1). T h e b a s i c f e a t u r e i n t h e i r a p p r o a c h is t h e s t r o n g c o r r e l a t i o n b e t w e e n c o n d u c t i o n electrons (') Supported in part by the U.S. Atomic Energy Commission, and in part by the National Science Foundation. (*') Work performed while a visitor at the Lawrence Radiation Laboratory. (1) j . BARDEE~', L. N. COOPER and J. R. SCIIRIEFFER: Phys. Rev., 108, 1175 (1957).

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A.E. GLASSGOLD and A. M. SESSLER

with equal and opposite m o m e n t u m and spin. This t y p e of correlation probably plays an essential role in other m a n y - f e r m i o n systems. F o r example, VAIN HOVE and EMERY h a v e shown how the usual p e r t u r b a t i o n t h e o r y for an imperfect F e r m i gas breaks down under just those conditions when the BCS a p p r o a c h is valid (2). Direct extensions of the BCS theory h a v e already been m a d e to finite nuclei (s), infinite nuclear m a t t e r , and liquid 3He (4-5). Of special interest is the prediction t h a t liquid 3He undergoes ~ phase transition at v e r y low t e m p e r a t u r e s to a highly correlated phase related to the phase change observed for superconductors (e,v). The predicted transition t e m p e r a t u r e is of the order of 0.08 °K, b u t so far no anomalous effects h a v e been observed in this t e m p e r a t u r e range (s). The theoretical description of this phase transition differs f r o m t h a t for the electrons in a superconductor in the following i m p o r t a n t respect. I f the F e r m i surface in a m e t a l is considered to b e spherically symmetric, then the correlation function in the original BCS t h e o r y is spherically symmetric. F o r liquid SHe, on the other hand, the correlation function is not t h o u g h t to be spherically symmetric. (This is a direct consequence of the fact t h a t the interaction at the F e r m i surface for two helium a t o m s in a relative S state is repulsive.) The possible existence of such solutions in the BCS t h e o r y was first noted b y ANDERSO~ (9). The anisotropic correlations contained in these solutions raise interesting questions of interpretation, particularly for liquid SHe, where there is no long-range order. I t is the purpose of this p a p e r to discuss qualitatively the physical significance of these anisotropic solutions in the BCS theory. We often consider liquid SHe as a specific example, although m u c h of the discussion is m o r e general. The interpretation is mainly given in t e r m s of two quantities, the e]~eetive mass ]or ~low through a straight channel, and the moment o] inertia

(2) L. VAN HOVE: Physiea, 25, 849 (1959); V. J. EMERY: Reaction matrix singularities and the energy gap in an in]inite system o] ]ermions, Lawrence Radiation Laboratory Report UCRL-9076, (February 8, 1960) (Submitted for publication to Nuclea Physics). (a) S. T. BELYAEV: Kgl. Danske Videqtskab. Selskab, Mat.-]ys. Medd., 31, No. 11 (!959). (4) L. N. COOPER, R. L. MILLS and A. M. SESSLER: Phys. Rev., 114, 1377 (1959). (5) N. l-~. BOGOLJUBOV, V. V. TOLMACHE¥ and D. V. SHIRKOV: A New Method i~ the Theory o] Superconductivity (New York, 1959), (6) V. J. EMERY and A. M. SESSLER: Phys. Rev., 119, 43 (1960). (7) K. A. BRUECKNER, T. SODA, P. W. ANDERSON and I'. MOREL: Phys. Re~,., 118, 1442 (1960). (s) D. F. BREWER, J. G. DAUNT and A. K. SREED~AR: Phys. Rev., 115, 836 (1959). A. C. ANDERSON, H. R. HART Jr. and J. C. WHEATLEY: Phys. Rev. Lett., 5, 133 (1960). (9) p. W. ANDERSON: Phys. Rev., 112, 1900 (1958).

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for the r o t a t i o n of a cylindrical container of the fluid. These quantities determine the ability of the fluid to t r a n s p o r t linear and angular m o m e n t u m . The outline of the remainder of this c o m m u n i c a t i o n is as follows: Bogol j u b o v ' s (~quasi-particle >>f o r m of the BCS theory is first reviewed in Section 2. The physical i n t e r p r e t a t i o n of the theory in terms of an anisotropie fluid is also given in this section. I n Section 3 the general formulae for the inertial p a r a m e t e r s are reviewed, and in Sections 4 and 5 the effective masses for flow and rotation are evaluated.

2. - Quasi-particle theory of superfiuid fermion systems.

