R EhbKIC/88/413

INTERNAL REPORT(Limited distribution)

?VP(J j International Atomic Energy Agency

United Nat^n^pfofucatlonal S c i e n t i f i c and Cultural Organization

ATIOHAL CENTRE FOR THEORETICAL PHYSICS

SYMMETRY BREAKING PHENOMENA

IN LIQUID HELIUH-A *

Mubarak Ahmad **

International Centre for Theoretical Physics, Trieste, Italy,

S.K. Tikoo

S.P. College, Srinagar, (Kashmir) India

and

T.K. Raina

Government College for Women, Srinagar, India.

ABSTRACT

Salient features of liquid Helium II are being analysed In the

light of symmetry breaking wechanisn. The excitation spectrum 1* derived

by scaling transformation. A relation between roton excitation and

Bogoliubov's quasi particles has been established by giving different

structures to the roton operators,similar to quasi boson approximation for

phonon-like spectrum. This structure of the roton operators enables us

to study the low temperature behaviour of superfluid He Hum-4 by introducing

the idea of pairing for roton-like excitation for the on set of energy-gap.

Superflutd He^ is divided into the states of condensed bosons,

oscillating condensed bosons, phonons, maxons, rotons and the states of

vortices. We consider the transition probabilities between these various

states.

Coherent states of various superfluid regions of He II are being

obtained and consistently analysed which have deep inter-relatlonshlp. As

higher excitations have also a group structure and their coherent behaviour

has been seen, this formalism seems to be more appropriate to deal with such

an interesting problem.

HIRAHARE - TRIESTE

December 1988

* To be submitted for publication.

** Permanent address: Institute of Kashmir, Srinagar, India.

- 1 -

I. Introdugt ion

Liquid helium because of its weak Interactive forces

and large zero point energy remains t liquid «ven in the

vicinity of absolute zero; gets a title of quantum /luld

which has a phase transition at 2.17k into two phases-

He-I and He-II. Helium-II behaving as a superfluid with

almost rsro flow resistance. The transition temperature

showing a merited anomaly in specific heat lends It the name

X point. Many Investigators visualized the transition

phenoaenon in different ways on the strength of their

While London l

interpenetrat ing

al fluid). Landau suggested

experimental and theoretical investigations,

visualized the two phases as two

fluids (superfluid and nor

theoretical basis giving the phase-I a status of elementary

excitation* named Phonon (P - 0) and Rotons (P • PQ) in

the background of a superfluid (Phase-II), the parameters of

the excitation energies of which were deduced from the

macroscopic properties like Sp.heat, first and second sound

through the liquid. Bogoliubov derived the phonon-spectrum

by replacing the number operator for particles in £ero

momentum state by a c-number. Several other modified

methods were used by many authors " for computing the

spectrum and

discussed in

other associated

Section III.

properties. This aspect is

Symmetry, a fundamental attribute of the natural

world helps in the study of the particular aspect of a

physical system. Symmetry and Invariance are synonyms in

physics, since the former concept relates more to the

intrinsic structure of nature while the latter to the

nathesutical form of the equations of motion. In many

body theory we find a non-invarlance of the entire Hilbert-

•pace of the system as a macroscopic observable which

behaves classically and corresponds to a non-Invarlance of the

-2-

14ground state. This ido first introduced by Heisenberg and

latter by Nambu and Jona Lasiniu, asserts that ths ground

or the vacuum ^tats might be 1 ems synmetric than tha basis

dynamical lawsi.e the -field equation. As a consequence it

lead to a remarkable theorem by Golds tone According to

which thero always is a zero mass particle associated with

•i broken symmetry of a continuous group. The significance

of the rglativistit theories though obscured by many m

facet is still a matter a* rigorous study after the

discovery of hadrans containing heavy quarks. ,A

non-relativistic analogue of this theorem implies that if a

continuous symimtry of the Hami1toni#n is not dealt with by

the ground state or the representative ensamble in the case

•f non-zero temperature, a gaplost excitation exists,i.a

there are states whose excitation energies vanish for

momentumt

Super-fluid helium does not exhibit all the symmetry

properties of its underlying Hami 1 tan tan, however in

general it is invariant under a gauge transformation of

first kind i.e it is particle number conserving. Evidently

superfluid He-4 may be characterised by the lack of gauge

invar ience or the lack of conservation Of particle number.

A special version of Goldstone theorem, theO

Hugenhultz-Pines theorem with interactions of finite range18 19

was verified by Lange as also by Katz and Frishman, but the

energy Spectrum determined with self consistent method of

first order exhibited a gap, contrary to Hugenholti

theorem. It was found however that such a gap was on

account of the approximation not taking a satisfactory view

of the long range ordering. Tha gap was however shown to

vanish under certain specified conditions. This symmetry

breaking aspect is discussed in the section-II of the

report.

The introduction of a weak external field in the23J4

system breaks the phase symmetry of the helium system, and

resulli. in thu translation of th» lioson field operators and25

iirtejt. i on of tohtr ent £»upe?rpoii>i t: i on of states. Coherent

»t .iti?s find an application in the theory of superfluidity

t)E?t.;t3u?5fr,» tht»y at & wel 1 suited to describe a system of

interacting particles whose low energy excitations are

-3-

base-modes and have a large occupation number. These

cohirint *tat» have bien i n n to behave classically. Thus

coherent states play the part of classical -fields which

describe * *et o-f many boions » a whole, just as the

classical electrodynamlc field describes the classical

limit of the quantum electrodynamics. In the theory of

liquid helium classical description implies superfluidity.

Again, the thermodynamical expectation value of the Boeon

field operator is conventionally called as the nrder

parameter 'f' for the super-fluid helium-4 and the

super-fluid phase transition is usually associated with the

loss o-f symmetry, which allows the expectation value to be

non-vaniahing. This order parameter y has been

conveniently chosen to be the coherent state representation27

of pure quantum states by many authors . This is discussed

in section-V.

In the light of the symmetry breaking the explicit

forms of the transition probabilities between various

states such as the condensed state, the oscillating

condensed boson state, the phonon states and the roton

states of superfluid He—4 have been discussed in

section-IV,. Approximation method making u u of the

Bogoliubov'a canonical transformations has been used for

transition between various states. From the forms af

transition probabilities, it has been visualized that these

states are all the probable ones.

II. SYMMCTKV MftKINB AND SUPERFLUIDITY.

Superfluidity is the outcome of the phase transition

( X-transition! which is characterised by a change? of order

in the sense of off -di .igonal long range order (ODLRQ) in

the particle density matrix. This ^ -Transition beinq a3S-34

second order phase transition is associated with

of the sysmmetr y of the Lagranqi art uf system and disci with

55!the appearance af the breaking symmetry co-ordinate 5!.a.

-4-

number of particles in the condensed phase of tho36

Superfluid. The broken symmetry co-ordinate is associated35

with the order parameter of the condensation. As the order

parameter is non-vani9hing due to the loss of symmetry

associated with the superfluid phase transition, mo a

broken symmetry state can be distinguished by the

appearance of a macroscopic order parameter. The

macroscopic order parameter obeys the 1 awe of classical

mechanics, and determines a continuum disjoint Hilbert

subspaces of the full Hilbert space of the system. The

breaking of the symmetry establishes multiplicity of the

ground states related by the transformations of the broken

symmetry group. The multiplicity of the ground states is

linked with the degeneracy of vacuum,thus,to the concept of

non-invariance of the vacuum and is a common feature of the

broken symmetry co-ordinates. The Hamiltonian of

euperfluid helium is invariant under gauge transformation

at first kind or invarient under the phase transformation

or is- particle number conserving. Thus the specific

symmetry which is broken in the system is the gauge

invariance or the particle number conservation or the phase

invariance. Accordingly the broken symmetry groups under

consideration are, the gauge group defined byi u («O = e*'

and the translation group1 defined by u(*O »= e ' where *N

and P *ra the particle number and momentum respectively.

The breaking of these symmetry groups is related by

Noethers theorem to non-conservation of a physical

quantity acting as the generators of the group (N or P).

From the statistical point of view thi* means that the

system is described by a density matrix jj which although

commuting with the Hamiltonian 'H* of the system, does not

commute with this group generator N or P the generator

however commut es wi th H.

[ V . H > O , (V,N] 1 0 Cf.rO f O[ N , H J > [ P , H ] =0 2-1

These ccjndi t inns are outcome of the condition that the 35order parameter is ram.vanishing when the symmetry breaks.

