Validity of many-body approximation methods for a solvable model

I

I.C

[

Nuclear Physics 86 (1966) 321--331; (~) North-Holland Publishiny Co., Amsterdam

I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

VALIDITY OF MANY-BODY APPROXIMATION M E T H O D S FOR A SOLVABLE MODEL (IV). The Deformed Hartree-Fock Solution D. A G A S S I a n d H. J. L I P K I N

The Weizmann Institute of Science, Rehovoth, lsrael and N. M E S H K O V

Catholic University of America, Washinqton, D.C. t Received 16 M a r c h 1966 Abstract: T h e previous treatment o f a model with particles in two degenerate levels interacting via a m o n o p o l e force is extended to the case of s t r o n g interaction. A " d e f o r m e d " Hartree-Fock solution is s h o w n to exist when the interaction strength exceeds the critical value where the collective m o n o p o l e vibration becomes unstable in the r a n d o m p h a s e approximation. Properties o f the d e f o r m e d solution are c o m p a r e d with the exact solution with particular e m p h a s i s on the physical interpretation as a simplified model o f the Bleuler, mixed-parity, Hartree-Fock treatment o f the nuclear shell model. T h e overall agreement between exact a n d Hartree-Fock properties is reasonable a n d improves with increasing n u m b e r o f particles a n d interaction strength.

1. The Model and its Symmetry Properties In a series of papers t) (hereafter referred to as I, II and IlI) a soluble model was examined describing a system of particles in two degenerate shells with a residual monopole-monopole interaction. Particular attention was given to the case where the interaction was relatively small, and could be treated either by perturbation theory or by the random phase approximation describing small collective oscillations about the unperturbed ground state. In this paper we consider the case where the interaction is comparatively strong and the methods described above are no longer valid. In this region the random phase approximation would give imaginary values for the excitation energy of the collective excited state. This is commonly accepted as an indication that the unperturbed ground state is unstable against these collective vibrations and that another solution of the Hartree-Fock equations exists which exhibits a finite deformation of this collective type 2). It is generally believed that the transition from spherical to deformed nuclei is of this nature. In the domain of spherical nuclei the spherical Hartree-Fock solution is stable against collective quadrupole vibrations which appear as collective t W o r k s u p p o r t e d in part by U.S. Air Force Office o f Scientific Research. 321

322

i).

A ( ; A S S [ ('t a l .

quadrupole excitations. In the deformed region the spherical solution of the l tartreeFock equations is no longer stable against quadrupole vibrations and an alternative solution of the Hartree-Fock equations exists which exhibits a finite quadrupole deformation. We show that a second "delbrmed'" solution of the Hartree-Fock equations exists in the monopole model for sufficiently large values of the interaction parameter. Properties of the solutions are examined and compared with the exact solutions which have been given in I. These may be of interest for the analogy between the transition from spherical to defl~rmed nuclei and also for the general case of instability of one Hartree-Foek solution, as indicated by the breakdown of RPA and the existence of a new stable Hartree-Fock solution. The model is also particularly relevant to the mixedparity, Ilartree-Fock treatment of nuclei by Bleuler et al. 3). The parity mixing used by Bleuler is just of the monopole type considered in our model. Thus many of the characteristic features of the Bleuler, mixed-parity, nuclear shell model appear also in this simple monopole model. We consider an N-fermion system with two energy levels, each having an N-Ibld degeneracy. Each state is described by two quantum numbers p and a, where a has two wdues: + 1 if the particle is in the upper level, - 1 if it is in the lower level. The quantum number p stands for all the other quantum numbers of the fermion. For each value of p there are corresponding states in the upper and lower levels. The particles interact via a " m o n o p o l e - m o n o p o l e " interaction which scatters particles only between upper and lower states having the same value of p and described by the Hamiltonian H

=

~.~: ~ ~.

(Ttd t,t a a pa q -

12V

p,a

E a p~a a p . , T a p',p,a

F _oap_a.

