The renormalized variational theory: Its structure, interpretation, and relation to other methods

1.C

[

Nuclear Physics A216 (1973) 365--385; ~ ) North-HollandPublishing Co., Amsterdam

[

N o t to be reproduced by photoprint or microfilm without written permission from the publisher

THE RENORMALIZED VARIATIONAL THEORY: ITS STRUCTURE, INTERPRETATION, AND RELATION TO OTHER METHODS LOTHAR SCH~FER Institut fiir theoretisehe Physik, Abteilung fiir Kernphysik, Universit~it Heidelberg, Deutschland Received 30 April 1973 (Revised 12 July 1973) Abstract: We apply the Ritz variational principle to a renormalized form of the Iwamoto-Yamada cluster expansion, restricting our discussion to infinite systems. The structure of the resulting theory is governed by the renormalization which keeps track of the normalization denominator in the expectation value. The single-particle potential for hole states (ul) is introduced as a Lagrange multiplier in the variational principle, and the self-consistent choice of ui guarantees that the renormalizationfactors are determined correctly. The importance of the renormalization is illustrated by a discussion of the two-body approximation to our theory. The general formalism is evaluated in more detail for the representation ~ r = exp(S)q) of the trial wave function. Very fundamental considerations show that the theory is especially adapted to that choice of ~r. In addition, if we use that choice of kgTthe self-consistent single-particle energies are directly related to experiment, and the theory is almost identical to renormalized Brueckner theory. Thus we are able to clarify many aspects of the latter. We also discuss the relation to the theory of Coester and Kiimmel.

1. Introduction A m o n g the m e t h o d s which are used to evaluate the ground-state energy E o of a m a n y - f e r m i o n system the v a r i a t i o n a l a p p r o a c h [Jastrow m e t h o d ] plays a u n i q u e role in that it starts from the expectation value ( H ) -- ( T w I H I T T ) (TTI T T )

(1.1)

of the H a m i l t o n i a n H with respect to a trial f u n c t i o n T T . All other theories from the outset involve the e x a c t ground-state wave f u n c t i o n T o . Thus the d e t e r m i n a t i o n of E o f r o m the variational m e t h o d involves two well separated steps. First expression (1.1) is evaluated in such a way that the n t h t e r m of the resulting 'cluster e x p a n s i o n ' gives the c o n t r i b u t i o n to ( H ) arising h o r n n-particle correlations. The cluster expansion folmally holds for a n y wave f u n c t i o n T T a n d for a n y n u m b e r ( A ) o f particles. I n a second step the exact wave f u n c t i o n T o is d e t e r m i n e d by applying the Ritz variational principle. 365

366

L. S C H X F E R

Recently we have established a renormalized cluster expansion 1, 2), [RCE] based on the theory of Iwamoto and Yamada 3, 4), and we thus have done the first of the steps sketched above. I n this paper we present the results of the variation of the RCE. We restrict our treatment of the resulting 'renormalized variational theory' [RVT] to infinite systems. In the first part of this paper we present the RVT. In sect. 2 we fix our notation and we summarize those results of I which form the basis of our method. The variation of the RCE is carried through in subsect. 3.1. We evaluate the resulting equations using the exp(S) representation 5-7) of ~T [subsect. 3.2]. A thorough discussion of the theory shows that all the important steps and concepts can be understood in detail [sect. 4]. In particular we explain the origin of the renormalization factors and of the self-consistent potentials. In sect. 5 we discuss the lowest order approximation of the RVT, and we demonstrate the importance of the renormalization in the variation of approximate energy functionals. In the second part of this paper we discuss the relation of the RVT to other theories. We show that the RVT is intimately related to renormalized Brueckner theory 8- ~o} [subsect. 6.1]. We believe that our discussion provides us with a complete understanding of many aspects of that theory, including the renormalization factors, the self-consistent potentials, and the mass operator variational principle. The relation to the theory of Coester and Ktimmel s, 6) is discussed in subsect. 6.2. Sect. 7 contains a short summary of our results. In appendix A we present the rules for the diagrammatic evaluation of the cluster integrals using the exp(S) representation of ~a'. In appendix B we prove some relations which are used in subsect. 3.2. In some respects our theory is a generalization of results presented by da Providencia and Shakin 7, 21). These authors discuss the first two terms of a renormalized version of da Providencia's cluster expansion. These first terms are nearly identical to the corresponding terms of the RCE. Useful discussions of the RCE can also be, found in the work of Ristig and Clark 12). Besides this a lot of work has been done on the lowest orders of the cluster expansion and especially on the connection with low-order Brueckner theory [see refs. 13,14) for further references]. 2. Results of the RCE [A ~ oo] The cluster expansion formally holds for any representation of the wave function ~x [see I, sect. 2 and ref. 4)]. For ease of writing, however, we here confine ourselves from the outset to the ansatz s - 7 ) A

15t':r) = exp (S)lq~) = exp { E S(")}lq~),

(2.1}

11=2 1~)

S (")

=

(n!) -2

~ il

. ..

in,

(b 1 bl

. ..

= .

a s+ •

.. a~[0),

. . b , IS(")li 1

.

.

bn

t T h e first p a p e r will be referred to as I in the following.

(2.2) .

"

ln)abl

+

.

.

.

a b+n a i , ,

.. a~,.

(2.3)

RENORMALIZED VARIATIONAL THEORY

367

Here we have introduced the destruction (creation) operators a~ (a +), and 10> denotes the bare vacuum state. In 1#> the hole states [1> to IA> are occupied. The unoccupied states [A + 1>, IA + 2 > , . . . are called particle states. Summations over indices i to m (b to e) always range over all hole (particle) states. We use a plane-wave basis, and thus S (t) vanishes by virtue of momentum conservation. The correlation amplitudes @1 • • • b,,]S(")lil • • • i , ) can be interpreted as matrix elements of symmetric n-body operators S (") with respect to normalized antisymmetric n-particle states ]cq . . . e,> = a + . . . a+10).

