Goldstone excitation and symmetry-breaking mechanism in superfluid-He4

IL NUOVO CIMENTO VOL. 14 D, N. 3 Marzo 1992

Goldstone Excitation and Symmetry-Breaking Mechanism in Superfluid-He 4.

M. AHMAD(1) (r S. K. TIKOO(2) and T. K. RAINA(3) (1) International Centre for Theoretical Physics - Trieste, I taly Institute of Mathematical Sciences - Madras, 113, India (2) S. P. College - Srinagar, India (~) Government College for Women - M. A. Road, Srinagar, India

(ricevuto il 18 Settembre 1989; manoscritto revisionato ricevuto il 12 Giugno 1990)

Summary. -- The form of Goldstone excitation in superfiuid He 4 is envisaged by the formalism of scaling transformation. This is achieved within the framework of symmetry-breaking mechanism which is akin to that of the relativistic quantum field theory.

PACS 67.40 - Boson degeneracy and superfluidity of helium-4. PACS 67.20 - Quantum effects on the structure and dynamics of nondegenerate fluids (e.g., normal phase liquid helium-4). PACS 65.90 - Other topics in thermal properties of condensed matter.

1. - Introduction.

Symmetry arguments are very powerful tools for bringing order into the very complicated set of data and the phenomenon of the universe. Symmetry and invariance are synonyms in physics, the former concept relating more to the intrinsic structure of nature, while the latter to the mathematical form of equations of motion. The primary theory relating symmetry considerations to physical consequence is Noether's theorem. This theorem asserts that ,,every continuous symmetry of a system implies conservation theorem and vice versa,,. In many-body theory, the noninvariance of vacuum manifests itself in the fact that there exists a noninvariant vacuum average for some observable for which commutator with the field operator is zero for spacelike intervals.

The concept of noninvariance of the vacuum was introduced first by Heisenberg [1, 2] and then by Nambu and Jona Lasinio [3]. Their basic assertion was that a vacuum cannot be a state invariant under the breaking of internal symmetries. According to Coleman's theorem [4] invariance of vacuum is invariance of the world. Thus every thing that is broken for the vacuum is broken for the world.

(*) During the editorial procedure we came to know that this author deceased.

229

230 M. AHMAD, S. K. TIKO0 and T. K. RAINA

There is a class of systems which is characterized by the phase transition into a state in which a condensed phase can be defined. The condensed many-particle systems like crystals, ferro and antiferro-magnets, superfluid helium and super- conductors have in common the feature that they do not exhibit all symmetry properties of their underlying Hamiltonians. These Hamiltonians are, in general, invariant under arbitrary translations, rotations, invariant under gauge transformation of first kind, i.e. are particle number conserving. Accordingly the specific symmetries which are ,,broken, in this class of systems are the translations, spin rotation, and the lack of gauge invariance. Thus the respective broken symmetry groups are translational, rotational and the gauge group.

Broken symmetry systems are subject to Goldstone's theorem[5] according to which there is always a zero-mass particle associated with a broken symmetry of a continuous group. The significance of the relativistic Goldstone theories though obscured in many facets is still a matter of rigorous studies after the discovery of hadrons containing heavy quarks.

Nevertheless this has led to look for an analogue of the modern concepts of this celebrated theorem for the superfluid-He 4, to get new insights into the mechanism of symmetry breaking. For helium system we have already Hugenholtz-Pines theorem [6], which is a special version of Goldstone's statement, however it has been found that the energy spectrum determined with the self-consistent method of first order[7] exhibits a gap[8] contrary to what is demonstrated by Hugenholtz theorem [6]. The validity or not of the approximations which lead to the excitation spectrum with a gap has been discussed by many authors[9-11]. The Goldstone excitation by analogy [20] between the planar ferromagnet and superfluid helium is identified as fourth sound [12].

In this paper we study the form of Goldstone excitation in relation to the symmetry breaking mechanism, which is akin to that of the relativistic quantum field theory. In this context the main idea would be the symmetry exhibited by the basic fields of the superfluid helium before the onset of 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 reaction. This is discussed in sect. 3. In sect. 2 we discuss the nature of Goldstone excitation where we use the formalism of scaling transformation.