BOGOLJUBOV has emphasized the quasi-particle nature of the BCS theory (~). B y a quasi-particle a p p r o x i m a t i o n , we m e a n t h a t the actual H a m i l t o n i a n for this p r o b l e m is t r u n c a t e d and t r a n s f o r m e d i~to the f o r m (2.1)

Ho = E o 4- ~ E(k)(~x~ k 4- fl~fl~,). k

The operators ~k and fl~ (~k and ilk) create (destroy) the excitations of the m a n y particle system. These excitations h a v e definite energy E ( k ) and monlenturn k. The quasi-particle operators obey the same a n t i - c o m m u t a t i o n rules as the corresponding operators for the actual particles m a k i n g up the system. (In order to avoid introducing a spin label, we use two sets of quasi-particle operators.) The linear t r a n s f o r m a t i o n between particle operators and quasiparticle operators is

(2.2)

%

u(k)ak+ - - v (k)a +.k

% -- u(k)a_k_ 4- v(k)a~+ or

ak~ = ~(k)* ~k + ~ , ( k ) ~ , (2.2a)

%

= u(k)*fi_ k - v ( k ) ~ _ k .

The operators a~o (ak~) create free-particle states of m o m e n t u m k and <~spin >> projection a - - ± 1 (to). The a n t i e o m m u t a t i o n relations are preserved for

(2.3)

iu(k)l~+lv(k)l~=l.

~t has also been assumed t h a t we h a v e u ( - - k ) =

u(k) and v ( - - k ) =

v(k).

(~o) Tire asterisks in eq. (2.2), as well as elsewhere in this paper', stand for (~complex conjugation ~>.

726

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According to eq. (2.3) we m a y write the two complex functions as u(k) = cos z(k) exp [i~(k)],

(2.4)

v(k)

= sin x(k) exp [i~(k)].

I t can be shown t h a t all physical observables depend only on the difference in phase,

F(k)=~(k)--~(k).

(2.5)

Hence the two real functions, z(k) and ~:(k), characterize the quasi-particle transformation. At absolute zero, BOGOLJUBOV determined the transformation in the following way. The Hamiltonian of the system is written in the new representation with all creation operators to the left. No quadrilinear terms are retained and the resulting truncated ]=Iamiltonian is diagonalized, i.e. forced to have the form of eq. (2.1.) This procedure is equivalent to the BCS variational calculation of the ground-state energy. At finite temperatures, the t h e r m o d y n a m i c potential is minimized instead (as discussed, for example, in ref. (6)). As a result, the theory is essentially determined b y the following coupled equations : (2.6)

C(k') C(k)=-- ½~(k,-klvlk',-- k')~,y,tgh½fiE(k),t~) k'

(2.7)

~(k)---- [ s ( k ) - - # ] - ~ ~ (k,

k't~lk, k'){/(k')~-[1--2/(k')]lv(k')]2}.

k'

The function C is defined as

(2.s)

C ( k ) = ~ (k, - - k Iv I k', - - k ' ) u * ( k ' ) v ( k ' ) [ 1 k'

-- 2t(k')],

where (2.9)

E(k)

= [~(k) +

l C(k) I~]~

and

(2.1o)

/(k) =

exp

[fiE(k)] ÷ 1"

The symbol/~ stands for the chemical potential and e(k) for the u n p e r t u r b e d single-particle energy. F o r a spherically symmetric Fermi surface s depends only on the magnitude k----Ikl. The m a t r i x elements of the two-body potential are (klk~lv Ik~k'~); the forward scattering of the quasiparticles, which

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727

appears in the expression for their energy in eq. (2.7), is

(2.m

(k, k'l~lk, k') = (k, k'lvlk, k3--(k, k'lvfk', k ) + ( k , - - k ' l v l k , - - k 3 .

We also note (2.12)

C(k)=lC(k)[ex p [ @ ( k ) ] ,

where q(k) was defined in eq. (2.5), and (2.13)

tg 2 z ( k ) --

lc(k)l ~(k)