(I.la)

where a~. and ap~ are respectively creation and destruction operators acting on a particle in the p, a state. It has been shown in I that the Hamiltonian can beTewrittcn in terms of quasi-spin operators as 11 = r J : + ½ V ( J + + j 2 . )

(I.lb)

: e.]:+ V ( J 2 - j 2 ) , ~vith J,. = ~at*, . , a p

, =: ,I x + i , l y ,

P

J-

=

~

"r i O p ~ ap_

l

= J ~. - iJy,

p J-

--

' Z ~a',°a,o.

2

p.a

(1.2)

where the operators J~, Jy and J: obey angular momentum commutation rules. Furthermore, the square of the total quasi-spin j2 commutes with the Hamiltonian, and its eigenvalue for the ground state of the zV-particle system is J = ~-N. Our treatment of the model is completely general and does not require any partic-

MANY-BODY APPROXIMATION METHODS (lV)

323

ular physical interpretation for the quantum numbers p and a. For comparison with the Bleuler model one can consider the a shells as represcnting subshells in the j j coupling nuclear shell model having the same value o f j and opposite parity. The treatment with the number of particles N = 8 then corresponds to eight particles distributed between f~ and gt- subshells. The quantum n u m b e r p represents the magnetic quantum number m. The interaction is the "magnetic monopole'" component of the residual two-body force, which can give rise to a pseudoscalar HartrecFock field. The model (1.1) can also describe several subshells. For example the case N = 12 could describe the f~ and pl subshclls for the lower state and g~ and d t for the upper state. Here the quantum number p would represent j, m. The use of the model for several subshells implies the following two simplifying assumptions for the nuclear Hamiltonian: (i) The monopole component of the interaction has the same strength V in all subshells considered (in this case, for pd and fg transitions). (ii) The energy spacings between the corresponding upper and lower levels (the pd and fg spacings) all have the same value e. Note that energy splittings of subshells within a single model shell (the fg and pd splittings) are not required to vanish. These splittings have no effect on the dynamics of the monopole interaction. These oversimplifications of the Bleuler model should be examined more carefully before applying detailed results obtained from the model (1.1) to specific nuclei. However, one can expect general qualitative features of these results to be relevant. If the model (1.1) is interpreted in the spirit of the Bleuler model, then any two states having the same value of the quantum number p and opposite values of the shell quantum number a must have opposite parity. We then see that the Hamiltonian (1.1) conserves parity as the interaction always involves a simultaneous changc in parity of two particles. The eigenfunctions of H thus split into two sets having opposite parity. Those having an even number of particles excited into the upper shell have even parity; those having an odd number have odd parity. This even-odd conservation law was noted in I and used in the calculation of the exact solutions without the interpretation of parity. When several nuclear subshells are represented together as the same shell in the model, these subshells need not have the samc parity. All that is necessary is that the two partner-states, having the same value of p, have parities opposite to one another. The parity operator can be defined formally in terms of quasi-spin operators as a 180' rotation about the z-axis in quasi-spin space. x = elnJ=e i¢~.

(1.3)

The choice of the phase factor e i* does not affect any physical result. For a particular set of configurations it can be chosen to make the definition (1.3) equivalent to the conventional definition of parity.

324

o.

AGASSI ('l a L

In addition to the operators j z and n the Hamiltonian also commutes with the operators nt, = ~ O p*a (dpa , (1.4) ~r

detined for all values of p. This expresses the peculiar property of the monopolemonopole interaction ( l . l a ) of scattering particles only between the upper and lower states having the same value of p. 2. The Deformed Hartree-Fock Solution

