dynamical effects in superfluid helium 4

D Y N A M I C A L EFFECTS I N S U P E R F L U I D H E L I U M 4

MUBARAK A H M A D

Department of Physics, Kashmir University, Srinagar-6, India

The close structural similarity between the commutation relations of harmonic oscillator operators and the operators for Bose fields is exploited to study the excitation spectrum in superfluid helium 4. By applying 'broken symmetry' condition it is shown how the creation of phonon gives rise to superfluid behaviour of liquid He 4. The energy gap needed for roton excitation is derived.

1. INTRODUCTION

This paper deals with the theoretical investigation on the problem underlying the description of the energy spectrum of superfluid helium 4 by exploiting the dynamics of the system. The aim of this paper is two-fold 2) to show how the creation o fphonons and rotons can be described microscopically and how these two energy spectra can be correlated; 2) to derive expression for the energy gap needed for roton excitation.

2. BOSE OPERATORS

For these purposes we define the helium field by the operator

_ 1 f d a k a k exp ( - i k r ) (1) ~/(r) (2n) 3,2 ,

where k is propagation vector.

The creation and annihilation operators ff § and ~O obey the equal time commuta-

tion relations as follows

(2) [~(r), ~ +(r') = r -- r ')

suppose that ~bo(r ), 4 ( r ) , . . . , form a complete orthonormal set of single particle states. It is occassionally convenient to expand 0(1") and 0 +(r') in terms of the operators al, a + that removes and adds particles with the wave function ~b~(r). These

expansions are

(3) O(r) = Fd,,(r) a , - O + ( r ' ) = E~p~'(r ' )a + . i i

The operators ai and a + obey the following relation

(4) [a, , a + ] = 5 u .

The structure of this equation shows that each normal mode has the same quantum mechanical structure as a simple harmonic oscillator [1]. Each such oscillator has

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Mubarak Ahmad: Dynamical effects in super fluid helium 4

its own Hilbert space with ground state characterized by

(5) ak Io> -- o

(6) <010 ) = 1.

The Hilbert space of entire field is the direct product space of oscillation. The energy levels of a three dimensional harmonic oscillator are known to be highly degenerate, which can be seen by analysing the problem in three-dimensional cartesian coordinates. Again the three-dimensional harmonic oscillator, which is also solvable in spherical coordinates, has the characteristic degeneracy of rotational invariance. In this case the states can also be classified by taking the eigen-functions of the orbital angular momentum. Thus the states in such classification are related to those in angular momentum classification. In both cases the angular momentum operator 13 is a good quantum number [2]. As is well known the symmetry algebra of the n dimensional harmonic oscillator is given by the symmetry group SU(n) [3] and from the above argument it is evident that its generators can be expressed in terms of n 2 bi-linear products.

3. O S C I L L A T O R O P E R A T O R F O R C O N D E N S E D BOS ONS

The crucial feature of superfluid He 4 is the existence of a condensate, having a single mode, macroscopically occupied. These zero momentum states are described

+ by the operators a,o, aye. In a classic paper on interacting bosons BOGOLIUBOV [4] proposed that the second-quantized Hamiltonian can be simplified, so as to take advantage of the assumed Bose- Einstein condensation by replacing a~+o and a,o

+ by a c-number Qo = a,oa~o. But for the purpose of getting an existence criterion for Bose - Einstein condensation, the operator a~o was replaced by EZAWA et al. [5].by the operator ave ~ 0o + a~o, where a~o is a new annihilation operator intended to describe small oscillation for the condensate.

The operator a~o and its adjoint a~o + obey the following commutation relations

(7) [a,~'o +,a(o ] = 5 , o , o = 1 when #o = % -

Conditions imposed on a(o are: while the shift of equilibrium point Oo is macroscopic, the matrix elements of ale are not proportional to the size of the system V, i.e. it is approximated that the amplitude of the oscillators represented by a(o remain finite and small even when the system becomes infinitely large, i.e. when V--* oo.

