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An extension of the random phase approximation
I.D.1
[
Nuclear Physics A108 (1968) 589--508; (~) North-HollandPublishiny Co., Amsterdam
I
Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
AN E X T E N S I O N OF THE R A N D O M P H A S E A P P R O X I M A T I O N J. DA PROVID~NCIA t Laboratory for Nuclear Science and Department oJ Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts tt Received 14 August 1967 Abstract: A theoretical scheme appropriate for generalizing the Hartree-Fock and RPA theories to include ground state correlations is developed. The Beliaev-Zelevinsky expansion is used for expanding the Hamiltonian in boson operators. The ground state wave function is defined as the independent boson wave function which minimizes the expectation value of the Hamiltonian. It follows that the ground state cannot mix with one- or two-boson wave functions, and this condition yields immediately equations generalizing the Hartree-Fock and the RPA equations. By consideringtheoscillations of slightly modified independent-bosonwave functions around the equilibrium position, the boson-boson correlations due to anharmonic effects are introduced. Finally, it is shown that the present approach possesses formal properties similar to those of the RPA theory.
1. Introduction
The r a n d o m phase a p p r o x i m a t i o n ( R P A ) has had a great deal of success in describing the b e h a v i o u r of spherical nuclei. W i t h i n its framework, m a n y nuclear properties have been explained t). However, besides the fact that the R P A fails to a c c o u n t for several experimental results, the p r o b l e m of defining a n d investigating the next order to the R P A remains a n interesting one in theoretical nuclear physics. Several approaches are possible in tackling that problem. I n a previous paper 2), the G r e e n - f u n c t i o n formalism has been used for extending the RPA. Corrections to R P A t r a n s i t i o n rates have also been calculated 3) with the a s s u m p t i o n that the twoparticle, two-hole correlation structure of the nuclear g r o u n d state is given by the R P A result for the g r o u n d state wave function. I n the present paper, we use the Beliaev-Zelevinsky m e t h o d for expanding operators defined in the space of m a n y - f e r m i o n wave functions in ideal b o s o n operators 4). We define the g r o u n d state wave f u n c t i o n as the i n d e p e n d e n t b o s o n wave f u n c t i o n which minimizes the expectation value of the H a m i l t o n i a n . This leads to the cond i t i o n that the g r o u n d state should n o t mix with one- or t w o - b o s o n states, which yields equations generalizing the H a r t r e e - F o c k a n d the R P A equations. By considering the oscillations of slightly distorted i n d e p e n d e n t - b o s o n wave functions a r o u n d the e q u i l i b r i u m position, the b o s o n - b o s o n correlations due to a n h a r m o n i c effects are t Present Address: Laboratorio de Fisica, Universidade de Coimbra, Coimbra, Portugal. tt This work is supported in part through funds provided by the Atomic Energy Commission under Contract AT(30-1)-2098. 589
590
s. DA PROVIDf~NCIA
introduced. Following the analogy with the Thouless s) treatment of small amplitude vibrations, it is finally shown that the present approach possesses formal properties similar to those of the RPA theory. The purpose of this paper is, therefore, to present a theoretically consistent scheme. The formal results obtained here generalize those of ref. 2), however, in spirit, this work is closer to ref. 3), where some actual numerical calculations are also presented. A very interesting preprint 6) where essentially the same problem is treated has recently appeared. In that work, the Hartree-Fock and RPA theories are extended by an ingenious variational method which yields simultaneously equations for the density matrix and transition amplitudes.
2. The Beliaev-Zelevinsky expansion We are interested in studying the behaviour of a system of fermions. It is our basic assumption that the Hartree-Fock theory provides a reasonable first approximation and a convenient starting point for a more elaborate description of that system. As usual, a single-particle state occupied in the Hartree-Fock ground state is called a hole and a non-occupied single-particle state is called a particle. We adopt the convention of denoting hole states by the Greek indices cq [3, 7, J . . . . and particle states by the Latin letters m, n, p, q , . . . . The letters 9, h, i , j , . . , may represent both particle or hole states. I f we denote by c + the creation operator for state i, the Hartree-Fock ground state of our system may be written N
I o> = 1-I
+
I>,
(2.1)
where ] ) represents the absolute vacuum. Excited states are then obtained by acting with the particle hole operators (c + c,) on I~0). However, due to the residual interaction, such states do not correspond to the normal modes of the system. In the socalled random phase approximation (RPA), the operators Cz+ c~ are treated as ideal boson operators, and the boson operators which generate the normal modes are given as linear combinations of the operators Cm+ Ca and c~,+ c m. In order to go beyond the RPA, Beliaev and Zelevinsky 4) have proposed an expansion of particle-hole operators in powers of ideal boson operators. We summarize now the derivation of the Beliaev-Zelevinsky expansion in order to present the main ideas involved. If we identify the operators c+~ c m with some ideal boson operators A,m, then from commutation relations like [ c + c = , e+ cq] = c+Cqamp,
(2.2)
I C+Cm, ca+ %] = - C#+ Cm ~ct~,
(2.3)
we arrive at the following expansions for the operators cp+ cq and cp+ %:
RPA EXTENSION +
591
+
c~ cq = ~ An, A~q, ),
(2.4)
~;c~ = ~p~- Z A~Ap,.