The unperturbed ground state in which all the particles are in the lower level is clearly a solution of the Hartree-Fock equations for the Hamiltonian (1.1), since the interaction term can produce only two-particle excitations on this state. This solution also respects the symmetries of the Hamiltonian; it is an eigenfunction of the total quasispin j2, the parity operator (1.3) and of all the np operators (1.4). In our search for a "deformed" Hartree-Fock solution which breaks the parity symmetry, we naturally try to retain all other symmetries, if this is possible. This is analogous to the retention of axial symmetry in the case of deformed nuclei. The requirement that our new trial functions remain eigenfunctions of np restricts them to a very simple form. We need only to consider single-particle states obtained by mixing a particular state having a given value of the quantum numbers p and a with its partner state in the other shell having the same value o f p and the opposite value of a. The two states for a particle having a = +1 correspond to eigenvalues of J; = + ½. A mixing of these two states corresponds simply to a rotation in quasi-spin space. The further symmetry requirement that our trial functions be eigenfunctions of j 2 simplities them still further. For the case J = ½N, which is relevant to the ground state, all the individual one-particle quasi-spins must be rotated by the same angle. We thus choose as trial functions a one-parameter set of states ~ , in which all the quasi-spins are oriented at an angle c~ with respect to the z-axis in the yz-plane *. This state is simply expressed in terms of rotated quasi-spin operators. Let us write J. = J~ cos ~+J2 sin ~,

(2.1a)

d~. = - J l sin 7+J2 cos :~.

(2.1b)

l-he state ~]/~is an eigenfunction of J~ with eigenvalue J = - )~N. The cxpcctation value of the Hamiltonian ( l . l b ) in the state ~b~ is easily obtained after expressing 1I in terms of Jt and Jz rather than Jy and J_. Thus

11 = e(Jl cos c~+de sin 7)+ V[d~-(dl sin ~ - d 2 cos ~)z],

(2.2)

T h e restriction to the yz-plane is evident from inspection o f the H a m i l t o n i a n ( I . l b ) and can be obtained formally by including the polar angle as an additional variational parameter. If the total quasi-spin o f the trial wave function is rotated, keeping J'-' a n d (J~). constant, then H is clearly minimized when (J~'-') is a m i n i m u m a n d (J~'-', a m a x i m u m ; i.e. when the spin is in the ),:-plane.

MANY'BODY APPROXIMATION METHODS (IV)

325

( ~ J H I~b,) = - e J cos :z + V[~zJ - j2 sin 2 ~x- ½J cos 2 0~]

= - e d { c o s ~ + !"~ ( d - ~ )e( l - c ° s 2 ~ ) }

(2.3)

The H a r t r e e - F o c k solution is obtained by choosing x to minimize the expectation value 42.3). This gives e

cosx - V(2J-I)

e

= V(N-1 i'

l I cos ztl =

(2.4a)

I {(N-l)V/e

Note that the solution (2.4a) exists only if ( N - 1)V/e, is greater than or equal to 1. This is just the condition under which the excitation energy as given by the simple r a n d o m phase a p p r o x i m a t i o n becomes imaginary (see eq. (2.10) of 11). C o m p a r i s o n of the H a r t r e e - F o c k ground state energy 42.4b) with the exact results given in I shows that the Hartree-Fock a p p r o x i m a t i o n is good for values of NV/e "~ 1. For example, the case N = 8 where an exact analytical solution is given in I t we obtain ( H ( N = 8)) = - e( 14 V/Q [1 + -4-t,~(e/V)z ], 42. 5a ) E[exact (N = 8)] = - e { 1 0 + l l8(V/e) 2 + 6 1 1 - 2 ( V / e ) 2 + 2 2 5 ( V / e ) 4 ] ~ } ~" = - e ~ 2 0 8 V / e [ l +~-5o(C./v)z]+higher order terms in ~,/V.