Thus i F the order -parameter 13 non-vanishing, the

Hami1 ton tan is invariant under the symmetry

tr *n«f or-mati tm»N - jf(x> d>: , but the density matriM is not

and we- have A broken symmetry.

-5-

For tha spontaneous symmetry breaking, tha

non-invariant macroscopic obaarvabla can ba described by

Mint of tha concept o-f quasi Average* introduced by

Bogoliubov, who introduced into Wie treatment an infinitely

weak axtarnal Ilaid which break* the symmetry of tha

system. For this purposa a niM tarn T)tv is lntroducid

into tha Hamijtonian^whereV1* a small parameter and h is

defined by

;3

a ek

2-2

Tha naw Huiltonlan H - na longer commuta*Hith the synMtry operation bacause of the unsymmetrical

added term. In this casa tha average of a dynamical

variable equal to zero for t> s 0, will ba different from

zero for V f O and it can remain diffarartt from zero even

in tha limit V — > 0 . Tha procedure of taking tha

limit V — >O is equivalent to the choise of the ground state

or what is the same thins, a concrete value of the

macroscopic parameter characterizing the representation. So

this state |V>is defined by *

4The state |V> is taken to be the state of lowest energy

of H^ - H - * h at finite uirunder the constraints <V|<g|V> -N

There is degeneracy of |V> with all states \< ,•>>> «e*<<'* )l?>

and this degeneracy of the (approximate) ground state is

taken synonymously for a broken symmetry.40

Ichiyanagi discussed the microscopic background of the

ansatz that superfluid flow i« the manifestation of the

local gauge invarianee and showed that the depletion

velocity fields are the Goldenstone fields due to

the breakdown o-f Galilei translation symmetry. Ferrel

and BhittacharJee showed thaL the scattering of light by

liquid h e l i u m in t h e v i c i n i t y of t h e 1 jmbiJ* p o i n t is

determined by the density-density correlation function of

the quadratic field LV , which is. the broken symmetry.

This function has a linear as well as a quadratic:

part.

gauge symmetry without

c o n d e n s a t i o n , w h i l e p h a s e II ( ̂ »1> in

4201 into while discussing the gauge symmetry breaking

and phase transition in strongly and weakly interacting

Bose-systems, introduced a renormalizable parameter C(p ̂ W

<; l) into the interacting Bose system Hamilton!an auchj

that the zero-mode amplitude satisfy the coaxnutation

relation l»t , /] = 1 -^The different phases of the «yste«

correspond to specific values of tf . The normal component

(phase I ) is obtained when % -0 and there arm two ordered

states according to the strength of interaction , phase II

(O< t^ < 1) is found in the strongly interacting system and

is characterised by a broken

Bose-Einstein

the weakly interacting system displays Bose-Einstein

condensation. The continuity equation for particle density

holds in both ordered phases to within an infinitesimal

term . A gauge transformation on the field operator shows

that I,, "twists" the phase angle related to the zero mode.

Recently we discussed the symmetry breaking mechanise

with an approach which is akin to that of the

relativistic quantum field theory. In this context the main

idea is the symmetry exhibited by tha basic fields of

superfluid -*He, before the onset of the field interaction

is seen to reappear in a different form when the theory is

expressed in terms of the physical field operators which

describe the system after switching on of the field

reactions. As a consequence there exist the generators of

the symmetry transformatjon^which when treated in terms

of the physical field operators yield a non-zero value,

however when treated in terms of the fourier components of

the field operators above the transition, they vanish

The Boss-field operator obeys the commutation

relatlonii

t y(«), V (Y' ] = i*<x-y> or [^IKI,? (yl 3 - it(x-y)

with ¥ (y) » y'(y> 2-4

42,44

when the interactions are switched on via the interaction

H^niltonian, We hive a translation o-f the operatorsA>«25

with the Londitions

.2-5

-7-

<0 t y <x) I O > = «< r 0 2-6

<O|JJ(K) |o > • O 2-7

where •< is a spacetime independent c-number because of the

tr«ntl*tiontl invariance of the system, | 0 > is the vacuum

or the ground state and the Eq. t.2-6) represents the

condensation of the single particles in the system. As

y a n d y » r e both Bose operators, Eq. (2-5) can be imagined

as an outcome of a unitary transformation of the type

y ( x ) - y < K ) + <•< 2-B

The theory baaed on the Eq.(2-4) is invariant under this

transformation.

If we restrict ft to the local -field, then the above

trans-formation represents an infinitesimal unitary

transformation In the Hilbert space.Hence

& (V(x) ) = •< 2-9

and with

J 3dx 1T<x) 2 - io

we have

- i t y<x) , KOI- - i [ y i» ) , *<J?T<y)dy]

- -1< Jty(M>,7T(y> Jdy

* •< ' 2-11

i.e KyiH))- < • -i[y(N),i<0] 2-12When we decompose the Bose-field y ( « > in terms of i t s

normal imdadto take account of the f i e l d reaction we have

1 X— 1 i k . x - l l k l t + - i k . x + i l k | tV<«>- p= I— —==• <a e + a e >

* 2-13so that

k->0 2 k k 2-14

This is because above is a Hermit ion -field

Now as Q is zero

ytyix)}" -iry(K),<o]= o 2-15

From this analysis we conclude here that i -f the generator Q

is valid tor the symmetry operation (2-5> then in

accordance with Eq.<2-ll)it should nut van 1 sh. Howe? ver as in

Eq.(2-14), • vanishes,then it cannot be treated as the

-8-

generator and hence it is pertinent to explain the

contradiction found in Eq.(2-11)and Eq.(2-14). Such a

difficulty is seen to exist in the analysis o-f spontaneous

breakdown of symmetry in quantum field theory.

Now to resolve the paradox we have arrived at

above,we in the light of scaling of the Fourier Components43 Wof the field operator write '

• '1

q<x> «~-t y<x)-i IT

and

.2-16-i

~wThe introduction of the coordinates'^" and its conjugate

momentum "p" here is ta study the helium system in the

classical limit,as the classical description of many boson26

system implies superfluidity. The fourier Componentsdefined by

q(K> =-ikx 3

q(x)dx1 y- ikx 1 r •

t-~ q e i q - -jMM- 1 e

1 f -ikK1 T- ikwp(«)= --^zr 1— P •

•PC y' k

so that q = q , p » p 2-17k -k k -k

and the expansion of the field operators y a n d 7T in terms

of p and*q'representation mrms

Z^ q (t)k k

+ikx

-Ikx.2-18

.2-19

Accordingly we have

r 3Q - JtT(H) dx =

Now i f 'd'vanishM for k —>O then p( t ) "«0

But p {t) = q (t >k k

so when pit) = 0 q(t) =« constantn-o km*

This constancy of q'is an account of the Bose-Einstein

condensstion.

-9-

In this way the vanishing of O ' *npres*M itsel-f

through the vanishing of canonical momentum for the k-0

aode of the massless Bose-field. This in other wordi means

that vanimhing of the integral of "IT IK) ov»r the whole

volume ia due to the fact th«t it pick* up only k«0 fourier

component of the integrand. Thus if P*°> then k«O mod* does

not represent

Therefore, qk-O

true dynamical degree of freedom

p represent the spurious coordinateskO kO

and the commutation relation:

Because

k=0

of

k=0.

this violation

is violated

the occupation number

p-Oa ap*=0

is replaced3

by a c-number in the Bogoliubovs'

prescription3, which neglects the dynamical behaviour of the

condensed state. Therefore k*0 mode does not represent a

true dynamical degree of freedom.

Since the massless boson p , (excitation) does not

carry energy,the linear homogeneous equation for the stable

massleaa field becomes invariant under the inhomogeneous

transformation(2-3). Hence the time independent symmetry

generator Q, can mix elements of :»ra mass spectrum and at

the same time induce a constant number by the

transformation (2-5). Such a constant represents the

Bosn-Einstein condensation of the massless bosons,which

break the symmetry in superfluid helium system.