(2.4)

The cluster expansion is used for the evaluation of the expectation value <0> of any operator (9. In I, sect. 5, we have proved that for an infinite system the renoIrealized cluster expansion of <(9> is given by <(9) = Z, + ,=2~.t i 1 il

, Z xh...i,[(9]llg,,: r=l

(2.5)

. . . in

The renormalization factors gi are to be calculated from the equations gi = 1 -

xiil...i,,gi n=l n. il...in

(2.6)

gi.. r=l

The cluster integrals x i l . . " i. and x~l ...i.[(9] have been defined in I [see also ref. 4)]. Roughly speaking x/ .... i. gives that contribution to <~TI~T> which arises from nparticle correlations among those particles which in I~) occupy the states i 1 to i,. Similarly x~.... i. [(9] gives a contribution of n-particle correlations to <~gTI0[~T> in which (9 acts on some of the correlated particles. Both xii.../, and x~,.., i. [(9] are symmetric functions of their indices, which vanish if any two of their indices become equal. For the representation (2.1) of kuT the detailed structure of xi~.., in and of n

X i . . . . in[(9 ] = X; . . . .

J ( 9 ] + X l .... ~ , ~ <01%(9a +10)

(2.7)

may be found from the diagrammatic representation given in the appendix. With the choice (2.1) the renormalization factors gi become equal to occupation probabilities g~ =

p, =

< ~T/T]a+ai] ~F/T)

,

(2.8)

' ~ <~BlVITfi>a= + a a+a a a ~

(2.9)

as has been proved in I, theorem 5.5. Specifying the operator (9 to be the Hamiltonian H = T+

t~a~+ a ~ + ~

V = a=l

af176

= l

of the system under consideration, and using eqs. (2.5) and (2.7), we find

,:2~ •

2 [Xh...i.--x,,..., . "

...in

i

s=l

tl,]

Here and in the following we use Xi .... i, to denote X i ~ . . . j H ] .

fl 9i,.

r=l

(2.10)

368

L. SCH.~FER 3. D e r i v a t i o n o f t h e R V T

3.1. RESULTS OF THE VARIATION In applying the Ritz variational principle to < H ) we note that both Xil... i. and #i are functionals of It/T. Following ref. 7) we treat eq. (2.6), which fixes g~, as a subsidiary condition in the variation of eq. (2.10). Introducing the Lagrange multi' pliers u, we thus vary the functional

~--[s(m); gi; Ul]

: -~- E nil ,

I1

1

E

(n-l)!

i~...s°

x, .... in H gir ,=1

= Z fi+ Z u i ( 1 - g i ) + ~ [ S(m); g,; us], i

(3.1)

S

~[s(m);gi;ui 3 =

1

Z

Oi~,

[Xq...','xi,...',,Ee'~}

n = 2 ~ . Sl...~n

r=l

(3.2)

s=l

(3.3)

es = t i + u i .

In deriving eq. (3.2) we have used the symmetry properties of the cluster integrals. The Euler-Lagrange equations of the variational principle take the form #.f

-

0~

0 <:> scgi = _ _

~ui

c~Y _ O ~ Og~ ~Y

+ 1,

(3.4)

,

(3.5)

Ou~ s~us

-

a~

#gi

a~

= 0 <:>

O*

= 0.

(3.6)

a*

All derivatives are to be evaluated at the point [soS(m); SCgi; S~ui] which is a solution of this set of equations (self-consistency problem). Clearly eq. (3.4) is equivalent to eq. (2.6), whereas eq. (3.5) fixes S=ui. Eq. (3.6) establishes a system of equations from which the correlation operators ~cs(m) must be calculated. In the following we omit the index sc. We express eqs. (3.5) and (3.6) in a more explicit form 1

u. = 0

~

.-~

g,~,

(3.7)

~" } i~i g,.

(3.8)

X [ x . . .s,.-~,,.. ,,.Z ~,~}

{ ( n - 1m)! s~ + Z, . . . s,, . O < b , . . .

~Xs~ ...s. bmlS(m)li~.., i,,,}*

ax,...s.

~*

r =1 ei" s = m + l

We again have used the symmetry properties of the cluster integrals. The variation of Xs~... s. or xil.., i. with respect to * yields non-vanishing results only if all the indicesjl.. "Jmare taken out of the set i~ ... i~ (see e.g. appendix A).

RENORMALIZED 3.2. E V A L U A T I O N

VARIATIONAL THEORY

369

OF THE EQUATIONS

The equations of subsect. 3.1 hold independently of the representation of ~T. More detailed equations can be written down if we limit our treatment to the ansatz (2.1). Here we only present the results. The details of the (essentially simple) proofs are given in appendix B. As mentioned in sect. 2 with the ansatz (2.1) the renormalization factors #i become identical to the occupation probabilities p~. A similar result holds for the Lagrange multipliers ui. Using eq. (3.8) and the diagrammatic representation of the cluster integrals we prove in appendix B.1 that eq. (3.7) can be reduced to ui = pi -1

~1 ~ ~, .. . n=2

~.)..,. 6u~} i

=

P~s"

(3.9)

s=l

Here y.(~o is represented by the sum of all diagrams which contribute to X h . and in which the upper point (it) is not touched by a link (see appendix A). Comparing eqs. (3.9) and (2.5) we find that PiUi is equal to the expectation value of an operator U (° which fulfils the relations (Olai U(°a+lO) = O,

1 <=j <= A,

X, .... ,,EU (0] = ~ X}i:)..i, 6u~.

(3.10) (3.11)

t=l

In appendix B.2 we prove that U (° can be chosen as U (') = a+Ea,, v ]

(3.12)

or as a symmetrized form of that expression. This yields p,e,

=

(a~/Ea,, n 3 >

(3.13)

or

el = Eo - ~ ~(°IHI 7s(°) (~(OlTS~o) '

(3.14)

17s(0) = ail 7%).

(3.15)

Here lk~o) denotes the exact ground-state wave function. We thus recover a wellknown definition of the single-particle energies. For a discussion of these quantities we refer the reader to refs. 15, ~ 6, 25, 26). We only want to mention that the expectation value (3.13) can in principle be measured 16). Thus the ei are directly related to experiment. They differ both from the separation energies and from the quasiparticle energies. In lowest order eq. (3.13) has been established in ref. 1~). (Note that in transferring these results to finite systems we become involved in the problem of defining physically meaningful single-particle wave functions 23, 24).) The expression for E o can be simplified, too. We discuss in appendix B. 1 that eq.

