Journal o] Low Temperature Physics, Vol. 23, Nos. 5/6, 1976

Noncompact Groups and the Excitation Spectrum in Superfluid Helium-4

Mubarak Ahmed

Department o f Physics, Universil v o f Kashmir, Srinager-6, India

( R e c e i v e d J u l y 28, 1975)

It has been shown that the underlying symmetry algebra in the superfluid 4He can be extended to the spectrum-generating noncompact groups. This method fi, rnishes the excitation energy of this system, exact in the.framework of the symmetry algebra.

I. I N T R O D U C T I O N

Noncompac t groups have been vigorously studied in particle physics. 1 This has stimulated us to use these groups for exploring the symmetry and the dynamical properties of superfluid helium. Here we shall at tempt to give the general technique for obtaining the energy eigenvalues, once the symmetry algebra is identified, from the relevant Hamiltonian of this system.

Following Ezawa and Luban, 2 let us assume that the condensed Bose particles in superfluid 4He are under small oscillation. The operator a 0 describing the condensed Bose particles, which was treated as a C number by Bogoliubov, 3 takes the form

q'o ~ ~o + c ; (1)

Here C~ represents the oscillation of the condensed bosons in the ground state. The operators C + obey the following commutat ion relations"

[ C , , C;, +] = fllt, p' t2)

[C . ,C , , ] = C + [ , , c,+,] = o

In order to develop the algebra we postulate the following conditions : 1. Let A1 -.- A, be the generators of the simple Lie algebra of rank 1

such that

[A i, ~9o] = 0, i = 1 - . . n (3)

A is called the symmetry algebra of the system.

6 7 3

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674 Mubarak Ahmed

2. Let ~r J2 . . . . , ~r be the l Casimir operators of the algebra A. Since Cr represents the oscillation in the stationary state of the system, therefore C, is the function of 4'o(Jl "'" ~-r of the algebra A.

3. In order to develop the spectrum-generating noncompact group, let us have a set of generators B i, where i = 1 --. m. These generators have the following properties:

A. The generators Ai,B i (where i ' = l . . . n , i = 1 . - .m) build up a noncompact Lie algebra L.

B. Ai, is the subalgebra of W, generating the compact group. C. The eigenstates of 4'o constitute a basis for the unitary irreducible

representations of L.

The operators Bi will have nonzero matrix elements between the different irreducible representations o f the symmetry algebra Ai,. In this way the group L generates the full eigenstates spectrum of the system.

The following two conditions arise.

[B i , 4'0] • i (4)

[Bi, 4'0] = 0 (5)

In Eq. (4), 4'0 is an element of the algebra L. Let L 1 . . . L g (g >_ l) be the invariants of the algebra L. These will depend on the generators AiB~, 4'0, and for every particular representation of L they become fixed C numbers. The equation

Li = Li(A 1 " ' " A, ; ~b o ; B i . " Bin) i = I . . . g (6)

will then establish the dependence of 4'o on the generators A f l i . Since the representations of L are given in terms of those of its compact

subgroup A, which in turn is found by the values of its invariants ~.r �9 �9 �9 ~.r it is expected that L~ will depend on A i only through the operators J l �9 "" J~, as is evident by the bilinear products of these operators.

Equation (5) contains the generators B~, which have nonzero elements between the different spectra; this makes it difficult to get the value of 4'0(Jl " " Jl) from Eq. (6). But A i appears in combination J l " " J ~ , and since Li are C numbers for the whole spectrum, we find B~ in (6) in combination in such a way that it is obtained for every kind of excitation of the subspace.

When the condition (5) is satisfied, 4'o is not a member of the algebra, which becomes now a noncompact algebra. Hence 4'o has the same eigen- values for every excitation in a given representation of L~. The whole energy spectrum is confined to one for every representation of L~.

Equation (5) can be taken as a limiting case of (4); and it is possible to find the linear combinations of B~ to constitute a raising and lowering operator for 4'o. The commutators of (4) written in terms of these are

Noncompact Groups and the Excitation Spectrum in Supertluid Helium-4 675

proportional to ~o eigenvalue spacing of the Hamiltonian, which contain only the discrete spectrum. When this discrete part becomes vanishing, we get the condition (5).