42.5b)

The difference between the two results is seen to be only a few percent. Numerical results for N = 8, 14 and 30 are given in table 1. These show that when NV/e is not much greater than unity the Hartree-Fock energy differs considerably from the exact ground state energy. However, at NV/c = 2 the difference is of the order of 20 '~'~,and for NV/e = 5 it is only a few percent. 3. Parity Doublets in the Hartree-Fock Approximation Let us now investigate the properties of the Hartree-Fock solution under the parity transformation 41.3). This s y m m e t r y is broken by the transformation (2.1) to the rotated coordinate system. Whenever a H a r t r e e - F o c k solution breaks a symmetry present in the original Hamiltonian a degeneracy of the H a r t r e e - F o c k solution is expected. This degeneracy is generally associated with a set of low-lying collective excitations which provides a n u m b e r of states nearly degenerate with the ground state. The degenerate H a r t r e e - F o c k solutions are linear combinations of these collective states and the collective spectrum can be obtained from the H a r t r e e - F o c k solution by prot There is a typographical error in eq. (3.5) of I. The result given here in eq. (2.5b) is corrected

326

D. A(;,'~SSI e l

al.

jecting out states which restore the broken symmetry. In the case of deformed nuclei t h e b r e a k i n g o f r o t a t i o n a l i n v a r i a n c e r e s u l t s in a d e g e n e r a c y o f t h e H a r t r e e - F o c k f u n c t i o n s w i t h r e s p e c t to t h e o r i e n t a t i o n o f t h e d e f o r m e d states. P r o j e c t i o n o f s t a t e s h a v i n g a g o o d a n g u l a r m o m e n t u m f r o m t h e H a r t r e e - F o c k s o l u t i o n gives a set o f s t a t e s w h i c h , if t h e a p p r o x i m a t i o n

is g o o d , c o n s t i t u t e t h e m e m b e r s o f a r o t a t i o n a l b a n d 4). TABLE

1

Ground state energy ,'~

NV r

Exact

8

I. 15

8 8

Excited state energy

HartreeFock

Projected H.F.

Variational

0.313

0.000

0.023

0.257

2

1.032

0.643

5

5.558

Exact

Parity doublet splitting

Projected H.F.

Exact

Projected H.F.

(I.382

0.982

0.7(15

1.005

0.750

0.633

0.533

(I.399

(I.217

5.207

5.208

5.439

5.206

0.119

0.002

--0.172

..- I).641

0.548

0.936

0.119

0.007

O.(X)4

0.000

8

/

14.422

14.286

14.286

14

1.15

0.376

0.015

0.295

14

2

1.636

1.385

1.388

1.517

1.381

14

5

10.268

10.004

10.004

10. 26,4

10.004

A5.5 0. 120

t).075

0.35 I

I).303

I).11112

0.000

o.oot)

o.000

0.270

14

-"

46.151

45.5

30

I. 15

I).471

0.1183

30

2

3.547

3.37")

3.379

3.545

3.3T)

30

5

23.049

22.809

22.809

29.049

__.~t) "" )

30

,-J., 218.83

217.5

0.228

0.280

217.5

The nt, mbers tabulated are energy shills from lhe unperturbed energy i.e. I / :

(.~\~- E).

In t h i s c a s e t h e b r o k e n s y m m e t r y d o e s n o t c o r r e s p o n d t o a c o n t i n u o u s t r a n s f o r m a t i o n b u t o n l y to a d i s c r e t e t r a n s f o r m a t i o n

with two eigenvalues. One therefore

e x p e c t s to find a t w o - f o l d d e g e n e r a c y in t h e H a r t r e e - F o c k

s o l u t i o n . T h i s is i n d e e d

p r e s e n t in t h e s o l u t i o n ( 2 . 4 a ) since t h e H a r t r c e - F o c k c o n d i t i o n d e t c r m i n c s o n l y t h e a b s o l u t e v a l u e o f ~. T h e r e a r e titus t w o e q u i v a l e n t s o l u t i o n s ~p, a n d ~,_~. T h e s e t w o s t a t e s a r c t r a n s f o r m e d i n t o o n e a n o t h e r b y t h e p a r i t y t r a n s f o r m a t i o n (1.31. T h u s t h i s d e g e n e r a c y o f 4-7 is j u s t t h a t r e q u i r e d by p a r i t y . S t a t e s o f d e f i n i t e p a r i t y c a n be o b t a i n e d b y t a k i n g t h e s u m a n d d i f f e r e n c e o f t h e s t a t e s ~'~ a n d ~;_~

Since ~

~+ = 0~+0-~,

(3.1a)

O - -- O ~ - O - ~ -

13.1b)

a n d ~._,, a r e n o t o r t h o g o n a l , t h e s t a t e s ~9 + a n d ~

are not normalized.