370

L. SCH.~.FER

(3.1) can be transformed by using the Euler-Lagrange eqs. (3.8). This yields E0 = ~ t,+ ~ u i ( 1 - p , ) + ½ ~ ptpj i

i

ij B

-

i

1

~ z..,

t~

n=4~l~.il...in

S~ z..,

..(t_l.~Fx(.t) ,,- ,1 . . . .. . _x(.t) . ,..,~,,

2<=t<~n

e~]) I I r=l

p~s"

(3.16)

s=l

We use the notation I~,,.,.,,,> = exp ( S ) ] i l . . .

(3.17)

i,>.

The quantities x~l).. ,. [X}',)...j represent the sum of all diagrams which contribute to xh... f. [Xf .... ~] and which contain t standing links (see appendix A). The last sum on the r.h.s, of eq. (3.16) contributes only in higher orders. Finally eq. (3.8) can be rewritten in a more familiar form. From the diagram rules given in appendix A, we easily find Ox, ....

~
. . . bd¢,,...,~>,

(3.18)

= < b l . . . binlHl~h, . . . . ~,,>.

(3.19)

=

,~

binlS(m)ti~ . . .

i~}*

gX,...,,. g ( b ~ . . . bm[S(m)]ia . . .

iin)*


We substitute these results into eq. (3.8). This yields •

•

x

•

~=1

g
.

=

g X i ~ . . , i,~ bmlS{=)li,..,

- -

,=m+l ( n - - m ) !

.•. . . . ...." ..

+1

~xi . . . . i. binlS(in)li,..,

iin>* - a < b ~ . . ,

Pi~ (3.20)

iin>* ~ e , ~

or equivalently in

(2 (m) Iqq, ... ,,.7 = - ( 2 ( m ) [ T -- E ~'~]- * V I e , , . .

,,>

r=l

qm! ,,=in+ 1

x

( n - m ) ! ,.,+" ...',. b .

~,

.

O < b , . . . { m ~ ( m ~ t . . . iin>* ,=t

.

.

.

.

.

. . . bin>

In+l

Pi,

O < b , . . . bmlS(m)li,.., ira>* "

Here we have introduced the projection operator Q(in) = 1

Z

]b~ . . . bin>
(3.22)

112! b l . . . bm

Eq. (3.21) establishes a system of coupled nonlinear equations. Solving this system by iteration we construct a second expansion in addition to the RCE. Some aspects of that procedure have been discussed in ref. ~3).

RENORMALIZED VARIATIONAL THEORY

37I

Since in eq. (3.21) there occurs Q(m)lozl" ..,m> instead of s ( m ) [ i 1 . . . i n > , it is obvious that for m _> 4 these equations contain disconnected products of lower amplitudes. The operator Q(4)[~Oi,.." ,,>, for instance, contains terms of the type [S(2)]zlil... i4>. Using the diagrammatic representation of eq. (3.21) we are able to eliminate these disconnected terms. 4. Interpretation and discussion of the basic assumptions In subsect. 4.1 we give an interpretation of the formalism evaluated in sect. 3. We explicitly assume that the cluster integrals are sensible quantities. This implies that the representation of the expectation value in terms of cluster integrals is a good procedure. A critical discussion of that assumption is postponed until subsect. 4.2. 4.1. INTERPRETATION The whole structure of the RVT is already inherent in the energy functional (3.1) which follows from the RCE. Aiming at an interpretation of the RVT we thus first discuss the derivation of the RCE and the origin of the renormalization factors. We have shown in I, sect. 7, that the RCE can be derived by expanding separately the numerator (~TI(9I~T> and the denominator <~//TI}[IT> of <(9> into unlinked cluster expansions. The unlinked cluster expansion [see I, sect. 2 and ref. 4)] expresses any matrix element by an uncorrelated contribution, plus contributions of all correlated pairs, plus contributions of three-particle correlations, plus contributions of all different choices of two correlated pairs, and so on. This yields

< ~r I~T> : 1 + ½ E xij+-~ E xiik+ } E XU Xk, + ....

(4.1)

Here we have used the symmetry properties of the cluster integrals, and each term must be summed over all indices. A tilde over a summation symbol indicates that the indices of different clusters must be different. The expansion of < ~TI (91~T> reads !

1

!

1

~

.!

= ½E X,j[(9]+~ E X,jk[O] + ~ E X,j[O]XK,+ . . . .

(4.2)

For definiteness we have assumed that (9 is a two-particle operator. Each term of eq. (4.2) contains exactly one factor x'[(9]. Combining all terms of <(9> which contain x~i[(9] we find

= ½ E ~;J[(9]{|-E (~,,+ ~J,)-½E (~,,,, + ~J,,)-...} ij

1

lm

= ½ E x,J[(glg,ay{ 1 + (9(I/A)}.

ij

In the same way we create the other terms of eq. (2.5).

(4.3)