4. S P U R I O N S A N D A S Y M M E T R I C R O U N D STATE

This operator a~o occuring in the modified operator of Ezawa, may be of quantum origin or it can be due to interaction between zero momentum particles and the walls of the container. As these small oscillators of the condensed state and the creation

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Mubarak Ahmad: Dynamical effects in super fluM helium 4

of phonon, resulting in this phenomenon, have the character of harmonic oscillators, we can well understand how the asymmetric ground states are created in superfluid system by drawing an analogy between the two systems. The bilinear operators appropriate for the system a.+,aj transfer a quantum from j direction to i direction. As one of the preferred directions can be picked up, asymmetric ground state can be understood by the behaviour of bilinear operators operating on superfluid system. To visualize the situation intuitively, the creation of massless boson, in our case the phonon, results in macroscopic asymmetry of superfluid helium-4. These phonons are interpreted in the language of long-range correlations. The phonon excitation in superfluid have vanishingly small probability at slow superfluid velocities so that it has small probability of disturbing the a(o oscillation.

Therefore, under such conditions, we can correlate these excitations at low energy to the oscillatory model of harmonic oscillator. Once this spurion (phonon) gives rise to the asymmetric ground state the symmetric quantum number of the ground state is carried away by such spurion, leaving the quasi-particles to be frozen. Accord- ing to UMAZAWA et al. [6] asymmetric quantum number of this basic field is dynamic- ally rearranged on the asymmetric or physical fields which describe the quasi-particle or a collective excitation of the system. In order to accommodate all the excitations of superfluid heluim-4, either we have to break the symmetry or we have to extend the group representing the ground state to an extended noncompact group related to it. At the moment, we want to deal with the spectrum of phonon and roton, which features predominantly in the theory of LANDAU [7]. According to his hydro- dynamic theory for superfluid He 4 the phonon region ranges from p = 0 upwards, and the roton region falls around p = Po. Accordingly, the whole spectrum of excitation of liquid He4 can be imagined as two parts of one and the same excitation spectrum. Phonon branch is rotation free and roton branch consists of rotation motion, which requires a non-zero excitation energy A called energy gap. The roton branch at very low temperature loses completely its rotation character and may be assumed to have a characteristic short wave-length elastic mode-type which is totally degenerate. But as the temperature of the system increases the roton spectrum becomes predominant and exerts its characteristics. Thus at very low temperature we have only the excitation mode of phonon, having an oscillatory character.

5. P H O N O N S T A T E

The superfluid behaviour of liquid helium 4 at low temperature is caused by the phonon spectrum, which can be treated as a dynamical effect, creating gapless excitation in the limit of vanishing momenta. In order to understand this particular phenomena, broken-symmetry condition can be applied. Mathematically, it can be said that, by imposing an asymmetric condition on the ground state, the phonons can be produced as mixed states composed of parts with broken symmetries.

The discussion on the subject of broken symmetries in field-theory had led to

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Mubarak .4hmad: Dynamical effects in super fluid helium 4

a remarkable theorem stated by GOLDSTONE [8], according to which there is always a zero-mass particle associated with broken symmetry of a continuous group. In the particular case of liquid He 4, the same theorem can be exploited to study, how the onset of phonons in liquid He 4 gives rise to its superfluid behaviour at very low temperature. Mathematically this can be stated as follows: if continuous symmetry of the Hamiltonian is not dealt with by the ground state, a gapless excitation is produced, i.e. there are states whose excitation energies E(k) vanish for momenta k --, 0.

In order to show this, the continuous symmetry group under question can be represented by unitary operators u(r) in N particle Hilbert space. The groups that can be considered in the particular case are:

Gauge group defined by u(r) = e i'N Translation group u(r) = e irp (translation about r)

The generators N and P are the particle-number and momentum, respectively. The Hamiltonian H is assumed to be invariant in the ground state

i.e. u+(r) Hu(r) = H .

In order to correlate the picture of broken symmetry with the physical situation in liquid He 4, the idea of Bose - Einstein condensation should be invoked. Because this degenerate condensed ground state enforces the existence of masslcss bosuns (in the present case the phonons), which breaks the underlying symmetry of the system. If the ground state is degenerate, the underlying Hamiltonian becomcs invariant to the above symmetry operations, giving rise to a conservation law. These transfor- mations change the ground state into different possible ground state. However, in the case of any dynamical effect taking place between two particles or subsystems in the ground state, in the form of creation of phonons, the above conservation law seems to be violated thereby breaking the above symmetry.