I n this brief r(,sum6 of the theory, we h a v e indicated explicitly the possible dependence of the properties of an excitation on its vector m o m e n t u m , in particular on its direction m e a s u r e d with respect to an a r b i t r a r y axis fi henceforth called the ~ quantization ~>axis. The original BCS t h e o r y of superc o n d u c t i v i t y for a sphericMly-symmetric F e r m i surface corresponds to the special case of isotropic properties. The possibility of this anisotropy stems directly f r o m the lack of invariance of the truncated H a m i l t o n i a n u n d e r an a r b i t r a r y rotation~ which in t u r n arises f r o m the direction d e p e n d e n c e of the excitation energies in eq. (2.1). This absence of rotational s y m m e t r y is associated with the t r u n c a t i o n process, since the original m a n y - p a r t i c l e g a m i l tonian describing the liquid is certainly i n v a r i a n t under a r b i t r a r y rotations. ( I t should be noted t h a t the quasi-particle t r a n s f o r m a t i o n of the original Hamiltonian leads to a new H a m i l t o n i a n t h a t is still rotation-invariant. This is t r u e even for the angular-dependent solutions, since eq. (2.3), the r e q u i r e m e n t t h a t the t r a n s f o r m a t i o n be canonical, is satisfied.) Despite the fact t h a t the model H a m i l t o n i a n is not i n v a r i a n t under arb i t r a r y rotations, there are physical situations to which the solutions correspond. F o r example, at absolute zero, the ground state corresponds to a fluid with a preferred direction c o m m o n to the whole sample and determined b y the walls of the container. I n this case, the arbitrarily small interactions with the walls (which are not usually included in the original rotationally i n v a r i a n t H a m i l t o n i a n ) play a crucial role just as in the f o r m a t i o n of a crystal. Other cases in which the walls serve to establish preferred directions are quasiequilibrimn situations corresponding to macroscopic fluid flow, discussed more fully in the n e x t sections. To arrive at a b e t t e r understanding of the quasi-particle model with angulard e p e n d e n t solutions, we recall t h a t the q u a n t i t y C(k) determines the paircorrelation function. The pair-correlation function in this t y p e of t h e o r y describes short-range order, with a correlation length of order fiJ~v~ (where vF is the F e r m i velocity and /~-~ is the transition temperature). I n addition, the

728

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GLASSGOLD a n d

A. M. SESSLER

particle density is uniform and isotropic, whereas the correlation function is angular dependent. I n other words, we are describing here an anisotropic liquid, as defined b y LANDAU and LIFSnITZ (11). The correlation length in the BCS theory is r a t h e r large c o m p a r e d with atomic spacings. F o r example, for 3He, for which the transition t e m p e r a t u r e is predicted to be of the order of 0.08 °K, the correlation length is a b o u t 200 A. F o r equilibrium at a non-zero t e m p e r a t u r e , this implies the f o r m a t i o n of a loose domain structure with a domain size n o smaller t h a n the correlation length. The existence of a d o m a i n structure for this s y s t e m was suggested b y BlCUECKNER et al. (7). W h e n the pair-correlation function is anisotropic, each domain has a preferred axis and, in first a p p r o x i m a t i o n , these domains are r a n d o m l y oriented. The existence of domains is inferred from the following energetic considerations. Particles in the liquid interact strongly only if t h e y are within correlation length of one another. Therefore the division of a domain in two has associated with it an increase in the total energy of the s y s t e m which is p r o p o r t i o n a l to the correlation length times the surface area in contact between the new domains. Thus a negligible change in the total energy of the sample is required for the sample to b r e a k up into a large n u m b e r of domains. At a non-zero t e m p e r a t u r e the n u m b e r of domains into which the fluid is subdivided is d e t e r m i n e d b y the condition t h a t the f o r m a t i o n energy of a domain is of the order of kT. As a consequence, at absolute zero, there is just a single domain, as was previously r e m a r k e d . On the other hand, as the transition t e m p e r a t u r e is a p p r o a c h e d f r o m below, the n u m b e r of domains increases rapidly, since the correlation energy approaches zero. F o r quasi-equilibrium situations corresponding to fluid flow, these energetic considerations m u s t be e x t e n d e d ; this is done in the following sections.

3. - General formulae for the inertial properties of a superfluid.

W e now discuss the superfluid properties of the system in a q u a n t i t a t i v e way, using the effective masses for uniform translation and rotation. Our discussion is clearly m o t i v a t e d b y L a n d a u ' s discussion of the superfluidity of liquid H e I I (12). F o r the special case of spherically s y m m e t r i c solutions,

(11) :L. D. :LANDAU and E. M. L~FSI~ITZ: Statistical Physics (:London, 1958), paragraph 126, p. 412 et seq. Professor W. D. K~IGHT has kindly informed us of some recent observations on anisotropic-fluid behavior in the melting of metals: Ann. Phys., 8, 173 (1959). These authors find that the short-range order in metals is often preserved in the transition to the liquid phase. (12) L. D. LAlffDAU: Journ. Phys. (USSR), 5, 71 (1941).