It is because the condensate involving macroscopic

occupation of the single quantum state acts as reservoir o-f

partricles and the particles may dissolve in the condensate

somewhere in the system and emerge at a remote point

leading there by to the non—conservation of particle

number. Thus He-II field is quantized in a gauge in which

q mode is cancelled. In this way Eq (2-lSt takes the

form

XL

r -ikx. 2-20

-10-

Accordingly the relation (2-4) is modified and with the

commutation relation between p's and q'% to

This relation follows

.2-3I

from the fact that ->* sode ora k=0

k»O mode has been neglected while ./»0 mode Is taken care

of by B.E condensation, which is represented mainly by

i-^t^rm in this expression. This equation la a modified

"delta" function,Thus we find that whenever the components

of the field do not represent true dynamical degrees of

freedom, we get a modified "delta" function and hence the

relationi

, p 1 =0 2-22

The commutation relation Eq(2-21) depends upon the

quantization volume. This volume is to be considered in the

infinite limit, as it is the characteristic of the

infinitely large systems that they condense into the state-I

oi lower symmetry. In this limit Si, does not vanish in the

Eq(2-21) because we have to ascertain that the Integral of

this term over the whole space ia unity. Accordingly we can

say that as <Sif~* °° ,the term >J"L has an infinitesimal

amplitude at every point in space such that Its integral

over whale of space becomes unity.This In other words means

that we have disjoint Hilbert sub- space* of a full

Hilbert s p a c e , determined by the volume as a broken

symmetry coordinate. The broken symmetry origin of volume

lies in the characteristic properties of condensation of

liquid helium system which is a second order phase33-34

transition. This is also clear fro* the fact that the

number of particles in the condensed phase taken as the3A

broken symmetry coordinate in superfluid helium, which in

the infinite volume limit is proportional to the total

number of particles in the system, la also proportional to

the volume of the system for the reason

'V = constant

The volume as a broken symmetry coordinate is

associated with the order parameter of the condensate from

the state with -full fcynimetry of the Lagrangian to the

state with lesser symmetry- The order, parameter describes

-11-

the dynamics o-f the condensate and consequently brings a

new d«gr se of freedom- This nsw degree of f r m d o m tide bot ;n

dynamical and hydrodynamical features. The hydrodynamical

aspect 1« characterized by local equilibrium. The dynamical

Aspect persists down to absolute zero. This means that the

dynamical mode of excitation persists in the col 1isionlese

domain. The condensed phase (second fluid) which is related

to the microscopic dynamics of the system below the

transition exhibits a higher degree of ordtr and develops45

a condensed phase of long range order at the transition.

Hence the infinitesimal amplitude of ifi, at every

point in space such that its Integral over the

whole quantization volume is unity,depicts the phonon

modes, which exhibit the Goldstone bosons and break the

symmetry. Accordingly the modified commutation relation

Eq(2-21> explains the long- range correlation which carries

away the symmetry of the ground state, leaving it frozen in

an asymmetric state.

ABELIAN TRANSFORMATION

The boson field operator appears for

superfluid-*He In the form Eq<2-13>, when Abel 1 an group

symmetries *ra phenomenological1y violated. As Vcont*in6

bath positive and negative frequency parts, the equation

for Tfik), the fourier transform of y <x> with

momentum vector 'k will be of second order- in time

derivative asi-

b t £ Ek= O .2-23

Where E denote* t h« energy of boson (!. ~~>0 as k - ->O>.k k

So the Lagrangian for the equat ion i s g iven by

- - - [ dlc/v<k)V2- -* I 1 t

Where a is an arbitrary c-number constant..

In the p,q representation we have

2-24

-12-

= - - - dk ) Jq - p + l ] p q + q p I2 J [ * ; \ k > \ k k k k J

Z 2 f 2 2 %

K.

* 2a . 2 - 2 5

Accordingly the Hamilton!an is given by)

1 f 3o t J

1 f 3 r_ f ,2 .2

--- j *d l E V " < k > Y1 J k x

<k> + c - numberAlso

.2-27

The Canonical transformations which keep the equation

invariant arei

-y(x> > y < x ) + c - number.

TT(K) > 1T(>!) + c - number.

Where TMx) = "KiTlx) + a' 2-29

f 3

Since J1T(x) dx commutes with both If7 and IT as -A —>«<> and N,

the particle number- is time independent. Therefore N

contains onlyTT1inearly.

rN = ->\ J

dx TTie

2-29

Where >\ is a c-number constant. This shows f i r s t l y that the

generators of the transformations Eq(2-2H) *re linear in

the boson operator 6 y(x ) and TT<><» and Secondly that the

constant "a'" in Eq(2-2B) is not arbitrary but related to

the ground state expectation value of the particle density

, J>= - - - - so that 7T<x> - y<H>

, Jthus a = - - - - so that 7T<x> - y<H>

2-30

Accurdinyly we see from Eq <2-28> that the canonical

conjugate of ' N ' is V < x ) / n . This leads to the

transformation of y(x> under the phase transformation;

2-31

as JjS < >: ) -' J^< :•: ) +"1!.* C-32

The ph^sori fic-ld obtained by the transformation Eq(2-31)

-13-

exhibits two fluid behaviour. The field*

the depletion or the excitation and

the B.E.condensation.•n.

describe

denote*

Whan the system in the ground state is givtn

fflonantun "p" without violating translational invarianceipH

through the factor a , than a ptriUUnt current of

momentum -flux (p.J»> appear* in the ground state. This gives

rise to the running of condensed phacon (boson) current. So,

if th« super-fluid system is contained in a circular

cell,the phase factor • in helium field starts the

running of the current _p^Xx) in the ground state. If the

system is to be rotationally invariant, the phase factor

"^" is represented byi

>(«) - $9 2-33

Where j It a constant denoting vortex strength and is

the angle of rotation. This phase induces a velocity field

v - --- V> - -£- ;s m mr 2-34

Where a. is the unit vector tangent to the circulation.47 48.3

According to Onsagar and Feynman y is quantized asi# - 2 TT n 2-35

Where n is an integer.

21T n, »Thus V ' « a 2-36

From this we find that the circulation of the

superfluid velocity is quantized. Since the ground state

momentum flux p has the f orn>/i>OOtht! velocity V bKoniBs

irrotational ie curl V, a O. This ie the Landaus condition

denoting potential flow motion.

Now the basic equations of superfluidity arHl

So from Eq (2-34) we have

2-37

2-3B

Where F is the total force? on the particle and /* i s; the

chemical potential. Thit it. the Landaus acceleration

-14-

31

equjtiijfi. From this we infer that superfluid helium

cannot sustain a true force in the sense of gradient of

thermodynamic potential without undergoing acceleration.

This means that in the absence of vorticity we must have a

flaw without ffictional daaping.

III. EXCITATION SPECTRUM

49-51Many authors attempted to extend the theory of

Bogoliubov and Zubarev by considering the interaction

amongst elementary exitat ions.However, in all these

theories, most of which had primarily been developed for

absolute zero, the temperaure dependence of the excitation

spectrum and structure factor of halium-4 was not taken upS3 54

rigorously, whereas the experiments of Woods, Hal lock,55 56 51

Dietrich et al Achtev and Meyer, Cowley and Woods etc

revealed a distinct temperature dependence of phonon

spectrum for liquid helium—4.This dependence, although

small at vury low temperatures became quite significant in

the neighbourhood of A - p o i n t ,

Samuloski and Isihara tn their microscopic theory

for liquid halium-4 for temperature variation of sound

velocity and structure factor attempted to Justify the

Landau phenomenologii_al theory and held the view that the

elementary excitations, being due to the collective

couplings of particles in the liquid state, should b«

temperature dependent. Their view was a soft potential for

He-He interaction. In view of liquid helium remaining

liquid even at absolute zero, a more realistic view was »

hard core mpulbivt interaction. Khanna and Das built a

Guacsian equivalent of the Leonard—Jonnes potential having

the required fourier transform, the more realistic

approach, and with this could derive the energy excitation

spectrum of interacting bosons resembling, both

qualitatively and quantitatively the experimental

on spectrum of liquid helium-4 . Many other authors50,60

have attempted their own approaches from time to time .

-15-

An excitation Bpectrue. VBS recently obtained by us

using a mealing transformation and using the Hamiltoni«ni

A/ yobeying the confutation relation of Eq.(2-4).

Takinqi

_1_ ra + / •

.3-2

such that

1 T- ikK +

" " 3-3

In this q, p representation we have.the Eq (2-B) <si

t t 1 -- / r-— \ -ikx

* 3-4

Accordingly the Hamiltonian (3-1) in terms of q , pk k

operators upto th» constant terms and terms of

higher than th» second term takes the

""Z- (im,-*̂ *. ". 3-S

Where N Is th« operator o* the number o-f particles and p

^v to the firsta

approximation in this formalism.

is the chemical potenti al,equal to

Now we make the following scaling change to cast

the Hamilton i an suit-able -for the iree field into the form

as that of interaction:

-16-

0 - q ; P => V(k) pk V(k) k k k .3-4

where V<k) is tha factor depending upon the nature of

momentum dependent interaction.