(2.5)
P
However, these results require also a consideration of the expectation values
<*olC;Cqla~0> -- 0,
(2.6)
<~0olc~-c~]~o> = 5¢~,
(2.7)
and the identification of the vacuum of the operators A~,, with the Hartree-Fock ground state 1~o>. Only then can the constant terms in the expansions be determined. We may now go one step further in the expansions of the operators c~+ Cr,. AS it may easily be seen, it is enough to write
c+ c,,
=
A~,,-½ E
+ A m A~p A~p
(2.8)
?P
in order to insure that the following equations are satisfied:
[c: cm, ~: cp] = c~+ ca fir,, - c,+ Cm3~#,
[C+~Cm,c-~c,] = 0.
(2.9) (2.10)
We have, for instance, if consistently we neglect quantities of second order in the correction terms
- ½Z {[A.., (A,+.A L A,~)] + [(A,~ A,. ~ ) , AL]} 7q
q
In a similar way it may also be seen that eq. (2.10) is satisfied. One might now ask if the improved accuracy in the expansion for c~ Cm requires for consistency the inclusion of new correction terms in the expansions for c~ Cq and c~ q. The answer is that it does not. We have, for instance,
7
-½ ~, [(A~,., A.,,.A..,,),
(A+ A~,q)]
= amp(A,q-- ½ E A~ra"A~,q A~m,), ~t'm"
so that eq. (2.2) remains satisfied up to first order in the correction terms; the same is true of eq. (2.3).
J. DA PROVIDF-NCTA
592
It was mentioned already that in the RPA, the boson operators describing the normal modes of a system are some linear combinations of the operators c+ c,. and Cm+ ¢~ ,,., (dt(r)~ + (r) + S~ = Z ,~',m~, Cm--t~r~CmC,), (2.11) ~tm
where the amplitudes ~/,.~9 ~'tJ satisfy the following orthogonality and completeness s.v) relations: Z (')* ¢~) "~ (2.12) ij
¢,,j(r)4,~,(s)o(,j).. = o,
(2.13)
~ t,l,(O*,t,(,) ,t,(~),t,(')*'~ t.7~'ij tFkl - - W j i tFlk ) : 61k6jtO(ij),
(2.14)
r
the quantity O(ij) being defined by O(ij) = 0 ~ - Oj ,
(2.15)
with O~ = 1
if
i is a hole state,
O~ = 0
if
i is a particle state.
(2.16) (We have therefore 0, = 1, 0 m -~- O, O ( ~ m ) = - - O ( m ~ ) = 1, O(~fl) = O(mn) = 0.) Our quantities ~/, (r) correspond to the J((') rctm -- ~tm and the a/,(') "tract correspond to the y(O - - ~m in the more usual RPA notation as adopted, for instance by Thouless s). The notation we follow is due to Fukuda, Iwamoto and Sawada 7). Our assumption about the HartreeFock ground state providing a reasonable first approximation to the actual ground state of our system means that the ground state correlations are in some sense small, so that the quantities ~-z~'/'(°(or the --~m,Y(O~are on the average small as compared with the ~b(° (or the Y(')~ for a positive eigenfrequency ~o~. We shall now show that the ground state correlations require a small modification of the Beliaev-Zelevinsky expansion. With the help of eqs. (2.12)-(2.14), we may invert eq. (2.11) (and its complex conjugate) and obtain +
(tb(r).~ "+ 4- Ib(r)*,~ )l
¢mect
~
Z ~Yam--r r
C;Cm
"-~ Z
- - v'm~t --rl~
(r)* S-r ) . .(|]/.(r)~+ ... --r "~-~Jatm
(2.17)
r
Now we remark that these equations are not consistent with the expectation values of eqs. (2.9) and (2.10) with respect to the phonon vacuum. We have indeed (01EC + C m , C+ ¢fl][O) = (0[ C+ CfllO)~mn - - (01 C+ ¢mlO)6~fl ,
(2.18)
(Ol[c+c,.,
(2.19)
c~-c,]lO) = O,