2. - Go lds tone exc i ta t ion in superfluid he l ium.

The Hamiltonian of the system of helium in terms of the field operators in the configuration space is given as

(1) H =-l - - j - f d~xV~'(x)V~(x)-U] d3x~*(x)~(x)+ 2m

1 f d3y @t (x) - y) ~b(x) ~b(y). + -2 f dSx @t (y) U(x

This is invariant under the gauge transformation ~ = exp [iO] ~, ~t = exp [-iO] ~t thus we have the conservation of current. ~, the Bose field operator, obeys the commutation relations

(la) [@(x), ~b t (y)] = i~(x - y) or [~(x), 7:(y)] = i~(x - y)

GOLDSTONE EXCITATION AND SYMMETRY-BREAKING MECHANISM ETC. 231

with

(2) ~ (x) = ~ (x).

Now to take the condensate into account we have

~(x) =/2(x) + ~, (3)

with the conditions

(4) <o1~o(~)1o>=o, <ol@(x)lO>=~o,

where [0) denotes the ground state or the vacuum state and ~ is a eonstant, for the reason that the system is invariant under translation. Equation (4) is equivalent to the existenee of the ODLRO (off diagonal long-range order) in the single-partiele redueed density matrix. As r and @0 are both Bose operators, we can imagine eq. (4) as an outeome of a unitary transformation of the form

(5) ~(x) --~ ~(x) + ~.

In the light of scaling of the Fourier components of the field operators we have

1 - 1 (6) q(x) = - ~ [~(x)+ iz(x)], p(x) = --~ [~(x)- i=(x)],

so that the Fourier components are defined by

I xf 1 ~ q(k) exp [ikx], q(k) = ~ exp [-ikx] q(x) d3x, q(x) = X/5 V~Z (7)

[ 1 ~p(k)exp[ikx] p(k)= 1 f p(x) - ~ , - ~ exp [-ikx] p(x) d3x,

such that

(8) qt (k) = q(-k), p* (k) = p(-k), also [q(k), p(k')] = iSkk.

The introduction of the coordinate q and its coniugate momentum p in our theory is justified by the fact that the study of helium system in the classical limit is needed for the reason that classical description of many-boson system implies superfluidity. Fur ther we write

[ - - - - 1 ak = ~ [q(k) + ip(k)] so that q(k) = --~ (ak + a*-k)

(9) and .

[ p ( k ) - ~ 2 (a!k-ak).

Now as ~(x) changes to ~(x) + a under a canonical transformation, the corresponding

operator in p, q representation changes to Iq(k)+ ip(k)+ a V ~ } so that

1 E [q(k) + ip(k) +a V'~} exp[ikx]. (10) ~b--. ~ + a - V ~

Accordingly the Hamiltonian equation (1) in terms of ,,q)) and ,p~, operators up to the

M. AHMAD~ S. K. TIKO0 and T. K. RAINA

1 (16) H = :

provided that

(13)

with

(14) 1 1 y-1 U~2 Uk = : {],(k) + ],-1 (k)}; U~ = : {v(k) - (k)}, U~ - = 1,

when we associated annihilation and creation operators with P(k) and Q(k) in the form

1 {bk t 1 + b-k}, P(k) = [b:k - bk]. (15) Q(k) = --~ - ~

Accordingly the Hamiltonian under the scaring transformation becomes

1 E I k 2 -a2V, a2k2t2} ~, ~k{Q(-k)Q(k)+P(-k)P(k)}- ~ k , o t 2 m m ' kr

k2 Vk) k2 (17) ~ m + 2a~ ~'~ - 2m '

where o~k, the energy of an excitation (quasi-particle), is given by

(18) o~ = [( k s 12+ ~2Vkk2] L\ 2m ] m

with h = 1. This is the form of Bogoliubov spectrum [13]. For small value of momentum o~k is of the form

(19) oJk = ]k a2Vk = clkl,

232

constant terms and terms of orders higher than the second takes the form

(11) H = +2~2Vk q(k)q(-k)+ -~--mm(P(k)p(-k))-

- E 2vk 2 , k r

where ~ (the chemical potential) = ~2Vo to the first approximation. Now we make the following scaling change to east the Hamiltonian suitable for the free field into the form as that of interaction

1 (12) Q(k) = - -~ q(k), P(k) = ,[(k)p(k).