The relation (5) includes all those symmetry properties that produce the noncompact groups that commute with ~o, while this extension will give algebras to generate ~0 eigenstates; they will not, however, describe the dynamical properties of He-II.

2. HAMILTONIAN EQUATION

Before applying the above algebraic technique to get the energy spectrum of this system and the relevant spectrum generating groups, we here obtain the same results from the relevant Hamiltonian of the system. In this way the efficacy of our algebraic method will be shown.

We write the Hamiltonian of the superfluid helium-4 as follows :

H : E e'ka;ak + �89 E Vkap+kaq+-karaq k k,p,q

(7)

where ~:k = h2k2/2m. Following Bassiachis and Foldy 6 we assume the sub- scripts p, q, k, p take the values 0, - 1, + 1. Hence the Hamiltonian (7) takes the form

H = �9 + ~:(ak a-k + a+ka-k) + V(ag + C +)

+ (ag + C;)2aka k + (ao + C,)2a;a+k] (8) • [a+ka k + a+ka_k +

For V = 0, the ground state would consist of all N particles in the condensed state under small oscillation. Let us assume for our weakly interacting system that the zero momentum state is occupied macroscopically and hence a o ~ a~ ~ N~/2 (Bogoliubov's approximation). Under this postulate the Hamiltonian (8) is written as

H = ~ N O 1 2 VoC~ ~_ ~ C ; Cu(r k -}- N O V k + N O Vo)a [ a k

+ �89 ~ (af a+_k + aka_k) (9) k

Here only the excited states are taken into account, with the oscillation of the condensed ground state.

If the occupation number in the excited state is given by N, the Hamiltonian takes the form

H - = �89 + ~ (ak + NVk)a;ak + �89 ~ Vk(a;a_ k + aka-k) (10)

676 Mubarak Ahmed

We write the bilinear combinations of the operators occurring in Eq. (10) by

eft;1 I l + + = - - ~ a k a_ k + a k a - k)

J z = �89 -- aka-k) (I 1)

~ 3 1 + = ~(a k a k + a+_ka_k + 1)

The operator ~.~ is introduced to close the algebra to satisfy the following commutation relations :

[j1,~r = -ior [o.r ~.r = i ~ , [j3,o.r = i~2 (12)

Since there are three operators, we have a Lie algebra of rank l, which is the same as the angular momentum algebra SO(3), which is isomorphic with the harmonic oscillator algebra SU(2). However, the combinations of the three operators occurring in Eq. (11) differ from those of the angular momentum algebra by a sign occurring in one of the relations (12). It is possible to have operators satisfying the commutation relations formally like angular momentum algebra by defining

Then,

tl = io.r t2 = i~2 , t3 = J 3 (13)

It 1 , t 2 ] = i t 3 , I t 2 , t 3 ] = i t 1 , It3, tl] = i t 2 (14)

The operator j2 now takes the form

Where

j 2 = + + t2) = - j 2 _ o. 2) = 2 _ l ) ( 1 5 )

A + + (16) a k a k - - a _ k a _ k

A represents the difference in the number of particles with the opposite momenta. Now we take the representation of J = - � 8 9 1 8 9 = - 6 . Writing the basis in terms of the eigenstates of J3 , where

rzg3Jnk) • (n k n t- tT )Jnk ) ( 1 7 )

so that

In1, n 2 " " Hk~ = I?'/1), I / ' / 2 ~ " " " Ink~, rt k = 0 , 1, 2 . . . .

we have therefore the energy eigenvalues as

E ( n l . . . nk) = ~ . (n k + �89 + �89 + Eo, E k = (2ekUV k + e2) 1/2 (18)

Noncompact Groups and the Excitation Spectrum in Superfluid H e l i u m - 4 6 7 7