To understand how this happens, the canonical commutation relation (2) is written as follows:

[~b(r), ~ +(r')] = 6(r - r')

Bose- Einstein condensation occurs when

;01 (r)10> = . , 0,

where (01 denotes the ground state; a denotes B. E. condensation, which is space-time independent c-number.

Hence, the Bose operator is written in the form

(8) = O~ +

with the condition that (01 O~ -- 0 .

] 130 Czech. l- Phys. B 25 [1975]


The field ~b~ describes the excited state. ~b~ contains the creation and annihilation operator of a massless physical particle.

The above condition [8] may be looked upon as a transformation which becomes canonical if ~b~ is a boson field.

As ~k and ~b ~ are both Bose operators, equation (8) can be imagined as an outcome o f a unitary transformation of the type

= +

If a is restricted to the local field then (9) can be represented as an infinitestmal unitary transformation in Hilbert space

i.e. ,~,(r)=~,=[~,(r) ,~, f~,+(r ')d3r'] .

By equal-time commutation relation, one gets

(I0) [~,(r),,,f,/,+(r ') dar'] = cr fv[~k(r), O+(r')l dar' = a.

The decomposition of ~b+(r ') in terms of its normal modes can be written as:

1 (akelk.,,_ilkl,+ a+e_ik.,,+ilklt). ~+(r') = ~V ~2 ~

In terms of this, I~+(r ') d3r ' takes on the form

(11) f~k+(r) d3r=lim~(k~2V)(ake-m-*-.o a~'eik') "

With this relation, the equation (10) is not satisfied. The contradiction between (10) and (11) resembles the broken symmetry .condition found in the theory of quantized field [8] From this analysis it is evident that whenever k --* 0 mode is produced i.e. phonons are formed as in the present case, the system does not exhibit all the symmetry properties of its underlying Hamiltonian and the symmetry becomes broken.

6. ROTON STATE

In order to correlate the phonon spectrum with roton state we choose the bilinear operators aiaf due for the phonon spectrum and construct out of them the angular momentum operator as follows

L,j = i(aia + - aja+).

We consider a set of eigen-states of ground state [0> of liquid He II having Lij = O.

Czech. J. Fhys. B 25 [19751 1131


F As there exist some levels composing the small oscillations represented by avo, we can represent the multiplets associated with this oscillation by ( 2 / + 1) states of well defined angular moment states .We create the roton state from the ground state [0), by the means of a operator T~

(12) T,+.,I~ = I * ' > �9

Ilm> is the well defined angular momentum state, formed out of eigen-values of operators of 1 z and 13.

The (2/ + 1) quantities T~+,, with the spectrum conditions - 1 < m < l transform under the rotation of the frame of reference as components of a tensor with rank I.

This excitation spectrum, as in the case phonon spectrum, is understood only in the limit V ~ oo or k --* 0. Here again a condensed asymmetric ground state is to be recognized for creation of roton. But when we go out of the volume limit, we are going out of the Fock space wherein our original operators were defined. In order to circumvent this difficulty BARDnEN et al. [9] derived a new ground state for superconductors by introducing the novel idea of pairs in the Bloch states, each pair having a certain probability of being occupied. We adopt the following formalism to circumvent the difficulty.

7. VOLUME LI'MIT

The above T operators related to the original operators by the following relation, recognized in the limit

T = T + = I(T~ + + T+~) d3x d

03)

where

(14)

This operator is recognized in the finite volume V

(15) T ; = Tv = fv(Ta+ + T+Q)d3x

and this Tv is a well defined Hermitian operator, though time dependent. In order to formulate a theory that can be put on sound footing, without going out of Fock representation of free physical quantum we recognize the rotational invariance for the production of rotons from the existence of condensed degenerate ground states that are distinguished from each other by the change of their phase therefore for each specified phase there exists the corresponding ground state and to each ground state is associated a Fock representation. But any specific mode of phase in the ground state means the ground state is asymmetric. Thus in the roton sector, we have a con-

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M u b a r a k Ahmad: Dynamical effects in super f lu id helium 4

densed asymmetric ground state. Now the invariance of the rotation transformation can be recognized by covering all the Fock representation by rotating the container of liquid He II itself or the same invariance can equally be obtained by rotating the axis of rotation, as is done here. This transformation is a passive one. This can be performed in any Fock representation in which ip is defined. The rotation of the entire vessel is an active transformation, but as the rotation of the entire vessel, when V ~ oo has no physical significance, the passive transformation is adopted.