MANY-BODY APPROXIMATION METHODS (IV)

327

The expectation value of the Hamiitonian (1. lb) can be calculated in the projected states (3.1). After considerable algebraic manipulations, the following result is obtained:

(_~-,,1I!!9 +)_ (~ .I~'.) (I#_IH[¢_) (I//_1~_)

=

ii + (cos

s n2

1 +(cos ~)2,, (~k~lnJ~k~) I i -

(3",a) '

!-c-°s:¢)2!-! .sin!--~7 1 --(COS ~)2s _] "

(3.2b)

When cos c~ differs appreciably from unity the corrections resulting from the projection are small. The two states then remain approximately degenerate and constitute a parity doublet. The prediction of the existence ot" parity doublets is one of the characteristic features of the Bleuler model. Thus it is of interest to check this property in the exact solutions. These indeed show a low-lying excited state of opposite parity. The energy of this excited state is also given in table 1 together with the energy calculated from the projected Hartree-Fock wave function. We see that for NV/c = 2 or 5 and N = 14 or 30, the splitting within the parity doublet is small in the exact solution. For N = 8 it is somewhat larger. For NV/e, = 1.15 the splitting has dropped considerably below the value unity which corresponds to the case V - 0 but is still not small. In all cases the projected Hartree-Fock solutions give reasonable values for the energies. However, the difference between the two Hartree-Fock energies does not give a good estimate for the splitting. This is to be expected since the HartreeFock procedure corresponds to a variational principle only for the ground state energy and not for the energy difference.

4. Occupation Probabilities in the Hartree-Fock Approximation In the Bleuler model each single-particle Hartree-Fock state is a mixture of two states in the conventional shall model having the same value o f j and different orbital angular momenta. The magnetic moment of such a mixed parity single-particle state thus lies between the Schmidt lines. At first sight this seems to be a spurious value. The Hartree-Fock state ~k, is a linear combination of two states tp+ and ~p_, eq. (3.1), which constitute a parity doublet. The expectation value of the magnetic moment operator in the Hartree-Fock state is thus a weighted mean of the moments of the two members of the doublet. If the two states have magnetic moments, each near a different Schmidt lines, the Hartree-Fock magnetic moment is between the Schmidt lines, but is not relevant to the magnetic moments of the parity eigenstates. The Hartree-Fock magnetic moment is useful only if both parity eigenstates have nearly equal moments. This point can be checked in the present model both by comparing results obtained from unprojected and projected Hartree-Fock wave functions and from the exact solution. For examination of the magnetic moment we consider the cases where the number of particles N is odd and N - 1 particles are coupled to a state of total angular roD-

D. A(;ASSI et al.

328

mentum zero. These N - 1 particles do not contribute to the magnetic moment, no matter whether they are in the upper or lower shell. The magnetic moment comes from the odd particle and is determined by the rclative probabilities of finding this particle in the upper or lower level. In this simple model the Hamiltonian is symmetric with respect to all the one particle states. Thus the probability of finding the "odd particle" in upper level is the samc as that of finding any particle in the tipper level. l-he probability of tinding a givcn particle in the upper or lower level is casily obtained from Hartree-Fock or exact wave functions. From eq. (I.2) we sce that the operator J: is simply relatcd to the number operators N+ and N_ fbr particles in the upper and lower levels

J. = ~ , ( N + - N _ ) = N~.-~,N = ~ N - N _ .

(4.1)

The probabilities P+ and P_ for finding a given single-particle in the upper and lower levels respectively is given by

(4.2)

P~.: = (,:V., ) = ,__' + {J:)

N

:\:

From the Hartree-Fock solution we obtain <~0:~]p+It/G) = ~ [ l ~ c o s : ~ ] = ~. I I ~ ( N _ ~ l ) i )

.