372

L. SCH~_FER

These considerations demonstrate 1) that the renormalization of x~... J ( 9 ] sums up all terms which arise from dividing out the normalization denominator in <(9> and which contain x~.... ~,[(9]: The renormalization yields a normalization correction t to the unrenormalized clusters X'[(9]. The renormalization factors are not related to the symmetry of the wave function and are not created by the process of solving the Schr6dinger equation. These statements form the basis of our interpretation of the RVT. They show that the renormalization is a sensible procedure since we sum up all the terms which ale created by the same important physical effect. The terms must be summed up into closed expressions for the following reason. If we increase the correlation amplitudes the cluster integrals xh... in will become large, as can be seen from the definition. As a result the iterative solution of eq. (2.6) almost certainly will diverge. At least there will be substantial cancellations among those terms of the not renormalized theory which involve the same cluster x'[(9]. (Note that systematic cancellations among terms involving different clusters x'[(9] would contradict our assumption that these clusters are sensible quantities.) In the RVT the single-particle potentials u~ are introduced as Lagrange multipliers: eqs. (3.5) and (3.6) define quantities S<~")[ul,..., UA], 9~[U~,..., UA] depending on the parameters u~. These parameters apriori have no direct physical meaning but must be chosen in such a way that the subsidiary conditions (3.4) are fulfilled. The self-consistent choice of u i guarantees that the normalization correction gi is calculated correctly. The structure of the RVT can be understood in a more intuitive way if we fix ourselves to the representation 17j> = exp (S)l~b). In eq. (2.10) we may try to increase the binding energy by building up a strong correlation Within the attractive region of the potential and thus decreasing the x'[H]. The enhancement of the correlations inevitably results in a decrease of the occupation probability within the Fermi sea. This effect, which works against an unlimited increase of the binding energy, is isolated in the occupation probability renormalization. Using the ansatz (2.1) and summing up parts of different cluster integrals x' [(9], we can introduce renormalization factors and single-particle potentials for particle states. This amounts to summing up a part of the long-range exchange correlations. In the context of the RYT this is merely and ad hoc procedure which by no means is of such a fundamental importance as the renormalization of the hole states. The unsymmetrical treatment of hole states compared to particle states is caused by the prominent role which in the construction of 7iT is played by the uncorrelated Fermi sea ~. As a consequence the RVT can be applied only to normal systems where the Fermi sea is distinguished from other states on physical grounds. There is no reason to criticize the gap in the single-particle energy spectrum occurring in eq. (3.21). This gap merely reflects the special role which in the determination of gt o is played by the Fermi sea. Some hints in that direction may be found in ref. ~7). C o m p a r e especially eqs. (9.4) and (9.9). See also ref. 27).

RENORMALIZED 4.2. EVALUATION

OF THE

BASIC

VARIATIONAL

THEORY

373

ASSUMPTIONS

In the discussion of subsect. 4.1 we have assumed that each cluster integral x;1.. "~, or xh... t, is to be evaluated as a whole. This assumption is by no means necessary. The characteristic feature of the RVT is the renormalization, and we may split the cluster integrals and reorder eqs. (2.5) and (2.6) in an arbitrary way as long as the renormalization is preserved. By such a procedure the interpretation of the theory will not be changed. We now examine whether there exists a resummation which cancels the renorrealization. The answer to this question depends on the ansatz for ~T. The lowest order normalization correction to the term ½Y,x'tj[d~] reads - ~ X r~ [0]x~k as can be seen from eq. (2.6). Using the form (2.•) of ~T we find from appendix A that this term contains parts which involve products of the form (bctS(2)[ij)*(delS(Z)Iik)*. Such products do not occur in any higher order cluster x;~ ... j(9]. Thus we cannot cancel the normalization corrections by splitting the clusters. The situation changes if we use the product form 1, 2) of ~T: ~T=d{fi[

n=2

1-[ 21< . . .

f~ .... ,.(il .....


i.)]I-['Pi(i)}" i

(4.4)

Here d denotes the antisymmetrization operator, and the argument (i) stands for the coordinates of the ith particle (see I, sect. 2). Trying to carry through the same argument as above we find that the lowest order normalization correction is cancelled [ref. 14)] by a part of X~jk[(9]. ~[hat behaviour can be traced back to the fact that for the product form of ~x three-particle correlations can be created by a product of two-particle correlation functions. Thus in X~jk[(9 ] there occur products of the form 13) fij(i,j)fik(i, k) and these have the same structure as terms contributing to x;j[(9] Xik. This shows that for the ansatz (4.4) the cluster integrals x~1...i,[(9] may not be sensible quantities. The corresponding effect occurs with the ansatz (2.1) if we assume that a part of the amplitude S (3) can be described as a superposition of two amplitudes S (2). From this discussion we conclude that the RVT is especially adapted to the ansatz [~T) = exp (S)[~). It is based on the assumption that in a neighbourhood of the exact wave function ~o the amplitudes S (") should be treated as independent quantities and should not be split into pieces. With that assumption the discussion ofsubsect. 4.1 applies, and the treatment of ((9) automatically leads to a renormalized expansion. Since in these arguments we have made use of our freedom to vary [~T), they apply only to variational theories. Other approaches are possible 5, 6 ) w h i c h may not favour the renormalization.

5. Low-order approximations We illustrate the content of the RVT by a discussion of the two-particle approximation 7) which is defined by the neglect of all clusters with more than two indices. With the expressions for X~j and xij taken from appendix A this yields for the rune-

374

L. S C H , ~ F E R

tional (2.10) (H> = ~ h + ½ ~ <$ij[H- t i - tjkkij>pi pj. i

(5.1)

ij

The corresponding approximations to eqs. (2.6), (3.9), (3.16), and (3.21) read

Pi = ['1 + ~

pj]-l,

(5.2)

J

u, = ~ pi,

(5.3)

J

Eo = ~, t,+½ ~ < / j l V I 0 ~ j > [ ' 1 - ( 1 - p 3 ( 1 - p i ) ] , i

(5.4)

~j

I~k,j> -- l ij> - Q(2)['T - ei- ei] -1VI~Pii>.

(5.5)

Here we have introduced the defect function Iz,j> = Q(2)IO~j> = S(2)lij>.

(5.6)

Taking into account that we use the language of second quantization, we find that eqs. (5.2) to (5.6) are identical to the corresponding equations of the renormalized Brueckner theory [see ref. 9), section IX E]. Eqs. (5.1) and (5.2) give the complete two-particle approximation of the RCE. This approximation takes into account the influence of any number of independent correlated pairs of particles. Besides shortrange n-particle correlations (n => 3) it neglects long-range exchange correlations between different pairs of particles, but it treats the normalization as well as possible. In that approximation we can illustrate the vital influence of the renormalization by a schematic example. We assume the existence of the expression (1/2A) ~ (ijlVlij> = a < m,

(5.7)

ij

and in addition we assume that there exists a function Ix> with the properties A = 1,

(5.8)

Q(2)Ix> = Ix>,

(5.9)

( 1 / 2 A ) 2 = c < 0.