This scaling transformation is a Bogoliubov

transformation of the form+ + /

b = u a - u *k k k k -k

with

o a - u aUK. * '*

I -1u = (Vlkl + V <k) }k 2

u' = { V (k) - V (kt >k 2

2 / 2u - u " 1k k

.3-7

provided we associate annihilation and creation operators

with P and 0 in the formk k

1 +D = y=-- [ b + b Dk / 2 k -k

1 +P = -=^r I b - b Dk -/2 -k k

So under the scaling transformation

Hamiltonidn can be written asi

3-8

where U> = ( - — 2 o^ V >"f(k>

2m'f<k>

Where l«> the energy of excitation with-ft^l ia given by.k

2 2W + 2 << V 1V 2m [_ 2m k)

•

[~ 2 2 J

- W -M + -V k

3-10

This is of the form of Boquliubov spectrum.45

When we define

-17-

q<K) - — [y(«i + tr (K)

P(>€) - 3-11

So that q --- C a -+ a 1

andi +

•JT k k 3-12

and N - % -k - a_k -_k

We obtain

2 2 2

y <k>3-13

_1Using the scaling change

0 - J"V < k > qk V k

1P • ,=» a . 3—14

k JV(k) HkNow in the phonon mediated interaction between 3He.

quasi particles in He - He mixtures a simple parametricform for V is represented by

k '. . / V <k) v

\ - - |VJ Co. (~£~) 3-15S o assuming that the interaction between phonon* in

4the excited superfluid He to be of the same form » that

between He quasi particles in He - He mixtures so

that we can take V(k> - Cos(k)

Such that V(-k) • Cos(-k) =• Cos<k) =V(I) 3-li

implying that V a Vk —k

Accordingly we have

U)?k> -ft---) * ---*— -V« 2' 3-17L V 2s> ' m J

With o •= --̂ -

Expanslon of C O B I U in the powers of k and taking the

Sq.root we get

<*><k> = c k C l - V k" + S l , J 3 - i a1 I

-18-

This to tha second approximation y i e l d s

2• ck ( 1 - r k )

l.3-JV

1/2

i th c «( ) and r = o.S i. 0.> m / 1

47 -2 -2 24992 X 10 kg M sec

This tallies with that obtained by Landau and61 ^i *1 -2 -2 2

Khali tnikov, Ulhere "*J - 2.8 X 1O kg M sec and with62

that obtained by neutron scattering data where. 47 -2 -2 2. '

>J= i 0.2 X 1O Itg M sec .For small values of momentum,

(3-19) reduced to the form:

fro* Eqs. C31O) and

. 3-2O

when from Eq.(3-10)

It is therefore evident that tJ^—>0 for k — > 0 in

accordance with the Boldatone theorem.lt is also clear that

the elementary excitations »ra the phonons propagated with

the speed of sound "c".

Many suggestions have been made to explain the

nature of roton minimum. Landau suggested some kind ofAH

vortsjx motion __ an idea latter taken up by Feynman and

Feynmn and Cohen. However recent work has shown that the

theory does not describe at all well ths behaviour of roton

minimum as a function of pressure. There seems to be no

convincing reason to believe that vortex motion is63 57

necessarily associated with a roton. de-Boer considered

rotons as short wave-length longitudinal elastic modes

which are completely dagenerate, because the wave length is

of ths order of the magnitude at average distance between

the neighbouring atoms. Chester and Miller et al regarded47

roton as the modified free particle. Oneagar called theroton as a ghcjst of a vanishing vort«x ring.

66According to AneJeroon vortex nucleation is the

biggesV puzzle in the theory of superfluid *He and hence

tht> dynamical formulation of the energy gap needed for the

i-oton excitation is still to be understood by a

rHined microscopic theory. It has been hypothesized that

-19-

the energy gap needed for rot on like i;::i.itilian,lile the

quarks in elementary particle theory can be generated67

dynamically and this energy gap is essential to68

describe the low temperature behaviour o-f superfluid 4He.

This idea became pertinent due to the fact that there

exists a complete analogy between the production of hadrons

and creation of excitations in condensed matter

physics . Encouraged with such a notion a theory of

roton-like excitation in He-II was formulated by

exploiting the 3-di i»ncional rotation group and the

introduction of raton operators -This way coupling of

roton like excitation including many states in the roton

region of the Landau spectrum of superfluid helium is

formulated. This model is Justified on two accounts.

(1) Liquid helium has a close resemblance to the

70system of solids

<2> Roton is the smallest of the vortices which Ara

produced when He-11 moves with a great velocity

and the mixture of quantized vortex lines and

the superfluid, components can mimic the rigid

body rotation.

On these lines recently a relation between the roton

excitation and Bogoliubov quasiparticles has been

established, by giving different structures to the roton

operators. This -formalism is similar to quasiboson or

random phase approximation for the phonon like

spectrum.This structure af the rDtori operators enables to

study the low temperature behaviour of superfluitf ^ He by

the introduction Of the idea of pairing for roton like

excitation for the ortnet of energy gap. This pairing is

passible due to • very large density of states of rotan

like excitations in the neighbourhood of roton minimum of

Landau's model. Application of this roton like excitation

modal, to describe the density dependence dnd temperature

dependence of tht> energy gap yitjlds tlie result uf the

temperature dependence of energy gap more or leys in73, Jb

consonance with thcist' obtained en per i mental ly. However ths

roton remains a nlstery and so dues a very targe part of the'

spectrum. The spectrum is vet-y sensitive to pressure

-20-

changes. Both tlit? vt'lucity of sound which controls the

small k-part of th« spectrum and the roton minimum changa

markedly at the pressure is increased. This requires Some

explanation. Further there seems so far no explanation to

what causes the spectrum to end.

76-79A number- of authors have in recent past worked

with the vortirity aspect of super fluid - ''He . However, It

still remains to be understood how actually the vortex

lines get into the fluid, since with the absence of

viscosity, the circulation is still an Invariant of motion.94

Venin suggested that vortex is a macroscopic excitation

which cannot be created at low superfluid velocity and the

observed breal down of pure super flow at very low velocity

cannot be well explained by stretching a few lengths of

vortex lines that may already be present in the apparently

undisturbed helium.

IV. TRANSITION PROBABILITIES

Super-fluid «He may be divided into the states of

condensed bosons, oscillating condensed bosons, the31 80 31

phonon states, the Ma::on states, the roton states and the48

state of vorticities. So there *rm different state to state

transitions in the whale sector of superfluid -*He .

Accordingly we consider the transition probabilities

between various states as is detailed below.

TRANSITION FROM CONDENSEDCONDENSED BOSON STATE:

STATE TO OSCILLATING

Bogoliubov's theory proposed the first atv-initio

theory for the excitation spectrum which gave a qualitative

explanation o-f superfluid helium-4. However the theory

failed to give converging results with higher order

correction to this spectrum. The reason is that he ignored

the quantum fluctuations of the zero momentum state .

Accordingly these fluctuations of the condensed bosons were

incorporated in to the Hamilton!an to remedy the anomaly.

-21-

To this end the zero momentum iingle particle operators a^

and a* taken equal to a c-number constant f have bean

replaced as i+ + •

a > a + c iO 0 /*

a > a + c0 O /*•

4-1

Where c Is a new annihilation operator that describes the

•mall oscillations of the condensed bosons in the ground+

state. These operators c and c obey the usual boson

commutation relations!

t^.c^l - 1 t

and Cc ,a 3 = 0 \ k f 0 4-2

However it is approximated that while the shift Df

the equilibrium point Ag3fA. — H is macroscopic, the matrix

elements of c -,are not proportional to the size of the

systemf ipJthe amplitude o-f the oscillations represented by

c *,, remain finite and small even when the system becomes

infinitely large. Further the operator c^ may be of quantum

origin or it can be due to the interaction of the zero

momentum particles with the walls of the container or with

k:=/0 particles. Th» vacuum |^lw> of £* and a (kj*G) is the

true ground state and is the eigen vector of ^ _ r^£

defined by.