(4.5)

The corresponding values for the projected wave functions (3.1 can be obtained aftcr some algebraic manipulations

<~0~ [P:I~_+) {~+-'0,)

=
1 ¥(_c¢?s ~)2Jl+(cos~)2J

~0-_IP_:: 0_-.)_ = (0:,[P -_[0,) !-+(c°s:z)2s- .... (0-10-)

'

{4.4a)

(4.4b)

1 --(COS x)2s

We see that tile effect of the projection is small if the number of particles is large and cos :t differs appreciably from unity. In this case both parity eigenstates have nearly equal values of P+ and P_. Numerical values of the occupation probabilities for the exact solutions and Hartree-Fock solutions are displayed in table 2. We see again that the HartreeFock result is good for NV:",: = 2 or 5 and N = 14 or 30. For smaller values of N and NV/c the approximations are not so good. Systems with even numbers of partides were used for this calculation since these particular numbers had already been chosen for other reasons. Although the magnetic moment interpretation applies to systems having odd numbers of particles onc expects thc results given in table 2 to be qualitatively valid also for the case of odd numbers of particlcs. We know that in this model there is only a single parameter which determines the characteristics of the solution namely the parameter NV'r or the equ?valent mixing

MANY-BODY APPROXIMATION METHODS (IV)

329

parameter cos ~t. Since both the occupation probabilities P+ and the energy shift or the ground state are determined by the single parameter, it is possible to eliminate this parameter and get the relation between the occupation probability and the energy shift of the ground state. In the Bleuler model these two quantities are related to the TABLE 2 /

i

e

p.

p_

p~.

p_

p.

p_

8

1.15 2 5

0.044 0.146 11.361

0.956 0.854 0.639

0.003 0.214 0.386

0.997 0.786 0.614

0.0~X) 0.208 0.386

1.000 (I.792 0.614

14

1.15 2 5

0.039 0.192 0.387

0.961 0.808 0.613

0.032 0.231 0.392

0.968 0.769 0.608

0.013 0.231 0.392

0.987 0.769 0.608

30

1.15 2 5

0.034 0.234 0.395

0.966 0.766 0.605

0.050 0.241 0.396

0.950 0.759 0.603

0.046 0.241 0.397

0.945 0.759 0.603

deviations of the magnetic moment from the Schmidt lines and the spin-orbit splitting. In the simple Hartree-Fock approximation this relation is expressed simply as AE = (~P,IHI~',) + ;N~: = - Nc _(p+!2_.

1-2P+

(4.5)

5. Properties of the Wave Functions and an Alternative Variational Method

The Hartrce-Fock wave function is easily expressed in terms of the unperturbed single-particle functions 2J+ 1

= T .-j+.0-j+.,

(5.1)

n=O

where J + n = m denotes the eigenvalue of the operator J:, eq. (1.2). Each particle has a probability P_ of being in the lower level and P+ of being in the upper level, and the probabilities for different particles are uncorrelated. The probability of finding r particles in the upper level and N - r in the lower level is just given by the usual binominal distribution. The projected parity eigenstatcs are obtained from the Hartree-Fock states by selecting those terms containing only an even or only an odd number of particles in the upper level. Thus the "mixed-parity" Hartree-Fock method is equivalent to a variational method with "configuration mixing" in the unperturbed basis. The trial wave functions include excited configurations with coefficients given by a binomial distribution with one free parameter, e.g. P+. This is illustrated in fig. I, which shows the projected Hartree-Fock and the exact wave functions, expressed in the unperturbed basis. We

l). A(iASSIet al.

330

see that the binomial d i s t r i b u t i o n furnishes a g o o d description o f the exact wave function for NV/t; = 2 a n d 5 but is very bad lk)r NV/e = 1.15.