(5.10)

it

Whether these assumptions can be fulfilled depends on the potential V. We discuss the approximate energy functionals

E(q) = __1 ~ (ij+rle~n)~lH(Z)_ti_tjlij+tle~o)~>, 2A i ER(,) = E(n)p2(,),

"

pQ/) = [1 + ,2p(t/) 1-1 _ Piirl)"

tl = > O,

(5.11) (5.12) (5.13)

RENORMALIZED VARIATIONAL THEORY

375

We have omitted the constant term ~t~. Choosing the phase exp (/6) in such a way that the relation e x p ( i r ) ~
(5.14)

holds, we find E(r/) = a + r/b+ r/2c.

(5.15)

Eq. (5.13) yields p(r/) = ~

[(1 +4r/2) ½.1].

2q 2

(5.16)

i

.21

iE ° )

I

+1 c

-1

I

I

.1 I

.3 .5 .7 1.0 ~

I

I

l/I I

I

2.0

10.0

I

I

~'tl

ER

-2 -3 -Z

E(2)II \ E Fig. 1. Different two-particle approximations of the complete energy functional (2.10) as functions of ~/. The scale of the abscissa is transformed according to r//(1 q-~7)-

In fig. 1 we present the functionals (5.11) and (5.12) for the choice a = - 2 , b = - 2 , c = - 1 . In addition we have given the first 'approximations' to ER(r/) which are defined by E(1)(r/) = E(r/)(1-2q2), =

- 2r/2 + 5¢).

(5.17) (5.18)

The most striking feature of fig. 1 is the completely differing behaviour of the different functionals. The unrenormalized functional is not bounded from below and shows no stable minimum. This feature, which is a consequence of the assumption c < 0, is known 18)as 'Emery difficulty'. In our example the Emery difficulty obviously is due to the wrong treatment of the normalization in the functional (5.11). If we take regard of the normalization correction the Emery difficulty is removed. The graphs of E°)(r/) and E(Z)(r/) show that it is not adequate to expand the normalization correction (5.16) in powers t of ~/. In our example this expansion converges t Note tha t r/~ corresponds to the parameter ,¢ of renormalized Brueckner theory 9).

376

L. SCH.~FER

only for r/ < 0.5 (which is equivalent top(r/) ~ 0.83), and the behaviour of successive approximations changes even qualitatively, as is obvious from fig. 1. On the basis of our interpretation of the renormalization factors, we suspect these results to hold quite generally for the representation [~) = exp (S)laS). It is not difficult to write down a three-particle approximation of the RVT. Such approximations yield results 7, 13) very similar to the Bethe-Faddeev theory. In general we can create approximations of any order by approximating the cluster integrals x and x'[H] in an adequate way. In that context we should mention that the arguments raised against the variation of a truncated energy functional do not apply to the RVT. Varying the truncated functional we find the same set of equations as is derived by truncating the exact equations of sect. 3 in the corresponding way.

6. The relation of the RVT to other methods

We here compare the RVT to the renormalized Brueckner theory (RBrT) and to the 'exp (S) method' 5, 6). We will show that the RVT is located somewhere between these two approaches: it has some features in common with the exp (S) method, and it is most intimately related to the RBrT. Throughout this section we use the ansatz (2.1) of ~x since this representation underlies both of the other methods. 6.1. RELATION TO THE RBrT The RBrT is complicated by the existence of overcounting corrections, and due to this fact it has never been presented in a form which is both rigorous and complete. For our discussion we must collect important aspects from several papers. In the following we denote refs. 8-1o) by II, III, IV, respectively. Ref. 19), appendix D, is denoted by V. For clearness we split our discussion into a number of separate parts. 6.1.1. Hole states: renormalization. In both theories each summation over hole states (i) carries a weight factor which is identical to the Occupation probability Pi (III, sect. VIII B-D; IV; V). 6.1.2. Hole states: single-particle potential. A detailed evaluation 20,26) of the diagrammatic definition (II, sect. VI; III, sect. VIII A; IV; V) shows that the hole potential of the RBrT coincides with that of the RVT. 6.1.3. Hole states: relation between Pi and ui. In Brandow's work on the RBrT the relation

pi = I1 - ~u~

1 -1

CO= t ~ + u B I

(6.1)

plays an important role (II; III, sect. VIII D). Here the function u~(oo) is defined by the set of all diagram s which contribute to u~. In the evaluation of these diagrams the energies (t~ + u~) of the external lines have to be replaced by m. In order to establish relation (6.1) within the framework of the RVT, we have to define a function u~(co) in analogy to u~(co). This obviously is achieved by replacing in eq. (3.7) the energy

RENORMALIZED

VARIATIONAL THEORY

377

ei, by co. In addition we must carry through this same substitution in those equations which determine those amplitudes ( b l . . . bmlS(")lilj2 • • .Jm) which are involved in Xij " or xix ... i,,. Evaluating the derivative of u~((o) by means of eqs. (3.8) and (2.6) we find . ..

in

do2

.=2 ( n - 1 ) ! 12... "

= 1-p/~ 1,

=

(6.2)

which coincides with eq. (6.1). We should note, however, that in the context of the RVT this relation is rather uninteresting since the function ui,(og) has no physical significance. 6.1.4. Particle states. In the RBrT the single-particle potentials and the renormalization factors for particle states are defined in exactly the same way as they are defined for hole states (II, IV). We can introduce these quantities within the framework of the RVT, too, and we believe that the resulting treatment of the particle states would be'identical to that of the RBrT. 6.1.5. Energy expression. The energy expression of the RBrT takes the form (III, sect. VIII E) E~ = E ti+ Z u i ( 1 - P i ) - Z ubpb+ ~ " . (6.3) i

i

b

Here ~B denotes a sum of skeleton diagrams which are to be evaluated with selfconsistent single-particle potentials and renormalization factors. Eq. (6.3) is very similar to eq. (3.1) or (3.16). From the resuks of subsects. 6.1.1 and 6.1.2 it is obvious that the first two terms on the r.h.s: of eqs. (3.16) and (6.3) coincide. According to subsect. 6.1.4, the difference in the treatment of the particle states should raise no serious problem. Neglecting this complication we find that also the remaining terms of both energy expressions are similar to each other since both represent a sum of renormalized skeleton diagrams evaluated with self-consistent energy denominators. A more detailed discussion meets with two difficulties. Firstly, in order to evaluate eq. (3.16) we must solve the Euler-Lagrange equations (3.21) (see subsect. 6.1.7). The diagrammatic representation of the solution will be very complicated. Secondly a general and precise description of ~ is lacking. 6.1.6. Variational principle. Within the framework of the RBrT there exists a 'mass operator' variational principle* which is expressed by the following equations (III, sect. VIII E): #E~ _ 0 ~ Pi = ONB +1, Oui Oui

(6.4)

(6.5) OPl

aPi

t This notation is somewhat misleading. Historically it is based on the wrong suspicion that the single-particle potential is connected with a part o f the field-theoretic mass operator.