FLOW PROPERTIES ()F SUPERFLUID SYST)~MS OF FERMIONS

~9

BARDEEN (l~) and KHALATNIKOV and AnaIKOSOV (~) h a v e already discussed the relation between the BCS theory and the two-fluid model. These authors h a v e calculated the density of normal electrons, which is simply proportional to out' effective mass for flow. I n this section we review tile general statistical formulae for the inertial p a r a m e t e r s . The explicit calculation of the effective mass for flow and the m o m e n t of inertia is discussed separately in succeeding sections. 3 " ] . E]]ective mass ]or ~low. - We consider the uniform flow of the fluid down an infinite channel. If v is the me~m drift velocity of the excitations and if ( P ) is the m e a n t o t a l m o m e n t u m per unit volume, t h e n the effective mass for flo~ is defined b y the equation

(P) = Mz(v)v.

(3.1)

The velocity v is, b y definition, the velocity (with respect to the l a b o r a t o r y system) of the reference f r a m e in which the quasi-particle distribution function is t h a t for a fluid at rest, i.e. eq. (2.10) for this problem. Unless stated otherwise, the effe(.tive mass for flow is t h a t obtained in the limit of zero velocity.

M ~ ( O ) - ~P(v)

(3.2)

~v

=o

W e conveniently define a superfluid as a s y s t e m with M~(0)< rim, where n is the density and m the particle mass. This characterization of a superfluid emphasizes the contrast with a classical fluid with respect to a liquid's ability to transfer m o m e n t u m . We note t h a t L a n d a u ' s n o r m a l density is just

,,,,

M~(O)/m.

According to ttle general principles of statistical mechanics, the m e a n mom e n t u m per unit volume is (3.3)

(P> =

Tr [ P exp [-- fl(H-- # N - - P.v)]] T r [exp [ - - . ~ ( H - t t N - P . v ) ] ]

The symbol Tr[...] indicates tile trace operation a p p r o p r i a t e to the grand canonicM ensemble, and H, N, and P are the operators for the H a m i l t o n i a n , the n u m b e r of particles, and m o m e n t u m density, respectively. Carrying out the differentiation indicated in eq. (3.2), using the fact t h a t P c o m m u t e s with H - - t t N , and t h a t ( P } = 0 for v = 0 , we obtain the formula for (1~) j. BARDE~': Phys. Rev. J~ett., 1, 399 (1958). (tl) L M. KIIALATNIKOV 3~Yld A. A. ABRIKOSOV:

Adv. Phys., 8, 45 (1959).

730

A.E.

GLASSGOLD

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A. M . S E S S L E R

the effective mass for flow: (3.4)

M,(0) = f l < ( p . ~ ) z } ,

where ~ = v/v. The statistical average is carried out in the rest f r a m e (v = 0). W e emphasize once more t h a t this is just L a n d a u ' s definition of the n o r m a l density. 3"2. Momenta o] inertia. - We now consider a cylindrical container of the fluid r o t a t i n g with angular velocity ¢o a b o u t uts axis of s y m m e t r y t~. I f J is the o p e r a t o r for the t o t a l angular m o m e n t u m of the system, t h e n the m o m e n t of inertia is defined b y the relation (3.5)

= I((@o.

W e discuss only the limiting value (3.6)

I(0) =

(J.¢3>

o"

B y a p p l y i n g the same statistical equilibrium discussion used a b o v e for M~, the f o r m u l a for the m o m e n t of inertia is found to be (3.7)

I(0) =- f i < ( j . ~ ) 2 > .

Again, the statistical average is carried out. This result is due to BLATT~ ~BuTLER~ and SC~AFgOTg (15). W e note t h a t this derivation takes for g r a n t e d t h a t = 0 for co = 0. ANDERSON and ~¢IOlCEL h a v e recently raised questions a b o u t this particular point w h i c h we shall c o m m e n t on in our discussion of the m o m e n t of inertia in Section 5 (~6).

4. - E f f e c t i v e m a s s for f l o w .

The a b o v e formulae, eq. (3.2) a n d (3.7), show how MI(0) a n d I(0) are rel a t e d to the statistical average of (P.~)~ and (j.¢~)2. The evaluation of these averages is carried out in the quasi-particle representation. This is precisely the procedure followed in a recent discussion of the m o m e n t of inertia for t h e low-density theory of ti quid ~He (17).