I

I

I

i

i

i

0.8-

I

0.6 IL~"

I

NV E = 1.15

0.4 i

I I I

0.2

°k~,ll !_

I

.J

i

_W.~9_..-.~-,_q_

..... 1__

'

I I i

E

o

0.4'I

NV _ 2.0 /

\

- - l O ~ 0 \\

o:

_._2_

.!

.

"1

!

I 0 . 4 ~.

NV

--;5.0 E

I i

o.2 l-

~

-13

-9

SOl~

-I

-5

+1

*5 m

Fig. I. Expansion of exact (solid line) and ~/'; (dashed. line) wave functions in the unperturbed basis

for

.,%'

- 3(').

This description of the wave function suggests possible extensions and alternatives to the H a r t r e c - F o c k procedure. Instead of a b i n o m i a l d i s t r i b u t i o n with one free parameter, one might try other d i s t r i b u t i o n s with more free parameters. Alternatively, one might note the i m p o r t a n c e o f the few unperturbed states at the peak of the binomial d i s t r i b u t i o n and simply vary their a m p l i t u d e s in the trial function independently. An example o f this p r o c e d u r e is given in table 1 for the case NV/r, = 1.15. where the H a r t r e e - F o c k result is not very good. The trial wave function q~ used was

MANY-BODY APPROXIMATION METHODS (IV)

331

t h e first t w o t e r m s o f the e x p a n s i o n (5.1) w i t h n e v e n :

C~ = a_sl]l_s+a_s+ 2¢_s+ 2.

(5.2)

T h i s p a r t i c u l a r c h o i c e o f t e r m s was m a d e a f t c r e v a l u a t i n g the a v e r a g e and s t a n d a r d d e v i a t i o n o f the d i s t r i b u t i o n function t

obtained

f r o m the u n p r o j e c t e d

Hartree-Fock

wave

T a b l e 1 s h o w s t h a t this v a r i a t i o n a l a p p r o a c h gives a g r o u n d state e n e r g y w h i c h is c o n s i d e r a b l y b e t t e r t h a n the u n p r o j e c t e d H a r t r e e - F o c k result a n d is c o m p a r a b l e to the result o b t a i n e d with the p r o j e c t e d w a v e f u n c t i o n . T h i s c a n be seen by e x a m i n i n g the f o r m o f t h e w a v e f u n c t i o n s s h o w n in tig. 1 for NV/e = 1.15. T h e e x a c t w a v e f u n c t i o n is seen to be well r e p r e s e n t e d by the v a r i a t i o n a l f o r m (5.2), w h i l e the unp r o j e c t e d H a r t r e e - F o c k w a v e f u n c t i o n c o n t a i n s large s p u r i o u s c o m p o n e n t s at m = -J+l = -14and -J+3 = -12. W e s h o u l d like to t h a n k P r o f e s s o r K. Bleuler a n d Dr. D. Schiitte for s t i m u l a t i n g d i s c u s s i o n s o f t h e i r m o d e l . O n e o f us ( N . M . ) w o u l d like to t h a n k J. J. C o y n e a n d R. J. M a c C a r t h y for assistance in c a l c u l a t i o n s . For this particular case the trial wave function (5.2) happens to be a linear combination of the unperturbed Hartree-Fock solution and two particle two-hole excitations. Such variational functions have been considered ~). However, the equivalence of this approach is accidental in this case, as can be seen from the forrn of the wave functions in fig. 1. '

References 1} A. Glick, H. J. Lipkin and N. Meshkov, Nuclear Physics 62 (1965) 118, 199, 211 2) D. J. Thouless, Nuclear Physics 21 (1960) 225, 22 (1961) 78 3l K. Bleuler, seminar delivered at Varenna Summer School (1965) unpublished; J. P. Amiet and P. Huguenin, Nuclear Physics 46 0963) 1:71 4) W. Bassichis, C. A. Levinson and I. Kelson, Phys. Rev. Bl36 (1964) 380; [. Kelson and C. A. Levinson, Phys. Rev. B134 (1964) 269 5) J. D. Providencia, Nuclear Physics 61 0965) 87