378

L. SCHXFER

By comparison with eqs. (3.4) and (3.5) we find that eqs. (6.4) and (6.5) are a part of the Ritz variational principle applied to the functional (3.1). A connection of that kind has been proposed earlier (III, sect. IX A, B), but a proof was not available. We should note that within the framework of the RVT the operation strictly equivalent to the derivative in eq. (6.4) is given by

"{- ~ (/Tl!)-2 Z ~U i

m=2

Z {~
w l ~ bmlS(m)~j

j l . . . j r n bl...brn

1

~U i

a +c.c.} x 8

(6.6)

rather than by 8/Su~ alone. By virtue of eq. (3.6) the last terms in eq. (6.6) yield vanishing results. The corresponding remark applies to eq. (6.5). 6.1.7. Basic quantities. The basic quantities (I1, sect. II) of the RBrT are the 'irreducible compact parts' [i.c.p. ]. In the RVT the most natural basic quantities are those functions z~"(o) .... ~.> which are defined by the following system of equations (compare eqs. (3.21)): n

O(")bb! . . . . . . . . . . . °)

• "> =

-- Q(")[T-

~2 ~i,.]- ~ Vl,#~,(o). . .

= exp { Z s o"°}li,

i,>,

~.>,

(6.7)

(6.8)

rn~2

(o) Z,,...,,,>

= S(o")li, ..- i,,>.

(6.9)

We can represent Z.(o) i l . . . i n \/ by the sum of all connected diagrams with n ingoing hole-lines il to i,, n outgoing particle-lines, and without any closed loop. By compari(o) ~.> constitute son with (II) it is easy to prove that the quantities [T - ~ , =n 1 ~i,.]l Z~,... i.c.p, of a very simple structure. Other i.c.p, will be produced by solving eqs. (3.21) (o) z,)by iteration starting from Z,1... 6.1.8. Conclusions. The RVT and the RBrT are nearly identical. Both theories have the same structure, the hole states are treated in exactly the same fashion, and in both theories there hold the same basic relations. There are two differences. Firstly, the particle states are treated differently. This, however, is not a serious problem and can be overcome. Secondly, the iterative solution of eqs. (3.21), which is implied by the RBrT, introduces a second expansion besides the cluster expansion. It has been argued 13) that this procedure is consistent with the cluster expansion. We are not sure, however, whether the iterative solution of eq. (3.21) will automatically create a representation of the wave function in terms of i.c.p. Since we must substitute the solution of the Euler-Lagrange equations into the energy expression (3.16), we are not sure whether the energy, expressions of both theories will automatically coincide term by term. Compared to the common features of both theories these problems clearly are of minor importance. Thus we conclude that the RVT, if Combined with the ansatz

RENORMALIZED VARIATIONAL THEORY

379

(2.1) for gT, gives an independent and consistent derivation of a modified form of the RBrT. The whole discussion of sect. 4 applies to the RBrT without change. This yields an interpretation of the energy-expression, the 'mass operator' variational principle, and of the renormalization and the self-consistent potentials for hole states. It shows that the renormalization of the particle states is of minor importance. 6.2. RELATION TO THE exp (S) METHOD The exp (S) method 5, 6) is based on the Schr6dinger equation which is rewritten in the form e-SHeS[#) = EolCb).

(6.10)

Multiplication of eq. (6.10) from the left by (q~la + . . . a + a b . . , ab, yields a system of coupled nonlinear integro-differential equations for the amplitudes S ("). The relation between the exp (S) method and the RVT is best illustrated by a comparison* of the basic quantities of both methods. We here give the respective energy expressions, renormalization factors, and single-particle potentials 5). RVT Eo

-



p~= < ~gola+a,I ~o>

exp (S) method

Eo - <~[Hl~o> <~l'eo> 1-

<~eol~o> p,u, = (~°[a+[a~' VJlgo) (~eolgo) Ut, =

0

<~la+ai[ go> <~1go>

! x u, = (q~[a+[a~' V ] l g ° )

(~lgo) ub =

0

These expressions show that corresponding quantities of both methods are expressed by matrix elements of the same operator. The expressions of the exp (S) method differ from those of the RVT only in the replacement of (~0[ by (~]. In thelanguage of ref. ~7), sect. IX, the RVT is based on the 'true' description, whereas the exp (S) method represents the 'model' approach. The formal expressions of the exp (S) method are simpler than those of the RVT, as can be seen, for instance, from ref. 5). However this is not a decisive advantage since the low-order approximations of both theories are of the same degree of complexity. Furthermore, these low-order approximations are very similar to each other 7, 13, 21). At present it is not possible to decide which of the two approaches is preferable. * This comparison can easily be extended to finite systemszo).