4-3

The number of physical particles ara to be counted by

N= TT a a the vacuum of which is defined by*- k k

a I 0> - O <O I O> - 10

4-4

Now the conservation of the particle number implies the

invarianca of Hemi1tonian under the phase Transformation!

k

Accordingly we imagine that we h-ive employed the

transfermed operators from the beginning and we make the

following change of variables

1/2a e0

N c n4-6

-22-

Although the replacement Eq (4-1) provides us with a

useful approximation scheme, this new form Eq (4-6) will

also work well both for fres particles 4a well as for the

interacting particles. The apparent forms of 3HJbefore and

after th» change of variables are different from each other,

but t h m r el gen values will Appear to coincide. The change

of variables can be looked upon as a unitary transformation

within the representation apace of a and a . In fact0 0

or"

and

Where.

cf-

c

+

•

B

exp

- U a

=• U a

1/2[-N

0

a e0

oe

eO

(aO

+

Va

Va

-

+

0

e0

+a

0

t

i/».4-7

1/2 +— exp E-i A N (a + a ) D

' o o o

~ exp t-2i A N0

+a ->- a0 ~ 0

1/2

u =and Ca J = [a

0 O o

....4-a

4-9

as operators.represent the commutation relation*of a ,*o o

Thus the true vacuum state is related to the vacuum state

o* * ^s I n V _ i I InS.

o H V > - ^l0** 4-10where fh with the vector /»f)L̂ t> is put to indicate a

superposition of the states with different occupation

numbers. This relation indicates a transition from the

states of the condensed bosons with zero oscillation, to

tha state of oscillation of the codensed bosons. To

compute the transition probability betweeen these states,we

first consider- the coupling co-efficient between the

&t<*ttf5 of C - oscillating condensed bosons and

k-osci1lating condensed bosons defined by t

! • : , ( '

» —! it:1.

O| a [~2iN

fiT< .4-11

-23-

Using the Eqs (4-7),<4-B> and (4-11) we qet for k and Iboth noo-tflcrusing th» -fallowing recursion formula*:

k Co»h( A )H ,< * ) =• -Sinh (

and

LCosh( A)H , ( *) -

k-2,1' k-i.e-j

with H ,( A) f O for k and I bath odd or even,k, I'

also H ( A) «= H A * ) - 0 , where N >0-N .£ ' k,-N 1

4-12

Let th» generating function be defined by:

H , < « ) a ti .4-13

Itc differentiation with respect to a. and subsequent use of

Eqs.(4-12) gives US!

. F ab 2 2 tanh^ 1H(a,b,/J) - H (/*> «Kp, -<a -b )r O,O I coshd 2 JL ' 4-1'14

Th value of H ( fii as computad from its0,0

definition,and on its differentiation with respect to

/> is equal to e • since H ( A ) = 10,0

So we have

2 2- (a - b > - 3

.4-15

Expansion of this generating function in powers of"a" and

"b" give* us the transition probability between the vacuum

state o* a^ and the true ground state w lth k osci1lating

condensed bosons «si

,<fi L,,^.>»^,- Z ) ! < Z ' )

even k.

^----)x J "^f^;;^;;Cosh^j

for odd k 4-li

From this expression we find that the transition

probability is a function of the condensate occupation N

-24-

TRANSITION ' FROM OSCILLATING CONDENSED BOSON STATETO THE PHONON STATES:-

The states of oscillation of condensed bosons

are harmonic type of vibrations. These independent

oscillations executed by the condensed bosons *rm with

random amplitude and direction but with the same

frequency,which are denoted by the index /*• In the operator29-80

C- Such a motion can be imagined as the collective state

that gives rise to a gradient of osci11ation.For the states

of oscillation by zero momentum condensed bosons,a state

having such a collective characteristic can be obtained by

choosing the appropriate linear combination of the

degenerate states. As this system of oscillating

condensed basons has motions of the harmonic oscillator

type,different types of correlations »ru obtained by

choosing different linear combinations. The lowest state

may have no collective dilational forms. However,the

highest levels in the system of oscillating condensed

bo&oris with t=O, form the excitation spectrum on this

intrinsic fundamental state. This collective behaviour will

finally lead to the formation of the phonon spectrum of the

cainpres«ional waves. The highest levels in the system of

oscillating condensed bosons is the result of the

oscillating condensed bosons having acquired different

frequencies. This way the ground state for phonon* becomes

asymmetric and changes into a different possible vacuum

state, distinguished from the former by the change of its

phase. This transition from one vacuum state to another is

described by the production of phonons, which break

the symmetry of superfluid helium ground state.

The variance in the frequencies of the oscillating

condensed bosons can be ascribed to the dynamical effect

that takes place between any two particles or subsystems in

the phonan vacuum state, thereby violating the conservation

law of the underlying Hamiltonian, under symmetry operator+

i« tr c +U ( » ) = e /j p i.e. , U ( « I HU(* ) / H ,

whiji-e LJ ( 0 ) is d unitjry operator representing a continuous

symmetry group.

-25-

Accordingly the trua vacuum s.tdte (state of

oscillating candented bosons) is related to the phonon

states u i

,4-ia

where 1o> im the vacuum phonon stats. The phonon operators+ *

a ,a which *rm the trinifornis of the oscillatingi i +

condtnud boson operators c and c are given by.r1 ^

a • U c + V ei 1 fi 1 >i

+ , * *a •= U c + V ci 1 ̂ 1 p

where

a c er< '

s + -sme e

4-19

S= i•c c U = Cosh 01

V » Sinhtf1

t a ,a 3 » C •i J »

+C a , a ]

i j i j . 4-2O

Adopting tha same procadura as in the previous section,we

Dbtain the transition probability from the zero phonon

state,to m phonon state aa

12 , r i i % m .

' I 1 tanh » Vt-

m ,0 r* 1 L J 2 '

2<4/Sinh fl )

from even1

m-1 tanh ( m,-l/m M-D*<< ) -X

Cosh A<4/Sinh & )

(2Z+I) ' (2+1/2) • {l/2<m-l)-'Zi •

for odd m

By the Theorem of compound probability, we have the

transition probability of having the oscillating condensed

bosons and phonons states simultaneously, as:

PhononG,< A)K,0

whareK,0

3 ( 9 )m ,u

is given by the er|.

TRANSITION FROM F-HONQN STATES TO ROTON STATESi

For law energy phonons,the velocity increases with

increasing energy .These phonons with increasing energy,

therefore always have velocities greater than the

velocities of all phonons of lower energy. Hence the decay

process is possible. However the difference in their

velocities is very small. For the phonons of higher

energy,the velocity starts to decrease during decay process

and some of the lower energy phonons have a higher

velocity. The decay possibility then becomes restricted

and above some critical energy E phonons become totallyc

stable against any decay. In the absence of any decay

process there Is only the interaction of phonons and the

interaction of attractive nature can cause the bunching of

phonons. This bunching of phonons results in the creation

of raton like excitations. Accordingly the critical energy

at which the phonons become stable against any decay

corresponds to the minimum energy required for the

creation of roton like excitations. This is borne out by

fact that the value of critical energy at which the phonons

become stable against any decay into a large number of

phonons,as found by Dyne and Narayanmurti ^ at lero

• 83 •

pressure is 9.SK Arid as found by Marie is 9.86 K while

the minimum energy required for the creation of roton as

calculated from the thermodynamlcal data is given by

= 9.B K

and this as calculated from Neutron scattering data i«73,74

— = B.6 K

To correlate the phonon states with the raton states

one can choose the bilinear operators

phonon state and construct out of them the angu

operators L given byi j

due far

lar momentum

i Ja - a a )i j J i

.4-23

-27-

The vacuum roton etate is tortsitiered as « s.et o-f eigen

•tatsi of the ground state with L =0 and th« roton statsi J

is a well defined angular momentum state formed out u{

eigen V I I U M of the operators of {, and /, . the operator

that we ust for the transformation o-f phonon to roton

st.t..