3. ALGEBRAIC CONSIDERATIONS

From the above analysis we come to the conclusion that the Hamiltonian describing the excitation spectrum at small k (weak coupling case) can be written in terms of the harmonic oscillator Hamiltonian, which is justified by the model of Chester ) The symmetry algebra A defined in Sec. 1 can be written in terms of S U(n), which describes the symmetry of the n-dimensional harmonic oscillator. 5 Its generators are expressed in terms of n 2 bilinear products a~ak of the creation and the annihilation operator. From this, the Hamfltonian commuting with all of them is subtracted. The operator of the system having SU(n) as symmetry algebra will be in general a function of its (n - 1) invariants J l " ' " J - - l - In the harmonic oscillator, the energy eigenvalue is found by any one of the quantum numbers. Hence the irreducible representations of the harmonic oscillator are characterized by the operator Yl # 0, J2""Y.-1 =0.

If the energy E is to depend on o.r we have to find a group in such a way that it satisfies the conditions (1), (2), and (3); and further it should satisfy Eq. (4).

To get the relevant group satisfying these conditions, in addition to the + of the harmonic oscillator bilinear combinations of the operators a k a k

Hamiltonian that generates the SU(n) algebra, we also include in the algebra the bilinear product of the operator B, containing the n(n - 1) generators:

+ + B = a k ak,, aka k, (19)

This gives the algebra of the noncompact group Sp(2n). But Sp(2n) is iso- morphic to SU(n). The minimal noncompact group satisfying the above condition is the noncompact extension of SU(n), namely, SU(n, 1). From the operators of the algebra the energy spectrum of liquid helium-II can be found. Hence if we consider the one-dimensional case (n -- 1), then we have the group SU(1, 1), which is isomorphic with S0(2, 1).

The generators of SO(2, l) are given by the relations (l 1). The unitary irreducible representation is the infinite-dimensional one based on the eigenstates In) of the operator a~a_ k belonging to the eigenvalues n - - 0, l, 2 . . . . corresponding to the Casimir operator

j 2 = ~ f 2 _ ~f22 - j 2 = ~ ( A 2 _ 1)

where (20)

A a~a k + ~-- - - 0 k a _ k

From (11) one easily deduces that the noncompact part A~I z + j 2 of the invariant j z is also diagonal in the subspaces of the compact algebra :

y2 x + j 2 = l[(ak+ak)2 + (a~ak) + 1] (21)

678 Mubarak Ahmed

Since H = J2Ek, where Ek is the energy spectrum, substituting (21) in (20) we get

H 2 = {rig + �89 + �89 where

E k = {2ekNV k + e2} '/2 (22)

Hence we infer from this formulation that both the energy spectrum and the eigenstates are obtained by the algebraic structure. The Hamiltonian written in terms of the boson creation and annihilation operators is not used.

4. CONCLUSION

Here we have shown that the noncompact extension of the symmetry algebras not only generate the eigenstates of the superfluid system but also furnish the energy levels, pertaining to the frame of the symmetry. From the algebraic structure the dynamical problems are solved, and in this connection we have seen that a prominent role is played by the Casimir operators. The connection between the properties of the higher algebras and the corre- sponding symmetry spaces is elegantly visualized. In this manner the whole analysis is kept at the algebraic level, and the Hamiltonian written in terms of the creation and annihilation operator is not used.

REFERENCES

1. A. Barut, Nuovo Cimento 32, 234 (1964); W. Ruhl, Phys. Lett. 14, 346 (1965). 2. H. Ezawa and M. Luban, J. Math. Phys. 28, 61 (1967). 3. N. N. Bogoliubov, J. Phys. (USSR) 11, 23 (1947). 4. G. V. Chester, in Elio-Liquido (Academic Press, New York, 1963), p. 57. 5. J. M. Jauch; Ph.D. Thesis (1939) (unpublished): R. C. Hwa and J. Nuyts, Phys. Rev. 145,

4, 1188 (1966). 6. W. H. Bassiachis and L. L. Foldy, Phys. Rev. A 133, 935 (1964).