(15) j. B. BLATT, S. T. BUTLER and M. S. SCHAFROTH:Phys. Rev., 100, 481 (1955) (16) p. W. A~-DERSON and P. ~OREL: Phys. Rev. •ett., 5, 136 (1960). (1~) A. E. GLASSGOLD, A. N. KAVF~AN and K. M. WATSO.~: Phys. leer. 120, 660. (1960).

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I n order to evaluate eq. (3.4) for M~(O), we need the expression for the m o m e n t u m operator in the quasi-particle representation

(4.4)

P= ~

k

t t (~-fi~#~)

.

k

W e n e x t write the average of (P.~)~ as (4.2)

<(P.V)~>

~ (t~..~)(t:'.~,)[<~]~,,>

-

<#,,#,,>][<~,,,~,, ~ ~ > - <#~,,fi,,,>] +

k#k'

k

Since the statistical averages of ak~k, ¢ + flkflk, and their squares are all just /(k), the first line of eq. (4.2) is zero and the second line leads to the following equation for M~(O):

(4.3)

MA0) = 2# ~ (k'~)~i(k)[~- l(k)]. k

As r e m a r k e d previously, this is essentially L a n d a u ' s expression for the n o r m a l density (~2). This formulu shows explicitly how the excitation spectrum, through the statistical factor f(k) determines the effective mass for flow. As a first simple example, we consider the spherically s y m m e t r i c energy gap C(k)----LJ, corresponding to the (( excitation s p e c t r u m ~>

E(k) = Lxl(!'22 n'4t%j Eq. (4.3) becomes co

(4.s)

M/(0) = 2tildE nm

E exp [fiE] [E 2 - A2]+(exp [fiE]

+- ])2"

I

The m o s t i m p o r t a n t contributions of the integrand c o m e f r o m the neighborhood of the F e r m i surface where E = A. I t is convenient to rewrite this equation as ¢o

(~.6)

M, (0) =---- ~

f;

)

ax

x +)~ exp [x +,~] . . . . . . . . . . . . . l,,"-' !- 2~.~1~ (exp rx ~ ,~] + ]):' .

.

0

where ~ = flA(fl). I t is now easy to establish the following a s y m p t o t i c limits of this integral, corresponding to the limits T -+ 0 (LI --> Ao) a n d T -~ T~

732

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A. M. S E S S L E R

(A -+0): (4.7) (4.8)

M: = [ (23rX)½exp [-- ),], nm [1,

-~- c~(T -> 0, A -+ Ao) , -~ 0 ( T - > To, A ->0) .

A more detailed discussion of M: at intermediate t e m p e r a t u r e is given b y KHALATNIKOV and ABRIKOSOV (14). As T decreases from To, M: decreases (linearly at first) to zero, v.anishing exponentially as absolute zero is approached. If the energy gap is set equal to zero for all temperatures, the case of the ideal gas is recovered. F r o m eq. (4.8) we see t h a t the effective mass for the flow of an ideal gas is the true mass. F o r asymmetric solutions, the angular-dependent factor (k.fi) 2 in eq. (4.3) is now important. We introduce the spherical polar co-ordinates (k, 0, T) for the quasi-particle m o m e n t u m k, with the preferred direction fi of the domain under consideration as quantization axis, and the angle z betwen fi and ~. W e assume here t h a t the excitation spectrum has cylindrical s y m m e t r y about ft. E = E(k, 0); and, for simplicity, t h a t C = C(O). In this case, eq. (4.5) must be replaced b y 1

(4.9)

ilC(0)nm-- 2flfd(cos 0) ~(cos~r cos 2 0 +

~ sin 2 ~ sin 20)-

-1 co

E exp [fiE] • d E [ E 2 _ I C(O ) 1~]½(exp [fiE] ÷ i) ~ . f

1¢(0)1

Current applications to liquid aHe make use of the form

(4.10)

C(k) = d,,,,(O) Y~,(O, q~) ,

and thus

It(k) 1~ = ~ , ( o ) P ~ A o ) . This function vanishes at several points, and the contributions to the integrand of eq. (4.9) from the neighborhood of these points are the most i m p o r t a n t ones. As a result, M: does not vanish as rapidly as T - > 0, as it does for a spherically symmetric gap. We now t u r n our attention to the question of the orientation of the preferred axis fi with respect to the flow direction fi in an actual experiment. Eq. {4.9) m a y be rewritten

(4.11)

M:(0) nm

-- cos 2 vK~ ~- sin 2 vK~ :

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733

where (x = cos 0) 1

--f dxx F(x) ,

(4.12(~)