L

i

tj 1 J j i4-24

where 4 i i * phase change as urn go from phonon states toL

roton state* . Thus the Bogoliubav transformation is e and

transforms c' and c' of the phonon opcntors+ k k

, L - L 'f /,+c " m a m • u a + v a

K i i i i J/+ L + -L • + #C » a a e • u a + v ai i i j

where

Hanc*

C. ,a + ]i J

u * u* « Cosh ( S ) ,i J

|V > -rot

I 0>r o t

J

S(<*)T

Si nh < * ) 4-25

.4-2t

Nhara |O> is the vacuum roton state.rot

Ma defined tha coupling co-e>f f i niant between two

possible roton states as>

( / .q' .r , .. P + L r + ,

<0 I a a e a a O>t j j i '

V q 7 r 7 " i ! H<*K ' f . . . 4 - 2 7T T P' ,q ,r ,e

Using tha M)tiatIons(4-25) and (4-:'7) , we obtain tha

-following relMibn* with indices al l non-vanishing.' •*> < / } , . -

P ,q',r,«p Co»h<

pi - l , q - l , r , s pp - l

+H- 1 , q - l [5 , q - l

rCosh(p . q ' . r - l . s - l

-28-

p ,q',r-l,«-l f5 .q'-^r.s-lP , q' , r , s

...4-28

As a consequence of momentum conservation and from tha

definition of H's we conclude that

unless a -a' «r-«H(0> , , = 0P',q^ ,r,B

= 0 i f p +q' +r+s' i s odd ,Here we take the generating functions asi

. ..4-29

Hta,b,c,d,P' , q ' , r ,

* b"

f / f

M ( « , b , c , d , * > <=H l<f>) Expr (cd-ab)tanhr 0 , 0 , 0 , 0

p' ,q' ,r ,l . . .4-30

Differentiating this with respect to "a" and making the

substitution from Eqs.(4-26) and (4-25) we obtain

*c+bd)+ JCosh f

...4-31The value o-f H ( O ) Is obtained -from its definition and

0,0,0,0the Eq (4-28).Expansion of tha generating function in thapowers of a,b,c and d gives tha transition probability from

phonon to roton states asi

rot I p* ,V ,0,O|

2 . 2 2n ,2«f

X

. . .4-32

P ' q' !

- Z) ' ( r - I ) 1 (q-p'+ * ) ! Z.

Its dependence on the condensate -fraction Is obtained by

the theorem of compound probability as i

(<0,rot) phonon rot

where *Y is given by Eq(4-22>

-29-

V. COHERENT STATES AND THEIR SYMMETRIC STRUCTURES

23,24According to Bogoliubov the symmetry of a

systes) is brokan by the introduction of an infinitely weak

external field into tha treatment. Tha Interaction term of

the total Hami1tonian of tha aystam braaks tha phasa

symmetry of tha fraa Hani 1 tonian. This condition of brokan

symmetry leads to tha translation of the boson field

oparatora by c - nuiaban , Eq (2-5) or equivalently

Eq(2-32). Under this situation tha Hamilton!an of tha

system becomes asymmetric sothat its conservation law seems

to be violated. Now to create a phase variance we have to

sand tha system through I O M external potential and the

magnitude of the phase variance equals the potential times

the time it is in the potential. Sanding the system

through some external potential n to set in the

interactions in the form of a dynamical effect taking place

between two particles or subsystems. This way we iitipa»

same asymmetric condition on tha ground state of the system

which changes into a different possible ground state.

These ground state* are distinguished from each other by

the change of their phase. Accordingly for each specified

phase there exists the corresponding ground state and to

each ground state is associated a Foci

representation. " ' This transition from one vacuum

(ground state) to another vacuum state can be described as

the production of quasi-particles (phonons).37

The term translation or

operator was first used by

transformation Eq (2-5) can be induced

operator| — t *

So that

di spl aceflivnt

Glauber because

of the fiald

the

by a unitary

U.5-1

The c- number constant has been generalized to

when the system is in a container with rigid walls.

un the coordinate space is related to

Eq(2-Ei> in the sense that the

- 30 -

This dependence <jf

the transformation

transformation can be viewed «H a nonlinear realization of

the gauge transformation which can be obtained by imposingi >• ( x ) •f

f(x)e , viz y y • t on

the Bosa-operatorB y and ^ r . The

generalization of c-number would ba the case even far the'

•yctem. However with

tha constraints of y<x>

nan-interacting

interaction having

Bason43,45

U(<K» - exp

i kx 3d

( x ) + i+ i q k x

aC(x) obeys a salf—consistent

*•< ( x )

X < (x > >

> ] I 3-2field equation. This ensures

that <JC(H> is flat spatially except at a microscopic

distance from the boundary. The field equations even with

periodic boundary conditions have solutions representing

the presence of spatially inhomogeneous fluid dynamic

patterns. Each such pattern Is associated with a different

spectrum, each of which can be determined by analysing the

quadratic form obtained after performing the shifty—>'

To this endVtx) - the Bose operator obeys the relation*

with | t > ] > - UC-O|o> 5 - 3

The states I r^l> which *ra the eigen statee of the operator

called the coherent states.

Thus broken phase symmetry leads to the translation

of boson field operator and the production of coherent

superposition of states.

Physically we can bring about a phase variance by

rotating the entire vessel of He-Il systea about a vertical

axim of its rotation called the active transformation. Such

a rotation of the entire vessel in the volume 1 imit »(lr»*t>*s

no physical HIgnificance,as the entire phonon spectrum in

liuper + luid He is to be understood in the 1 imi t al/**9 and if

we go out of the volume limit we are going out of the

Foci; space in which the original bason field operators

are- defined. Therefore we take recourse to passive

transformation, which is a coordinate transformation in the

D|ipu<iil« seri«,e and can be performed in any Fock

representation in which y is definad. This ensures the

-31-

rotational invariancB by the product, ion of phonon coherent

states. Accordingly coherent superposition of states in the

phonon region of superfluid helium is obtained by applying

a "Block rotation" to an extremal (vacuum) state in a

••unbounded unitary irreducible representation of the25

non-compact group SLJ (1,1)

-ii* J,t»

Z S

(- l )

/Sn/2

with the ground state defined by

- i ~fJ | j , o > - O e | j , o > - 1 j , o >

and the generators of the group SU(1,1)97

obeying tha commutation relationsi

< V J2:-"1J3 ' tJ2' V 1J1

.5-5

3 1i J

with J "

i tJ , J

c i J a t J i

- J

The norm o4 the state is given byi-

3n2n

c ™

s tj , J = ± J

.5-6

From the finite nature of the norm, we conclude that

the coherent Stat»» |«C^^ under discussion, far all values

a-f n do «pan the Hilbert space, as is the case with|HK>

which -form* Che) basis of the number operator N and entails80

the complete orthonormal set. After the formation of

coherent superposition of states the boson field operator

isi

i ky. p""] .5-9

-32-

Accordingly the density25

becomes

>>

ikx -Ik x

) 1/2+[n(K)] e

c

) ,CntK

c

1/2 -l#(xI e

.3-9

This is the form postulated by Ponrose and Onsagar^and

discussed by Hohenberg and Martin and Anderson.90 The

first term of this expression describes the normal

uncondtnud phonons in the system. This term, on account of

oscillatory nature of the exponential, vanishes as x-x

The second term depicts the phenomenon of DDLRO. According

to yang, if the fundamental definition of DDLRO is

taken to be the existence of some finite degrees of

factorization of an nth order reduced density matrix in the

coordinate representation then definitions of ODLRO »nd

coherence became identical. Accordingly the second term in

the expression also depicts the coherence phenomenon in

phonons in superfl<_iid helium.

of the Eq<5-3> the particleFurther because

density V*<x>« <"5f <x> ,y(x)>, the Hamiltonian describing

the helium system in fock space defined by Eq(3-M and the

associated current density defined byi

joo • -i^have their C-number counterparts

.5-10

COHERENT STATES IN OSCILLATING CONDENSED BOSONS

the condensed bosons showing high

SU(n) in

Oscillation of

degeneracy is described by the symmetry group

N-dimpnsitjris. The basic N-componemts a-f quanta emerging

from high zero point energy of the symmetry group ara

-33-

trans-farmed among themselves and accordingly the frequency

of the oscillating condensed bosons is assumed to be eii

Thus the ground state of tupcrfluid helium consists of

oscillating condensed bosons and it seen to contain SU(2)

•yfflfflttrit structures. If the frequencin of the oscillating

condensed bosons Arm different then the broken sysmmetry

condition is visualized and leads to excited states of

80

phonon type, roton-type and even the higher ones.80

Const dor ing the quantum fluctuations in the

condensate as a classical source which excite the single

node, the oscillating condensed bosons can be treated as a

system of driven oscillators

1 2 2Hamitonian H = ( p + q -1 ) 5-li>

o 228.92

represents the vacuum state \0 > of superfluid helium

with vanishing mean position It momentum

i.» < O| q \0 >-<0j p |0 > - • ....5-12a

Introduction of c~number terns due to the quantum

fluctuations gives the new Hamiltonian that represents the

true ground state of superfluid helium. This state is the

coherent state of the oscillating condensed bosons with the

exact fequencies . This is realized by the operation of the

unitary operator

on the ground state. As this operator

translating the operators q & p

its

has the effect of

by c —numbers, the

coherent state obtained by its operation has its

coordinates and momentum displaced by q I* p from that of

the state |0>.