(16)

(17)

(18) (19)

Here

(20)

(21)

8. C A N O N I C A L C O M M U T A T I O N R E L A T I O N S

After the preliminary remarks for the justification of the above model the following equal-time commutation relations between the operators T and I are defined:

[1+, 7"/+] ' T + = bm+l l , m + l

[/_, r,+.,] , = bmTt ,m_ 1

[ix, r,+.] = mr,: [12, ~,+] = ix,, r , ]

(22)

(23)

(24)

b~ is defined by the relation

/+ = /1 ---+ /2,

T, + = T,,.,

/+ and /_ are shift angular momentum operators

(21 + 1) x (2t + 1) matrix It has the following non-vanishing matrix elements

I , . , , , = 2 m 1 3 - m 2 + m - ( b ~ ) 2

, = b m + l l _ , Im m + l 1

I m m - l . = blml+ ,

(25) bl = 4 [ ( / + m ) ( l - m + 1)] .

In the operator T~ + all the states appropriate near p - Po are incorporated. T~ + and T/m are Hermitian conjugates.

As these operators have the rotation character, the direct product of two irreducible representations of the rotational group give rise to the following states [10]:

(26) ,,,2 1112/ , ] / , - mS+ + 11--12<l<11+12 \mlmlm/

Czech. J. Phys. B 25 [1975] l 133


(27) Tlzm, Th+.,l [O > h h \ m t m 2 m / [[1211] It -- m>_+

(28) T|2ra2TllrnllO) = ~/e|tml l[/2lt] 11 m > + _ . \ m l m 2 m /

Here a, b, c are numbers depending on l~, 12, l and l[12ll] tin> stand for the wave function with definite angular momentum quantum numbers.

For roton spectrum, the states as defined on right-hand side of equation (12) coin- cide with right-hand side of equation (26)-(28). As for the states of other modes like phonons or vibratory states these states are orthogonal.

Now we impose the following subsidiary condition on the operator T

(29)

with

(3(3)

and

(31)

T + T + T+ T + =Zalh,~( 11 lz l~Ttm 12m2 llml ~ Jt llttll /2rn 2 lm \ m l m 2 m /

' (2l + 1 ) { ~ 1 2 1 ~ •lll2 = 0 0 J

Tim = ( - 1)" T , ; .

Equations (29), (30) and (31) imply that the operators Tl, ~ have the form of spherical harmonics of some collective angles.

It will be seen that these operators which excite roton modes in the superfluid system, yield an energy-gap, thus giving the same physical situation in the dynamics of superfluid helium.

9. ENERGY GAP IN ROTON SPECTRUM

Landau in his paper (1941) [7] imagined some sort of rotational or vortex motions in superfluid helium. But his ideas were based on macroscopic theory and no elemen- tary quahtized excitations of roton which are unique to He I1 were postulated. This lack of quantum version reflects itself in the postulated energy spectrum which Landau proposed viz.

p2 (32) ero t = - - + A.

2~

Here A is the postulated energy gap between the roton and phonon ground states. The ensuring ~rot/~ fraction will decrease exponentially with temperature and becomes negligible at lowest temperature. As temperature increases a n d near p ~ Po, the excitation of the rotons exert themselves their peculiar characteristics. In addition to this non-quantum version of the roton spectrum, the proper statistics of this

1134 Czech. ]. Phys. B 25 [1975]


spectrum is also not decided, due to the uncertainty of the model. Owing to the presence of large energy gap A in the energy term, some Boltzman-like statistics was assigned to this spectrum. But, as discussed before, both the phonon and roton have been treated as two branches of the same excitation. Therefore, there is no justification to give a classical statistics to an excitation whose dynamical behaviour is of quantum origin.