This scaling transformation is equivalent to a Bogoliubov transformation [13]

bk t = Uk a~ - U~ a -k , bk = Uk ak - U'k aC-k


where

( 2 0 ) c = k-*0 u m

Thus we find that our formalism of scaling transformation leads us to ~k --~ 0 for k-* 0 in accordance with the Goldstone theorem.

One-particle Green's functions for the superfiuid helium-4 have poles at energies of elementary excitations of first and second sound quanta [14]. The hydrodynamics of superfluid-helium[15] also includes two modes first and second sound quanta, the frequency of both going to zero with the wave number. The Green's functions describing the density oscillations have poles for elementary excitations of one kind, i . e . for energies of the first sound quanta [16]. The energy spectrum determined with the self-consistent method of first order[7] exhibits a gap [8]. This gap has been demonstrated [17] to be on account of unsatisfactory account of the long-range effects in the approximation of Coniglio et a l . [18]. The expression for the Goldstone mode frequency in terms of the order parameter and long wave-length forms of static susceptibilities obtained [12] on the basis of nonrelativistic Goldstone theorem [18] and the theory of collective motion of dynamical variables [19] shows up this mode as a pole of the retarded Green's functions of two variables connected with broken symmetry. This leads to the excitation which, by the analogy [20] between the planar ferromagnet and superfluid helium, is identified as the fourth sound[21] with velocity

1 (21) c~ ---- ~- (PS Cl 2 "~ ~n C22) �9

This way the Goldstone mode interpolates between the first sound (at T = 0, Ps = P), i . e . c4---)cl and the second sound c4--~ c2 (for T = T~pn >>Ps ). An analysis of the presence or not of the gap in relation to the breakdown of symmetry reveals that any approximation method by which the properties of a system of interacting bosons at T = 0 can be calculated gives rise to a gapless energy spectrum if it conserves current, exhibits condensation of the states of one or more particles and satisfies the con(~ition [8]

(22) L t k g ( k . o J) = O. k--*0

Our formalism does not yield an excitation spectrum which would exhibit a gap nor we had to impose any preliminary condition for the nonexistence of the gap in the excitation spectrum as has been done by Coniglio and Marinaro [8].

3. - Symmetry breaking.

In this case we consider the Bose field ~b(x) and the equal-time commutation relations:

(23)

where

[~(x), r:(x)]t = ~, = i ~ ( x - x ' ) ,

= ( x ) = 4 r


q~(x) is taken as a scalar field. The theory based on (1) and (2) is obviously invariant under the symmetry operation (5), Here a is a real constant and represents the condensate, which breaks the original symmetry. Now we restrict our attention to infinitesimal a, we then represent the symmetry operation (5) as an infinitesimal unitary transformation in Hilbert space

1 (24) 4~(x) = a = _ [~(x), aQ] I

with

Q = f =(x)dax. V

By virtue of the equal-time commutation relations

1 [~(x), aQ] = - ia f[~(x), =(x)] dSx a. (25) - = V

However, if we decompose the Bose field ~(x) in terms of normal modes, we have

~(x)= ~ ~ 1

We insert this into the generator

(26) Q = f =(x)d3 -- f 3t (x)dSx =

(ak exp [ik .x - ilklt]) + (a~ exp [ - i k . x - ilklt]).

. lv, -~ ~ T ~ak exp[-i]klt] - a~ exp[i[klt]) = O.