. . . .5-14

Thus in the state

not vamuh

<r,«v

f,^^ the mean position S. momentum do

5 1 b

the density matrix

f „ _L (-}<<- 5-16

describes the average over phases of condensed bosons and

provides the particle number I

5-17

and the probability of having "n" oscillating bosons in the

state i

....5-18

This Is poiseon'e law for the oscillating condensed

bosons representing coharant behavior and confirming that

the state is the coherent state of the oscillating

Londen»t?d bosons. It is seen that the displaced oscillating

condensed boson ground state \ Pt^/ "̂ evolves in time from

another state of the same nature under the action of

displaced Hamiltonian, meaning that only its mean position

l> the momentum changes in accordance with the calsslcal

mechanics. Thus the coherent states formed in the

oscillating condensed bosons of super-fluid **H» are

physically the rel event ground states far the driven type

oici11 at ions.

DISCRETE AND COlNTINUOUS REPRESENTATION OF COHERENT STATES

Bogoliubav showed that the problem of

superfluidity reduces to that of finding the spectrum and

wave functions of the Hamiltonian that is quadratic in

boson creation and destruction operators. He also indicated

how the problem can be solved by diagonalIzinq the

Hamiltuman with the aid of the linear canonical

t.r antif or mat ions, now called Bogol iubov's canonical

transformations. The set of linear canonical

trari*formatians for this; problem forms a certain group and80

indeed a direct product of SU(1,1) groups. This group

SlK.1,1) is isomarphic to SQ(2,1> and SL (2,R) groups. The

generators of this group as visualized from the

L'ogoliuhov's truncated Hamiltonian and written in tewrms of

creation and annihilation operators obey the commutation

relation as indicated in eq.5—6

-35-

The group under consideration has several series

of irreducible unitary repre»nt«tians, the discrete, the

continuous and the supplementary. It is passible there-fore

to construct uveral u t i of coherent states associated

with the group.

Physically it can be said that the group can

describe a large number of states which can be visualized

here to be the property of interaction in super fluid

helium. This spectrum generalng nan-compact group

describes the excited states upto the phonon level.Again

•IB there exists no non-trivial finite dimensional hermitian

representation, the excitation can be represented by a

unitary representation i

....5-19

5-20

In terns of the SLM 1,1 > t*narator« the reduced Hami 1 tonUn

of the Bogoliubov is given by t

and for the unitary representation we have

H5-21

consider ing the Cast mier operator* * • i • »•

3J4 C J \ - 0 JJ

5-22

. ...3-23

From the equations 5-22 and 5-23 it it. evident thatj

belonging to compact group gives a discrete spectrum which

is integer spaced, while the generators j( and J belonging

to the non-compact group give a continuous spectrum.

The Hamiltonian Eq.5-21 which deals with the

interaction leading to the cut i terf st.«ten, bears a

symmetric structure, so the ccjhprent stales abs,ui.iated with

. . 103this group also havp a symmetri c structure.

-36-

Let us now consider the set of coherent states

associated with the representations of the discrete series

and the ctmtinucjus series of this group. The basic vectors

of the group are the vectors | J)Hv> forming a complete

orthcinormal set.

DISCRETE SERIES REPRESENTATION

The dibcrete class of unitary repre»entations in

Hilbert space with baaic vectors )j /'m > mrm given byt

where E = ffL° " .... 5-24

and is related to the universal covering group property

of the superfluid system.

^ There are two discrete series o-f representations

U(j) and LNJ> for the group under consideration. Although

it is enough to consider only U<j>, as all the results

obtaitiBd with this, transfer to the other, however, we

consider them even separately.

As the Casimier operator " 0 " is not Independent for

the discrete series of representation therefore:

a) far tha positive discrete representation U<J>

""ere the spectrum is bounded from below, we have

i ) j + E » 0

o) I . E

m oO for J<0

iii) j - E. = j + J » O,1,2,3 5.23

3 o .5

Accordingly thr coherent, states tak»n as the el gen states

of the Ladder operator J are given by

i-27

-37-

BO that the norm is givan by

) is the confluent hyper-geometric

function. From Eq- 5—28 it is found that the eigenvectors

MSare complete, but do not form an orthonormal ««*t, Tha

states t*O a n d \ • ( ^ become orthonormal only if the

function F

vanishes.

become orthonorma1 only if

2 *'*< ) has a zero at the 2 **'•< where it

Thus it can be said, that the states as formulated

above become the Guassian-wave packets centred around tha

point «C«( and zero at tha point 2KK and are vary narrow

in the classical limit. This result copes with the usual

definition of coherence and thus to the phenomenon of

superfluidity.

b> For the unitary negative discrete series of

representation (J) bounded from above, we have,

i) j -E -0o

ii) I E " 0 for j < (0-2jtm o

so that j - E - 0,-1,-2, ...3-293 o

Accordingly the coherent states are given by

.... 5-30

where

. .. .3-31

From this it is clear that coherent behaviour is depicted.

-38-

CONTINUOUS SERIES REPRESE-NTftT lONt

the

The continuous bases -for BO(2, 1 ) —• SU( 1 , 1 > -»•' BL <2,R)93

been studied by many authors. In this representation

states are taken as the el gen state*

....3-32

and the norm is represented by

/ )

....3-33

Front thift -finite nature of norm, It can be concluded that

the coherent states |«f]Wor all «( do span the Hilbert

space, as is the case wi th 1*1,,̂ t <N -0,1,2. .. > 'forming the

b<i&>b of the number operator N' and entail inj| the complete80 %

orhtonormal set. By the systematic analysis, it has been

established that sub-set o-f all coherarit states |^> for

which |K > i« represented in terms o-f wall defined quantum

numbers as in eq.S-32, span the Hilbert space.

Experimentally it has been verified that if a subset of

coherent states already span a Hilbert space, and forms a

complete set, then, tha full set of coherent states may

become totally complete. This indicates that a certain type

of linear dependence should exist among the coherent states

of the supsr-flLiid helium system. Again from eq. 5-32, it is

evident that ̂ \<n\ < (*> . T h i s reveals that the

allowed sequences i -' > themselves form a Hilbert space with* ft'

an inner product *»• ̂ . tio the norm can be written as :

, t- cm , X

....5-34

Evidently the arguwnont of -{*f*v } oi H i l b e r t space vector

U-f^ I 1 > -€t, 16 itself a vector in a different but

<itill in infinite dimensional Hilbert space, i.e,

-39- ....5-35

From this it follows that

5-36

which corresponds to writing the number operators for the

•uperfluid helium—4 by

I n , n , n , n > • I N >1 * • J K K

. . . .5-37

Thus it is clear that such space contains not only the

coherent states of the oscillating condensed bosons but

also the states | •< ^defined by :

....5-38

The above argument leads to

....5-39

This relation indicates that -for higher values of 'n' i.e.,

in the higher energy range of photon like excitations, each

coherent state tends to look like an oscillator type ground

state with an arbitary higher value of •< ̂ . The basic

functions in the coherent state continuous representation

•re contained in eq.S-38 for each «<n >•€."£• since

| ̂ >i \ > arm complete In the analysis of Bogoliubov's

quasi-particle spectrum, so too ara the coherent states

emerging -from It. As in the quantum field theory for a

single degree of freedom analyticity argument forms basis

to get complete proper subset of coherent states, so does

our analysis generalized to several degrees of freedom.

COHERENT ROTQN STATES

According to Feynman rotons are the small Bet

vortices . The velocity distribution found by analytical

means is similar to that round a vortex ring which may be

small . If the curvature of the anulleit vor tex (rotori) is

increased beyond s certain fiwed radius which is assigned

by quantum number "j" , energy i«i needed. This enables us80

to assign a rotational level j<j + 1) tor the roton .Thus

it can be said that the rolon stale's are merging tn form d

continuous distribution of statts. This is coherent

superposition of roton states a<i intutivfiiy one meaning of

-40-

the statement, that the states are coherent is that there

is, in some sense "one state", which is sum Df the states.