In order to get the energy gap from our model, let us split the Hamiltonian of superfluid helium into two parts:

(33) H = H i n t + I L 2 = H i n t + hro t .

We assume that the intrinsic Hamiltonian Hin t is angular momentum free, therefore its eigen-values and eigen-function do not depend on the quantum numbers of hro t which governs the roton spectrum. With this condition the spectrum of the system will be purely rotons, built upon the states of intrinsic Hamiltonian. Hence I, the moment of inertia is introduced in the 2nd term of (33).

10. C A L C U L A T I O N F O R E N E R G Y G A P

We presume that the Hamiltonian H is preassigned and therefore write it in the form:

(34) H = T + V = H o + v ,

where T is kinetic energy, and V is potential energy. H denotes a one-body operator chosen in such a way that the remaining part v = H - H 0 can be iterated. Ho has the axial symmetry and therefore the angular momentum quantum number can be assigned to it,

(35) i.e. /3lk> = m, ik>

where Ik> are eigen-functions of H o and m k are the eigen-values of 13.

(36) H o l k ) = w k ] k ) ,

wk is the eigen-value of H o.

The equation for roton excitation follows immediately from the commutation relation between H and Tl+,,.

Making use of equations (33), (19), (20)-(25), we find

(37) [H, Tt +] = [hrot, Tt +] = I ( I t . T~+).

Evaluating the matrix elements of (37) between eigen-states we get

(38) (wk -- Wm)(Tt+)km +<k[ [v, Tz +] [rn) = I ( I , . Tl+)km .

Czech. J, Phys. B 25 [19751 1135


This equation is written in the form:

(39) (Tt+)kra = I < k l I t . T f f lm> for W k - - W m

The matrix (k I I t . T,[rn) satisfies the equation

(40)

k e e m .

<kl~. rt+lm> = <kl*. r,+l~> - E n # k , m

{ ' t I I ' } vk. < , I . r+l~> - < k / . r+m> v.m �9 W n - - W t W k - - W m

Diagonal matrix elements of T~ are not defined by equations (39) and (40). We impose the condition that equations (16)-(19) are satisfied.

From equation (29) we infer that the normalization condition on Tcan be written as

l

(41) E E (T+-,.)Ok (T,m)kO ( - -1 ) = = 1. k m = --1

O denotes a particular eigen-state of H o which is associated with a ground state of superfluid He 4.

Equation (30) implies that the ro ton can be expressed in terms of three operators Tt+,,, which give the standard representation of rotation group. We write in the form:

it1 for m = - 1 (42) Tt + = I + to for m = 0

- i t x for m - 1.

Here t 1 and t o are hermitian.

If the effective interaction on Tt + are not very long and the energy occuring in equation (39) and (40) is large compared with the first excitation of a roton state, then the norm of tl and to is small, i.e.

(43) Zl(q)o,I 2 ~ 1 ; 210o)o,I 2 ~ 1. n n

Therefore, to define matrix elements of (k[ I . t, I m> in the first approximation we can drop components of t~- = t + including to and t 1 out of the first term on the right-hand side of equation (40).

This gives

(44) <kit. Tit> X ~. < ' q t ' T ? l l > - - < q l ' r + l " > - - ~ . , = n * k , l W n - - W t W n W 1

l for m = - 1 = 21 0 for m = 0

/ l+ for m 1 L

1136 C z e c h . I. P h y s . B 25 [ l g751


From the equation (44) and (34) we get

(45) f ( l _ ) k l / ( W k - - W l ) f o r m = --1

(T,, .), , + = ~/(2) I ~0 for m = 0 [(l+)kt/(Wk W,) for m = + 1 .

The equations (16)-(18) show that the excitation for a roton resembles that of a rigid rotator. This can be understood when we know that the rotation of the superfluid is maintained by the establishment of quantized vortex lines. The phonon branch of superfluid moves irrotationally, the curl of the fluid velocity is zero except within the core of vortex lines. Such vortex cores are composed of normal fluid. This quantized vortex core can be assumed to be due to the dynamical effect produced by the roton operator. Thus the mixture of superfluid and quantized vortex lines can mimic rigid body rotation. Since the cure of the fluid velocity averaqed over many vortex lines is equal to twice the fluid angular velocity [11].