Hence Q vanishes identically, in contradiction to (24). Such a difficulty is encountered in the analysis of spontaneous break down of

symmetry in quantum field theory [22]. Now to resolve the paradox that we have arrived at as above, we have recourse to

the breaking symmetry coordinates, which are associated with the order parameter of the condensation from the states with the full symmetry of the Lagrangian to the states with lesser symmetry. They also determine the disjoint Hilbert space of a full Hilbert space [23]. For the helium system, the number of particles in the condensed phase of the superfluid is the broken-symmetry coordinate [24] represented by a. However, in the infinite volume limit the number of particles in the condensed phase of the superfluid is proportional to the total number of particles which in turn is proportional to the volume of the system. Thus we consider the volume as a broken-symmetry coordinate. The broken-symmetry origin of the volume is also evident in the characteristic properties of the condensation of the liquid helium-4 [23] which is second-order phase transition. If Q is the generator of the symmetry (5) in accordance with (26), then clearly it cannot vanish. On the other hand, if Q does vanish, then it cannot be the generator and we explain the contradiction with (26) as follows.

To this end, let us investigate the argument which led to the vanishing of Q. To elucidate we expand the Bose operators in terms of q(k) and p(k) representation so


that we have

(27)

I 1 ~ qk(t) exp[ ik 'x] , ~(x) = X/~ ~

1 ~ Pk(t) exp [ik.x]. [~(x) ~ k

We now notice that the integral of =(x) is just proportional to the canonical momentum of the k = 0 mode.

Accordingly we have

Q = f ~(x) dSx = tgpk = o (t),

since Pk (t) = qk (t) and q(t) is a constant, we have Pk = o = 0. k=O

In this way the vanishing of Q allows itself through the vanishing of the k = 0 mode of the massless Bose field.

But, if Pk = 0 vanishes, then by the well-known argument we cannot regard k = 0 mode as a dynamical degree of freedom of the field. Indeed the vanishing of Pk = 0 would contradict the canonical commutation relation

[qk=O,Pk=o] = i.

There is a close parallel relation here with the well-known problem seen in the quantization of the massive vector field and the electromagnetic field. In both cases the field canonically conjugate to the fourth component of A,(X) becomes zero. Therefore, we reject A0 (x) as an independent degree of freedom.

In the electromagnetic case A0 can be set equal to zero by means of gauge transformation, on the classical level. Hence in our case of superfluid helium by making the analogy we can view eq. (3), namely ~(x)--* ~(x)+ ~ as a gauge transformation which affects only the constant part of the field represented by the coordinate qk = 0 that is Bose condensate. Since we can select a in order to cancel qk = 0, it is evident that k = 0 mode of the condensate cannot represent a true dynamical degree of freedom. Therefore, Bogoliubov treated a as a C-number [13]. Hence the time-dependent symmetry generator can mix elements of zero mass spectrum and at the same time induce C-number constant by this transformation.

This constant represents the BE condensation of the massless bosons which breaks the original symmetry. After getting physical intuitions by this picture we now quantize our Bose field in a gauge in which qk = o mode is zero. Then eq. (23) is replaced by

(28) f ~ ( x ) = - -

~(x)= - -

1 ~ qk (t) exp [ik. x] V ~ kr

1 ~ pk(t)exp[-ik'x]. V ~ kr

An immediate consequence is that the equal-time commutation relation (la) is not appropriate for this case. Indeed the canonical commutation relations for the q and p


take the form

[qk (t), p(t)] = i~kk,, k, k ' r 0.

This now takes the form

1 ~ ~ [qk (t), Pk' (t)] exp [ik. (x - x ')] = (29a) [~(x), r:(x')]t = t' = ~ k ~ 0 k' ~ 0

= 1 ~ exp [ik(x - x ' ) ] , t) k~o

o r

(29b) [~b(x), =(x')]t=t, = -~ ~ ( x t - x ' ) - ,

since the k = 0 mode is missing from the completeness sum. Although (29b) looks strange, but by substitution into (24), we see that the initial

paradox is resolved. The commutation of ~ with Q is now zero. We further note that we need not assume that qk = 0 vanishes to get the above

commutation relation. Pk ~ 0 = 0 means that

(30) [qk = 0, Pk = 0] = 0,

so that the k = 0 mode cannot contribute to the completeness sum in (29a). However the key-point is that (29) is incosistent with the interpretation of qk = 0 as a dynamical coordinate.

The modified commutation relation (29b) does not affect any other consequence as follows from the usual one.