-1 1

(4.12b) -1

and co f

,

E exp [fiE] /~'(x) = 3 f l t d E [E 2_ !C(x)i 2]~ (exp [fiE] g- 1)~" 5

(4.12c)

T h e mass for flow, and therefore the t o t a l energy, is a m i n i m u m for T = 0 a n d z for T ~--~/2, depending on w h e t h e r K2 > / £ i or K1 > K2 holds. I n the special case, K1 = K2, the effective mass is independent of T and all directions of the preferred axes are equally probable, energetically. I n this i m p r o b a b l e ease (KI= K2), the fluid would m a i n t a i n its domain structure a l t h o u g h the orientation of the various domain axes would be essentially uncorrelated. I n the m o r e likely situation, with K1 ~ K2, the preferred axes a n d the flow direction are, on the average, either perpendicular (K~ < K~) or parallel ( K I < K 2 ) . (There is no difference between 7----0 and T = ~ . ) There is, of course, a statistical distribution of the directions a b o u t these average values. Which of the two directions is m o s t p r o b a b l e depends on the relative m a g n i t u d e of K1 and Ks. A general conclusion on this point depends on obtaining more complete solutions to the basic equation (eq. (2.6) and (2.7)). This question can, of course, be discussed in the a p p r o x i m a t i o n of equation (4.10) (6,7). As T - + 0, the different energy gaps A ~ for the various m values are generally distinct, and the lowest energy is obtained with the largest energy gap. The integrals K1 and Ks can t h e n be evaluated for this value of m and the parallel and perpendicular directions distinguished. F o r e x a m p l e , the solution t h a t gives the lowest energy for / = 1 is C~-AI~Y~, and a simple calculations gives K I > K~. This m e a n s t h a t the preferred direction in the fluid is perpendicular to the flow direction in this case. As the t e m p e r ature is increased, the fluid breaks up into domains, and there are B o l t z m a n n distributions b o t h for the domain directions and for the various solutions characterized b y the different m values.

5 . - M o m e n t of inertia.

t~efore e v a l u a t i n g eq. (3.7) for the m o m e n t of inertia, we recall t h a t , in the derivation of this equation, it is assumed t h a t ( H - - # N ) and J-¢~ comm u t e . Since J . ~ is the projection of the t o t a l angular m o m e n t u m along 46

- Il

Nuovo

Cimento.

734

A.E. GLASSGOLDand A. M. SESSLER

the axis of rotation, it follows t h a t the operator H - / i N must be invariant under rotations about to. This condition is fulfilled for quasi-particles whose excitation energy does not depend on ~, where k, 0 and ? are the spherical co-ordinates of the quasi-particle m o m e n t u m k with to as polar axis. This p r o p e r t y is possessed b y the approximate solutions to eq. (2.6) given in eq. (4.10), which are valid just below the transition temperature. There is a wider class of functions t h a t v a r y as exp [imp] and which, therefore, correspond to an axially symmetric model Hamiltonian. Since little is known about the general properties of the solutions to eq. (2.6) and (2.7), however, we cannot exclude even more general solutions. In any case, the calculation of the m o m e n t of inertia in this section is confined to axially symmetric solutions for which the general formulae, eq. (3.7), is valid. This corresponds to the physicUl situation in which there is a single preferred direction in the fluid parallel to the axis of rotation. We now evaluate eq. (3.7) for the m o m e n t of inertia following the m e t h o d recently used for the low-density t h e o r y of liquid 4He (~7). The operator for the projection of the total angular m o m e n t u m along the rotation axis is, in the n o t a t i o n of second quantization,

(5.1)

J.~

=

~: ~ kk'

Ll, k, al,oak,o.

o

We ignore the negligible contribution of the intrinsic spin of the particles and the effect recently discussed b y ASTDERSO~N and MOREL, which is similar to an intrinsic angular m o m e n t u m of one particle about another. The symbol L stands for the projection of the orbital angular m o m e n t u m of one particle along fi = ~ . Its matrix elements in m o m e n t u m space satisfy the relations (5.2a)

Lk~ = L~,k,

(5.2b)

Lkk, = L_k_k, ,

(5.2e)

Lk~r = - - L ~ k , ,

and (5.3a)

L~k, ---- L~, ~kk'~k.~,k .~,

(5.3b)

Lkk --~ 0 .