The phjse variance is brought about by the interaction

terms of Hamiltonian. The rotation of the system, that

produces a^phase variance can be regarded a* sending the

system through some external potential and the change In

phase is equal to the potential times, the time, it is in

the potential. Sending the system through same external

potential can be thought of as any dynamical effect taking

place between two particle* or sub-systems in the roton

ground state. This way the ground state changes into the

coherent superposition of roton states. The roton state

hag been created from the ground state by means of the

operator T

T* |O >= I j,m>jm

...5-40

where | J,m > is the well defined angular momentum state

farmed out of the eigen values of the operators J* and J.

The (2j + l) quantities "J^with the spectrum condition

-j<m' < +j transform under the rotation of the frame of

reference as components of a tensor with rank j . The

coherent superposition of roton states »rm defined by

jm

where

the operator T-»£incorporates "al 1 the statesnear p»p.

operator T . ^ bears with j and J » J( ± Jt

following commutation relations^9

the the

5-42

(2j + l) (2J + 1) matrix t. has the following non-vaniahing

trij; elements.-41-

—<i; .!>.•

, , - 2.7»C jx : a.

I , , ^ is , j

5-43

Th« coef4icients"J* In •q. (5-41) can be

obtained by the determination of the mitrix alinentE of the

tensor operator T: _» , making u u of the Wigner - Echartisor operator T. *

theorm .

The inner product of the state vectors <j,m |jm> is

independent of "m" . Thus the matrix elements of the tensor

operator factories into two parts. The directional

properties *rm contained In the Clebsch - Gordon

coefficients and the dynamics of the system appears only

in the scalar matrix element <j' II T.+ ( j> called the

reduced matrix element given by t

3-44

The time evolution of the coherent roton states is

obtained by solving the Schrodinger equation,

5-45

.... 3-46+ 1L = H , +where H *• H.I*

So the time evolved non-standing states are given by

...5-47

where E is tht* energy of the rotun »;:ci tation .At time t-0,

the states have the properties characteristic of a

coherent rotational state like those of coherent angular100

momentum states as disriissed hnre :-42-

a) The absolute value of the amplitudes a- ' ar»

peaked around n mean value of angular momentum J.

b) The energies E of the states ( j,m' > to a good

•approximation obey the rule j<j + l) while in the

case of reference <•••) the states lj,m > are

members of an ideal rotational band.

c) The phase ^ ^defined by «.,- m•^ exp(-i+.A

of the amplitude a. j ̂ appears roughly equidistant

while in the case of coherent rotational state

of reference <*>•» they are precisely equidistant.

Now if the phases ^*m'are exactly equidistant at t«=0

and energies obey the rule j(j+l> perfectly, the coherence

will disappear after a short time because of time evolution0

but it will appear again periodically. This is related to

the fact that the spread of the coherent states over the

spacial angle oscillates In tine with the same

periodically. However if the roton spectrum E would follow

the rule j(j+l) only approximately then the coherence will

recover quasi periodicatly.

VI. CONCLUSION

The preceding section elucidates how the symmetry in

liquid '•He system is broken. Exploiting the displacement

of the boson field operator, we arrived at a paradox. This

paradox has been resolved by using the coordinate q and

its conjugate canonical momentum p representation

for the boson field aperators.lt has been accordingly shown

that the He-11 field is quantized in the gauge in which

q mode is cancelled. This has resulted in a modified

commutation relation Eq (2-21). From this relation it is

found that whenever the components of the field operator do

not represent the true dynamical degree of freedom, we get

a modified "delta" function. The commutation relation

of btise field operators contain! the term »*** ,which in

the limit of X L — oo hdta been found to have an

infinitesimal amp]1tude at every point in spacetso that its

integral aver all splice becomes unity, thereby depicting

the disjoint Hilbert subspaces of a full Hilbert space.

-43-

This relation elegantly explained the long range

correlationwhich carritu twty the symmetry at the ground

state leaving it -frozen in the asymmetric state.

Representation theory of groups shows by group

decomposition technique that coherent state* identified at

different regions of excitation spectrum of superf1uid-He

become Gaussian wave packets centered at103

coordinate "q". This identification validates the context

of our symmetry breaking mechanism.

Using the displacement of the field operator and a

•caling transformation Me obtained the Bogoliubov spectrum,

as Hell it, the Landau and Khalatni Irov excitation spectrum

of superfluid '•He. From the forme of the spectra

obtained with our formalism, it is seen that the lowest

modes of excitations in the superfluid ''He are the Phonon*

and these excitations vanish in the limit k — > 0 in

accordance with the Baldstone theorem. From our formalism

it is evident that it is not necessary to bevel ope the104

cumbersome theory of weakly repelling bosons or hard

sphere gas methods. This formalism is simpler in the sense

that the states of bosons in superfluid helium are

treated in such a way that the macroscopic condensation is

effected by an invariant inhomogeneou* trincformatiom

Treating it as a gauge transformation and analysing

it in the light of scaling transformation, we visualized a

formalism which gives a general vehicle to the study of

special features of superfluid helium. Thus it bacame

evident that such an approach is a more general framework

than the dollactive variable theories for the study ofa. MB

superfluid ~*Ht>. Our analysis can lead us to new insights

in the superfluid system , establiehing a new connection

with the infrared divergence problem, quantum

electrodynamics and soft pion theorem in high energyphytti ci.

From the study of Abelian transformation,it is spun

that in the absence of vorticity we must have a flow

without fr i t t ional damping in accordance with the Landau's

acceleration equation.*

-44-

In the light of the symmetry breaking , we have found

the explicit forms of the transition probability* between

various states , such as the condensed states, the

oscillating condensed boson state,' the phonon states and

the roton states of superfluid^He. The forms of the

transition probabilities reveal that the states of

superfluid -*He considered here ,are the probable ones.

These transition probilities hava been obtained as the

functions of the condensate fraction. Our formalism of the

novel extent ion of the Bogoliubov method will enable us to

determine the transition probabilities amongst the other48 80

states , like the Vortices, Maxon states, roton pairingstates 106>107and Maxon pairing states.106

The phenomenological phase variance is the broken

symmetry condition for superfluid^He. As phase variance

causes a translation of the bosonfield operator and

produces the coherent superposition of the states, the

phase symmetry of the free Hami1 Ionian is brokewn by the

interaction terms of the Haml1tonlan. Setting in of the

interaction within the system is treated as the dynamical

effect taking place between the particles or the subsystems

resulting in the creation of phonons and violating thereby

the law of conservation of the Hami1tonian thus breaking

the symmetry of the ground state.

The phonons produced can decay into a number of low

energy phonons or interact among themselves, in which case

the phdnDns mre considered stable against any decay and

the possibilities of the bunching of phonons leading to the

formation of roton-like excitation* has be»n considered.

The coherent states obtained in the phonon region bear a

deep inter-relationship with those in the other sectors of103

the liquids-helium system.

A systematic analysis of coherent state phenomenon in

superfluid helium-4 has been conducted by exploiting the

group theoretic methods. This h*c been done by laying

emphasis on the theory o-f the grciup representation obtained

from the? truncated Ham i 11 on i an of Bogoliubov, incorporating

these in the oscillating condensed bosons.

-45-

Several coherent states visualized are seen to have

deep inter-relationship . As higher eatitJtionE hjvp also a80 i0

group structure and their coherence behaviour has been seeni

this formal ism seems to be more appropriate to deal with

such art interesting and challenging problem in the system.

The approximation method using the canonical

transformations of Bogoliubov to determine the probabil i ty

of transit ion from phonon to roton state yields coherent

rotational states identical to the angular momentum states

Hith

i) absolute value of amplitude peaked around a mean

value of the angular momentum,

i i ) the energy of the states to a good approximation

obeying J(j+1) rule when in the case of coherent100

rotational states the states *r& the members of

an ideal rotation band,

i l l ) The phase of the amplitude appearing roughly

equidistant while in the case of coherent100

rotational states I t Is precisely equidistant.

In the case of the phase* exactly equidistant at t=-0,

and energies obeying the rule j(J+l> perfect ly, the

coherence w i l l disappear after a short time because of time101,102

evolut ion, but i t w i l l appear again p e r i o d i c a l l y . This is

related to the spread of the coherent s ta te over the

spacial angle f o s c i l l a t i n g in time with the same

per iod ic i ty . However, the roton spectrum E following the

ru le j ( j + l ) only approximately implies coherence recovering

quasi periodical l y .

ACKNOWLEDGMENTS

One of the author* (H.A.) would like to thank Professor Abdus Salaths International Atonic Energy Agency and UNESCO for hospitality at th«International Centre for Theoretical Physics, Trieste.

-46-

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