To cite an example the Hamiltonian equations of motion yield

1 i f 1 f dSx , ate(x) = _ [~b(x),H] = --:- [~(x), ~2(x ' ) ]dSx '= =(x ) - ~(x') = r~(x) "t "t "V

and

at =(x) = V 2 ~.

These are the usual results. The right-hand side of (29b) is not simply a delta-function; this happens whenever

the modes of the field do not all represent dynamical degrees of freedom. For such Bose field, the coordinates of the k -- 0 mode can be gauged away and the equal-time commutation relation must be corrected by an addition te rm (-~9) -1 on the right-hand side.

For electromagnetic field the usual commutation relation for the radiation gauge is

vi ) [Ai(x), Ej(x ' )] t=t , = - i ~j V2 ~ ( x - x ' ) .


Taking into account our present analysis the correct commutation relation now becomes

[Ai(x), Ej(x')]t=t,=-i ~ij V2 ~ ( x - x -~ .

Here also the commutation relation depends on the quantization volume. We have to see what happens when we let ~ go to infinity.

If~9 ~ ~, then 1/D on the right-hand side of (29b) cannot be taken as a vanishing quantity because it should be taken that the integral of this term over the quantization volume remains equal to one. Therefore, as ~9-o ~, we should take this term as having an infinitesimal amplitude at every point in space such that its integral over all space stays equal to one. Objects of this kind do occur in quantum field theory.

Rather than dealing with such exotic functions we prefer to carry out the analysis in a finite quantization volume, where mathematics remains well defined. This picture is in consonance with the idea of having disjoint Hilbert subspace of a full Hilbert space determined by volume as the broken symmetry coordinate.

Thus the infinitesimal amplitude of t~ -1 term which emerges in our formalism, such that its integral over all space is unity, depicts the onset of phonon modes, which is the Goldstone boson in this case, which breaks the symmetry. This is because the Goldstone spectrum of excitation underlies the unique properties of the phase transition at which the broken symmetry coordinates make their appearance. This also explains how long-range correlation is visualized in our formalism where k = 0 mode has been treated by us as spurious degree of freedom which was gauged away. If we could embed it along with other modes, then in a way we have to enlarge the Hilbert space. This gives the situation of electromagnetic field where the Hilbert space is enlarged to accommodate the longitudinal mode and the timelike photon. This gives new insights into superfluid system.

4 . - C o n c l u s i o n .

In this work we exploited the translation of the field operator to get the form of the Goldstone boson in superfluid He 4, making use of a scaling transformation. The formalism shows that the excitation spectrum does not exhibit a gap, as was envisaged by Coniglio and Marinaro [8]. Further we had not to utilize any condition on the excitation spectrum, as was imposed by them for the nonexistence of the gap in their approximation. We obtained a broken symmetry for the superfluid system, from which we exhibited the formation of long-range correlations. Our formalism is easier than the cumbersome theory of weakly repelling terms [25], or hard-sphere gas models[26], in the analysis of liquid helium-II.

In our formalism the microscopic condensation is effected by an invariant inhomogeneous transformation and, analysing it in the light of scaling transformation, we visualize a formalism which gives a general vehicle to the study of special features of superfluid helium. This method seems to be a more general framework than the collective variable theories [27]. To this end we have analysed the structure of the Hilbert space of supel:fluid boson field and arrived at the crucial conclusion that the k = 0 mode is not a real dynamical degree of freedom of the field. A consequence


of this statement is that the equal-time commutation relations of the fields cannot have the simple form which is usually assumed but must be corrected by an additional term t~ -1 which does not vanish when t) --) ~. Our analysis leads us to new insights in superfluid system having obvious connections with the infrared divergence problem and soft-pion theorems in hadron physics. The quantum theory of free fields can still give new insights. With regards to the application of our analysis to the theory of spontaneous symmetry breakdown, we say that when expressed in terms of physical fields the symmetry generator vanishes identically, as does its commutator with the field, therefore Q cannot be regarded as the generator of the symmetry ~ - ~ + ~ .

One of the author (MA) would like to thank Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, and the Institute of Mathematical Sciences, Madras, for providing all the facilities of work.

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Goldstone excitation and symmetry-breaking mechanism in superfluid-He4