Eq. (5.2a), (5.2b), and (5.2c) follow from the requirements of hermiticity~ inversion inval~iance, and time-reversal invariance. The last relation, eq. (5.3), expresses the p r o p e r t y of L as the generator of infinitesimal rotations about ~. Upon transformation to the quasi-particle representation b y direct substi-

FLO'W

PROPERTIES

OF

SUPERFLUID

SYSTEMS

OF

735

FERMIONS

tution of eq. (2.1), eq. (5.1) becomes

(5.4)

J.~t = ~ Lkk,[u(k)u*(k' ) ~- v(k)v*(k')] ( ~t ,

+

~,~,)

.

kk'

We note t h a t J - f i involves only ((diagonal operators >), i.e., operators involving the same n u m b e r of creation and destruction operators. T h a t no other operators occur (such as products of two creation or two destruction operators) is a direct consequence of the axial s y m m e t r y of the quasi-particle transformation. Another consequence of this s y m m e t r y is

J'alo> = o, where 10> is the e x p e c t a t i o n value volves the t e r m s The square of

g r o u n d - s t a t e or quasi-particle v a c u u m . F u r t h e r m o r e , the or the ensemble average of J . t$ is always zero, since it inin eq. (5.4) for which k = k' and Lk~, = O. J . ~ which appeurs in eq. (3.7) is

(J.~t) '~ = ~ ~ Lk,,L,~h;[u(k~lu (kx) ÷ v(k~)v (k~)]. k~k[ k.,k ,

I

~

?

t

I n the averaging of this expression, the terms k 1 = k'l and k 2 ~--k~ do not occur because the corresponding m a t r i x elements vanish. The only non-zero t e r m s are those involving four ~ or four fl operators.

(5.5)

<(J.fi)~> = ~ Lkk, Lk,~ lu(k)u*(k') + v(k)v*(k')[3. kk'

• [ < ~ ~ > (1 -

< ~ ,+~ , > )

+ <~>(1

-

)] .

According to eq. (5.3), the only non-zero t e r m s in this equation are for k and k' differing only in their a z i m u t h a l angles ~v and F'. Since the quasi-particle t r a n s f o r m a t i o n does not depend on the a z i m u t h a l angle, the u's and v's drop out completely (when eq. (2.3) is used) and all the statistical factors are the same: <(J.a),>

= 2 ~ l(k)[1 -/(k)]Z~,S~,,k, kk'

or, using closure, (5.6)

<(J-a)~> = 2 ~ / ( k ) [ 1 - / ( k ) ] ( L ' ) k ~ . k

F o r the diagonal m a t r i x element of L ~ a p p r o p r i a t e to a cylindrical container,

736

A. 1~. GLASSGOLDand A. M. SESSLER

we h a v e (L2)kk = ½(k X fi)2, where =

8r(x2 + y2).

The m o m e n t a of inertia is therefore

(5.7)

I(O) = 2fl ~ ½(kx fi)2/(k)[1 - - / ( k ) ] . k

F o r a spherically s y m m e t r i c gap the angular average of ½(k × fi)2 is equal to the angular average of (k'8) ~, which means

(5.s)

1(0) _ M,(0), Io

nm

where Io is the rigid-body m o m e n t of inertia. F o r an ideal gas, therefore, we h a v e I(0) = Io. This result for the spherically s y m m e t r i c case has been o b t a i n e d previously b y m o r e tedious m e t h o d s (ls,19). The statistical approach e m p l o y e d here is more a t t r a c t i v e because it emphasizes the role of the energy s p e c t r u m of the system. I t is p a r t i c u l a r l y easy to a p p l y to quasi-particle models, which encompass a large class of a p p r o x i m a t i o n s to the m a n y - b o d y problem. F o r the a s y m m e t r i c case [E= E(k, 0)], eq. (5.7) m a y be t r a n s f o r m e d to (5.9)

I(0) _ K2, Io

where K2 was defined b y eq. (4.12) and (4.13). This result is easily u n d e r s t o o d b y recalling t h a t , for the case considered in this section, the flow velocity is always perpendicular to the quantization axis. Hence we expect t h a t eq. (5,8), originally w r i t t e n for the spherically s y m m e t r i c case, should now be valid when we use eq. (4.11) for Mf(O)/nm with 3 = z/2. I t m u s t be emphasized t h a t the results of this p a p e r are b a s e d on the quasiparticle a p p r o x i m a t i o n and t h a t the interaction between quasi-particles has been ignored. These interactions m a y be i m p o r t a n t for the calculation of the m o m e n t of inertia (19), b u t the investigation of their effect has not y e t been completed. Similarly, the p r o b l e m of viscosity has not been discussed. H o w ever, we do expect the viscosity to vanish at low t e m p e r a t u r e s in the limit of small flow velocities. This follows f r o m the fact t h a t in this limit only a v e r y limited class of excitations are possible in view of the modified energy (is) R. D. AMADO and K. A. BRUECK~P.R: Phys. Rev., 115, 1778 (1959).