In the light of this physical situation, taking the big components in equation (16) to (18) which are the diagonal matrix elements of equations (16) and (17) and averag- ing with the ground state wave function 10) we get

1 (46) x/2 (01 [I+, T__+l] 105 = 2Ik,o2 ( l + )ok ( l_ - Wo

This gives a model for the moment of inertia. We now take the ratio of non-diagonal to diagonal matrix elements of the operator

1 (l+),k(t_)~,+ (l+),~(t_)~j (47) ~ ~ , wq ~ w7 wk - w,

k# j

This is very small.

The right value for the moment of inertia is fixed by the condition (46).

The norm of the roton exciting operator is obtained as follows. We have reached its value, satisfying the physical situation. It is written as

1

(48) 1 = <o r 2 ( -1 ) m r:r-mro> = 1 - 2i(a) -1 , m = - - 1

where

(t+)o~(l_)~o / v (l+)o~ (t_)~_o (49) A =k.O2 Wk -- WO /k,O" (Wk -- Wo)2 "

The energy denominator in (48) is not large, therefore the excitation energy for the roton spectrum is spectracular; the second term containing the energy gap A can not be neglected.

C z e c h . L P h y s . B 25 [19751 ] 137


11. DISCUSSION

In the above formalism for the theory of excitation spectrum for liquid He 4 ive have shown how, starting from the bilinear operators, we can generate the smaU oscillations in the condensed ground state and how it gives rise to the vibratory phonon modes which, breaking the symmetry of the ground state, transfer it to an asymmetric ground state. These spurions are necessary for the onset of superfluid property of the liquid helium 4.

By imposing the subsidiary condition on the operator for the small oscillation in the ground state of condensed boson to be finite throughout the volume, by choso- ing certain commutation relation between this operator and the roton-producing operator T, we obtained the physical nature for the roton states, which has been performed by making use of the irreducible representation of the direct product of two representations of the rotation group. By applying this model in the relevant Hamiltonian we have shown these operators gave the energy gap, appropriate for the production of rotons in superfluid He 4.

The author is thankful to Prof. N. N. RAINA, and Prof. R. S, SHARMA for constant encourage- ment.

Received 25. 11. 1974.

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[1 ] BARTON C., Introduction to advanced field theory. Intersc,, Publ., New York 1963, p. 118. [2] LIPKIN H. J., Lie group for pedestrians. North Holland Pub. Co., Amsterdam 1965. [3] PAULI W., Z. Phys. 36 (1926), 336.

FOCK V., Phys. 98 (1935), 145. BAROMAN V., Z. Phys. 99 (1936), 576.

[4] BOGOLIUBOV N. N., J. Phys. (USSR) 11 (1947), 23. [5] EZAWA H., IUBAN M., J. Math. Phys. 28 (1967), 6. [6] LEPLAE L., SEN R. W., UMAZAWA R., N. Cimento X, 49 (1967), 1, 35. [7] LANDAU L., Zh. Eksp. Teor. Fiz. 11 (1947), 91.

LANDA~J L., Eksp. Teor. Fiz. 5 (1951), 71. [81 GOLDSTONE J., N. Cimento 19 (1961), 54.

GOLDSTONE J., SALAM A., WEINBERG S., Phys. Rev. 127 (1962), 968. BLUDMAN S., KLEIN A., Phys. Rev. 131 (1963), 2364. JONA-LASlNIO G., N. Cimento X, 34 (1964), 1790. STREATER R. F., Phys. Rev. Lett. 15 (1965), 475.

[9] BARDEEN J., COOPER L. W., SCHRIEFFER J. K., Phys. Rev. 108 (1957), 1175. HAAG R., N. Cimento 23 (1962), 237. TAKASHASHI Y., UMAZAWA H., Physica 30 (1964), 49.

[101 HAMMERMESH M., Group theory and its application to physical problems, Addison Wesley, Reading (Mass.) 1962.

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