380

L. SCH~FER

7. Summary We have presented a general theory of the ground state of a normal Fermi system. Our approach is based on the application of the variational principle to a renorrealized cluster expansion. The alMmportant feature of our method is the renorrealization which governs the structure of the theory. We have shown that the renmmalization resums the normalization corrections in the energy expression. These corrections play a decisive role in the variation of ( H ) with respect to a trial wave function and thus the renormalization is justified. The single-particle potential for hole states is introduced as a Lagrange multiplier. The self-consistent choice of that parameter guarantees that the renormalization factors are determined correctly. Thus the self-consistent choice of ut and the renorrealization are closely related. The renormalization is the more deeply rooted concept, and the choice ofu~ is a consequence of it. We cannot use one of these concepts without using the other, and both reflect the influence of the normalization denominator on the expectation value. Formally the RVT is independent of the representation of TT. However there are good reasons to believe that the theory is especially adapted to the choice I~T) --exp(S)l~). We have argued that a basic assumption of the RVT is expressed by the statement that the correlation structure of TT in the neighbourhood of To is best represented by that choice. This implies that we are not allowed to split the amplitudes S (~). Besides this with that choice of TT the renormalization factors become equal to occupation probabilities and the single-particle energies become observable quantities. Usually the latter result is reached only by introducing 'rearrangement energies' which by definition account for the difference between experimental and calculated results 22, 15). We want to stress, however, that the single-particle energies (3.14) should not be compared to separation energies. If we use the representation I~T) = exp(S)l~), the RVT is almost identical to renormalized Brueckner theory. Our derivation is much shorter (and in our opinion also simpler and clearer) than the usual treatment which is based on perturbation theory. Using the diagrammatic methods of I we can generalize the RVT to finite systems. We expect that most of our results remain essentially unchanged. Furthermore the RVT might contribute to the understanding of the self-consistent choice of the singleparticle basis, just as it has clarified the role of the single-particle energies. The author is indebted to Professor H. A. Weidenmfiller for many valuable discussions. He wants to thank Professor H. Kfimmel for an interesting discussion on the relation between the RVT and the exp (S) method.

Appendix A DIAGRAM RULES For completeness we here give the diagram rules (I, appendix A.2) which are used in the evaluation ofx~ .... i, and Xh.:.,.[(9]. : .

RENORMALIZED VARIATIONAL THEORY

"J~

PlX.'~. ~2 "'" ~3m ff >

i

"¢1 "¢2"" " "¢~

a)

bl b2"'°br ilt?/i2t~" : :". ir'~

b)

I¥

381

e)

i2t'b2 b'~i3 i4fb4 d)

Fig. 2. (a-c) Basic ingredients of the diagrammatic representation of the cluster integrals, defined in the text. (d) Example of a diagram contributing to Xq ...t4[O]. This diagram yields the contribution (A. 1).

Rule A.I: Construction of diagrams for Xh... i, [(9]. (i) Draw n pairs of points, each pair connected by a vertical line (fig. 2a). If g) is an m-body operator, draw one horizontal beam with m points (fig. 2b). (ii) In each point of the beam there ends one (directed)contraction line starting at the lower point of a pair, and there starts one contraction line ending at the upper point of a pair. In the lower (upper) point of each pair there starts (ends) exactly one contraction line. (iii) Each lower (upper) point is touched by at most one hanging (standing) link (fig. 2c). It must be touched by a link if it is not connected to the horizontal beam by one contraction line.

Rule A.2: Evaluation of the diagrams. (i) Label the pairs of points from left to right by the indices i x to 1",.Label the lower (upper) point of the pair i, by b, (G)(ii) A hanging (standing) link connecting the points ikl to iks contributes a factor

(bk .... bk, lS(S)ii~l...ik s) ((ek,...

%lSCS)[ik,... iks)* )-

(iii) The line starting (ending) at t h e j t h point of the horizontal beam is labelled by fl~ (yj). The beam represents (171 .. • 17mlCI71 • • • 7m). (Note the notation (2.4).) (iv) A contraction line connecting a lower point i, to an upper point i, contributes Jb,c,- A line connecting a lower point i, to the point ~/j contributes 6b,rj if and only if i, is touched by a link. Otherwise it contributes 6~rj. In the same way a line starting at flj and ending at & contributes either &pjc=or 6hA~. (v) We sum all indices 1?j, 7j (b,, c,) independently over all the states (particle states). We multiply by ( - 1)~-la - ~. Here l is the number of closed loops consisting .of contraction lines and lines connecting a pair i~, and a denotes the symmetry number (def. a.3).

382

L. SCH*FER

These rules are illustrated by fig. 2d. This diagram represents the contribution

--½ ~

~,

fllf12~l~2

b2b3b4cl

(ClC31S(2)1ili3)*(c2c4[S(2)1i2i,>* . • • ca

× (ill flzl GI~172)(b2 b3 b41S(3)1i2 i3 i4>3i,~, 603~2(~#tc,3#2c2(~b2c3Ob4c4.

(A.1)

By virtue of the symmetry of (0 two diagrams which differ only in the ordering in which the contraction lines are fixed to the m-point beam are taken to be identical. The symmetries of S (") yield the following prescription for the symmetry operations P which leave the contribution of a diagram unchanged: (i) Permute the labels a,, b, among themselves in such a way that we interchange only labels which are attached to the same link. (ii) Shift the points along the links with the contraction lines fixed to them in such a way that the original labelling is restored. Two diagrams y and y' are equivalent if there exists a P with Py = y'. (Remember that the ordering in which the lines are fixed to the horizontal beam does not matter.) Def. A.3. The symmetry number o-(y) of a diagram y is equal to the number of operations P with Py = y. A simple rule for calculating o- is based on the notion of equivalent lines. Def. A.4. Two contraction lines are equivalent if they both start at the same link (or at the operato r ) and end at the same link (or at the operator). Lemma A.5. If the diagram y contains r groups of 2s, s = 1 , . . . , r, equivalent lines, then a(y) is equal to 1-I~=12s !Our final result is contained in the following statement. Lemma A.6. The cluster integral Xij... ~,,[(9] is given by the sum of the contributions of all not equivalent, connected diagrams containing the operator 0. The cluster integral x, .... ~, is given by the sum of all such diagrams not containing any operator.

a)

d)

b)

c)

M R d ~)

e)

Fig, 3. (a) Diagrammatic representation of Xill2. (b) Representation of X/lf2[T]. (C-C) Representation of Xq iz[V]. We can reduce the number of the diagrams by allowing permutations of the pairs ia to in. The resulting modifications of the diagram rules are obvious. In fig~3 we have

RENORMALIZED VARIATIONAL THEORY

383

given some examples which represent the following contributions (see eqs. (2.4) and (5.6)): (a)

½ ~, (bl b2]S(2)lil i2>*


=

bib2

~ *(b2 b31S~2)1ili2>

(b)

--

,

blb2b3

(c)

(il i21V[il i2),

(d+d*)

½ ~, +c.c. = +c.c.,

(e)

¼ ~

bib2

* = .

blb2b3b4

Diagram (a) represents xqi2 and diagrams (b) to (e) give Arh i2[H]. Appendix B ADDENDA TO SECT. 3 B.1. PROOF OF EQS. (3.9) AND (3.16)

We define a set of operators Dq... i, by ~,, 1

011...~, =

<12...i,,>

-m=2

~

/7'/! j 2 < . , .