(19) R. M. ]~OCKMORE: Phys. Rev., li6, 469 (1959), and private communication.

FLOW PROPERTIES

OF S U P : E R F L U I D S Y S T E M S OF F E R M I O N S

737

s p e c t r u m in t h e s u p e r f l u i d s t a t e . I n a n y case t h e v i s c o s i t y s h o u l d b e d r a s t i c a l l y r e d u c e d b e l o w t h e v i s c o s i t y in t h e n o r m a l fluid w h i c h , in t h e l i m i t T - ~ 0, v a r i e s as T -2 (P). A c c o r d i n g t o ANDERSON a n d MOREL (~6), t h e m o s t r e m a r k a b l e p r o p e r t y of t h e s u p e r f l u i d p h a s e is i t s i n t r i n s i c a n g u l a r m o m e n t u m . T h e y t h i n k of t h e a n i s o t r o p i c p a i r c o r r e l a t i o n in t e r m s of s m a l l c i r c u l a t i n g c u r r e n t s w h i c h c a n c e l e v e r y w h e r e in t h e i n t e r i o r of t h e f u i d b u t le~d t o a n o n - v a n i s h i n g c u r r e n t on t h e fluid surface. T h i s c u r r e n t is p r o p o r t i o n a l t o t h e s u r f a c e are~, a n d h e n c e i m p l i e s a t o t a l a n g u l a r m o m e n t u m p r o p o r t i o n a l to t h e v o l u m e . This a n g u l a r m o m e n t u m is, h o w e v e r , so s m a l l as to d e f y e x p e r i m e n t a l o b s e r v a t i o n . E v e n a t e x t r e m e l y low t e m p e r a t u r e s , i t is a l w a y s v e r y m u c h less t h a n t h e a n g u l a r m o m e n t u m c a l c u l a t e d h e r e . l q e v e r t h e l e s s t h e d i s c u s s i o n of ANDEtCSON a n d MOREL rasises a n i m p o r t a n t q u e s t i o n of p r i n c i p l e . T h e w o r k of this p a p e r does n o t t o u c h on t h i s p o i n t b e c a u s e i t is b a s e d on a m e t h o d a p p r o p r i a t e o n l y t o a n i n f i n i t e s y s t e m , a n d h e n c e does n o t i n c o r p o r a t e a n y s u r f a c e c u r r e n t p h e n o m e n a . A r i g o r o u s t r e a t m e n t of b o u n d a r y effects a p p e a r s to b e necess a r y for a n u n d e r s t a n d i n g of t h e s e effects.

The authors have been helped by conversations wiry numerous colleagues a n d owe s p e c i a l t h a n k s to D r . P . W . ANDERSON, D r . V. J . EMERY a n d P r o f e s s o r 1~. L..~IILLS for t h e i r c o m m e n t s . (a0) A. A. ABRIKOSOV and I. M. KHALATNIKOV: SOY. P h y s . U s p e k h i , i, 68 (1958).

RIASSUNTO

(*)

Si dh una interpretazione fisica in termini di un n~odello di fluido anisotropo alle sotuzioni sfericamente asimmetriche della tsoria, di Bardeen-Cooper-Sehrisffer. Quests soluzioni sono state precedentemente usate da EMi~RY e SESSLER e d~ BRUECKNER, SODA, ANDERSON S MOREL per predire una transizione di fase nel 3He liquido. Si esegus una rieerea delle earattsristiche di fiusso di tali sistemi, ehs comporta il ealcolo della massa sffettiva per il flusso in un canale diritto e del momento d'inerzia di un cilindro sontenente il liquido. I1 gap di energia dipendsnte dall'a.ngolo caratteristico di questo tipo di teoria p s r t a ad una massa effettiva per il flusso she dipende dall'angolo fra Passe di simmetria del fluido e la direzione del flusso. La massa effettiva per il flusso si annulla al tendere a zero della, temperatur~ ~ssoluta, anehe se non ~ltrettanto rapidamente ehe per un gap a simmetria sferiea. Quando la dirszione di simmetria del fluido coincide con l'asse di rotazisne, il momento d'insrzia sta in un rapporto semplics csn la massa per il fiusso. (*) Traduzione a cura della Redazione.