~

*


× a ( b l . . , bm]S(m)lia JE...Jm)*"

(B.1)

The symbol x? / ,Jz< ...
1

E

n=2 (n--l)! 12...i

D1....

,=1

s=l

(B.3)

Combining eqs. (B.2) and (B.3) and substituting the result into eq. (3.7) we find

ul,

=

=

Z

[1-O....jXi

,=2 ( n - l ) ! ~2...i, ~=2 ( n - l ) ! i~...i.

z~ . . .

i,,

Pi~.

. . . . i.

~=2

Pi~

(B.4)

384

L. SCH,~FER

Here ah~"(/o...i, is represented by the sum of all diagrams which contribute to Xh... ,, and in which the upper point it is not touched by a link. Eq. (3.9) follows from symmetry considerations. In the proof of eq. (3.16) we start from the operator

Z

D = ~ (m!) -2 m=2

Jt ... Jm,bt

bmlS(m)lj~...j,,>*


O(bl . . . bm[S('n)[j 1 . . . jm)* "

(B.5) Eq. (3.8) yields

n~2 ~

~' {DXII""z"-(Dxi .... i.) ~ eir}

. i l . . . in

r=l

Pcs = 0.

(B.6)

S=I

We define x!°,...i. L--,I...,.j[Y'(°. a to give the contribution of all diagrams which contribute to x h . . . i. [Xi .... i.] and contain t standing links. With these definitions we find

Dx,~ ...i, = x,,.,./,+

X

( t-Dx!°, ,,.,. ,,,

(B.7)

1 <=t<=kn

DXh.../. = Xi,...i.+

Z

(t-1)X}t,)...i."

(B.8)

O<-t<--½n

Substituting these equations into eq. (B.6) we find

1 Z a=2t/!fl...'tn

r=l

s=l

=½~..Xi°)PiPJ- ,=2 n! h . . . i , ~=a l i p / , { E ( t-lhi'Y`O _(t) 2 ~t___~, " - " a ' " i " - ' % ' " i " Z e,=1 ld}" ~J

(B.9)

This proves eq. (3.16). B.2. CLUSTER INTEGRALS FOR U") The ant/symmetric matrix elements of the operator U ") = a + [ae, V] are given by

We denote by K(i,, is)= K(i~, i,) the set of all those diagrams contributing to X~.... /.[U (°] in which the operator U (° is connected with the upper points (i,, i~) by contraction lines. By virtue of eq. (B.10) and rule A.2(iv) at least one of the upper points (it, i~) must not be touched by a link. We denoteby K(/~)(i~, i~) that subset of K(i,, is) in which the upper point i~ is touched by a link. The subset of diagrams in which neither ofth e upper points (i~, i~) is touched by a !ink . is denoted by K(°)(i,, is). With these definitions we find <1...

Xi .... i.[ U(O] = Z

n>

[g)u~K!i")(i~", i~)-I-6ilsK(i'?(ir, is)+(6u,.+g~i/.)K(°)(ir, is)]

P'~ 8

'

= r 2= l

y,

s=l,s¢r

EK"%I is)+

(Bll)

RENORMAL1ZED VARIATIONAL THEORY

385

The last sum in eq. (B.11) represents the set of all diagrams contributing to Xil.." i, [U(°], in which the upper point ir = i is not touched by a link. By virtue of eq. (B. 10) the contribution of this set is equal to (5.. ~.(ir). This completes the proof of eq. (3.12). References I) L. Sch~ifer, Diagrammatic methods and renormalization in the Iwamoto-Yamada cluster expansion, J. Math. Phys., to be published 2) L. Schafer, Phys. Lett. 41B (1972) 419 3) F. Iwamoto and M. Yamada, Prog. Theor. Phys. Jap. 17 (1957) 543 4) J. W. Clark and P. Westhaus, J. Math. Phys. 9 (1968) 131 5) F. Coester, in Lectures in theoretical physics, vol. 11, ed. K. T, Mahanthappa (Gordon and Breach, New York, 1969) 6) H. Kiimmel and K. H. Liihrmann, Nucl. Phys. h191 (1972) 525 7) J. da Providencia and C. M. Shakin, Phys. Rev. C4 (1971) 1560; C5 (1972) 53 8) B. H. Brandow, Phys. Rev. 152 (1966) 863 9) B. H. Brandow, in Lectures in theoretical physics, vol. 11, ed. K. T. Mahanthappa (Gordon and Breach, New York, 1969) 10) M. W. Kirson, Nucl. Phys. A l l 5 (1968) 49 11) J. da Providencia and C. M. Shakin, Phys. Rev. Lett. 27 (1971) 1069 12) M. L. Ristig and J. W. Clark, Nucl. Phys. A199 (1973) 351 13) L. Schiifer, Nucl. Phys. A194 (1972) 497 14) J. W. Clark and M. L. Ristig, Phys. Rev. C7 (1973) 1792 15) A. E. L. Dieperink, P. J. Brussaard and R. Y. Cusson, Nucl. Phys. AI80 (1972) 110 16) A. E. L. Dieperink and P. J. Brussaard, Z. Phys. 261 (1973) 117 17) B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771 18) V. J. Emery, Nucl. Phys. 6 (1958) 585 19) B. H. Brandow, Ann. of Phys. 57 (1970) 214 20) L. Schafer, Nucl. Phys. A, to be published 21) K. H. L~ihrmann and H. Kfimmel, Nucl. Phys. A194 (1972) 225 22) K. A. Brueckner, Phys. Rev. 110 (1958) 597 23) L. Sch/ifer and H. A. Weidenmiiller, Nucl. Phys. A174 (1971) I 24) L. Schiller and H. A. Weidenmiiller, Nucl. Phys. A215 (1973) 493 25) D. S. Koltun, Phys. Rev. Lett. 28 (1972) 182 26) D. S. Koltun, preprint 27) C. M. Shak/n, Phys. Rev. C4 (1971) 681