and in appendix 1 the following expressions for the expectation values (OIc+ %10)
RPA EXTENSION
593
and (OIc~+ Cr~lO) with corrections due to ground state correlations included have been derived (°lc+cal °) -- a = t - Y~ ~',t't'<~)*'t'{~'V'q=, (2.20) rq
(Oi4cmlO) = Z ~~/I,<~ ~ . ~//"~* , •
(2.21 )
r7
It is shown in appendix 2 that in order to satisfy eqs. (2.18)-(2.21), one must replace eqs. (2.17) by the following equation together with the corresponding complex conjugate c r+a g = ~ [ ( ~ c t(.> (2.22) m "~- Z.(,)~c+ , m J O r 4_(,/?r)*+ -- ",'rmo~ - - Z.¢,)*~c m . J O r Jq, r
where the correction factors/,~tm " ( ' ) and Zm. "(') have the following expressions: .(r) 7.,m = ~
On ~(r)
m. = Z
[ ~,(fln)d,(r) \A~tm W'an _L~,(a,,)*,t,¢,)a ~ Am~t Wnfl],
(2.23)
[,~(tn),l,(r) ± ,,(fln)* ,l,(r) x \l.m~: '/',an V A a m Wnl3)"
(2.24)
Here, the quantities .(t.) are Hermitian matrices given by (2.25) with = -
Y, V'q~'Z'(r)*'l'(')V'qt,
(2.26)
rq
".,m ---- (OlCm+C.lO)
(2.27)
= Z '1/')*'1'(') w m 7 "Yn7 ' r7
and the quantities Z~ff) are symmetrical matrices in the following sense: Z(~,) ma
=
.(,m),
(2.28)
~n t
which otherwise remain at our disposal. Following a discussion of Shakin and the author 8), we consider now the expectation value (Ore+ c t c + G]0) in order to specify the quantities Z~"). From eq. (2.22) we have
(Ole~+cpC+m¢,lO) \Writ
T Anti
] A kT"*atmT
r "~ Z [ d ' ( r ) * ' l ' ( r ) ~ ~(r)*'l'(r)'~ "~\ W n t Worm T A n t Wotml r -~ Z Wnfl'b(r)*'lt(r)Watm.~,,(¢tm)@znlJ r
,
(2.29)
594
J. DA P R O V I D E N C I A
where small quantities like V ~(,),t,(,)* and ~/ ,r /~(r)~(,). /_.,,~,mv,# have been neglected. Now, , a m Anti we have (01c~+ c# C c~l0) = -- (01c2 c~c~+ c0 0), so that the following condition should be verified: Z "t'(r)*'t'(r) -t- ,,(~m)*
,I,(r)*d,(r) ± ,/am)*~
r
r
,,(~ra)
A convenient choice for ,~,# , which is in agreement with this condition and symmetrical in the sense of eq. (2.28), is Zt~m)
,#
= _ ~ ~',='t'(')'l'(')*~'#~.
(2.30)
r
In terms of the operators A~m = S" (dt(r)R + 4-~b(r)*R r
A + = ~ (,I,(~)R + .4-,h(')*.~ ~
(2.31)
r
it is possible to give the following simpler form to eq. (2.22): +
Cmc~
~
A + + ~V\ ~ m:,,(0.)a ~ a O+n V_L%~,)*A¢,)"
(2.32)
an
The quantities t/,,# q,, m and Z ~") have also simple forms in terms of the operators A,m, namely q.,p = - E (0lA~Aaql0), (2.33) q
ft.,,. = E (oIa+.,a,.lO),
(2.34)
7
Z(") n0
=
-(OIA#,,A,.IO).
(2.35)
These results may be combined with eqs. (2.25) and (2.32) to give %+c , ~ A + m - Z { ½ ( 0 1 A + . A~,IO)Ao,.+~(OIAt3,,Ao.IO)A.,+(OIAm, + ' + + + A,.IO)Ao. + }.
(2.36)
On
Next we look for higher-order terms in the expansions of the operators c + ci in powers of the operators S, and S +. These higher-order terms must be constructed in such a way as to preserve the commutation relations
[c~cj, C?Ck] = C+Ckfjl --Cl+Cjfik
(2.37)
up to first order in the corrections. Although eqs. (2.4), (2.5) and (2.8) are basically acceptable, a slight modification of these equations is imposed by eq. (2.36), which specifies the form of the terms linear in the operators S~ and S +. The equations which replace eqs. (2.4), (2.5) and (2.8) are
RPA EXTENSION
+ Cp Cq =
+
(~,(a")A+ A + 4-v(am)*A A~q),
1
(2.38)
7am
7 + C a C., =
595
6 ~ 7 - ~ A~pAap+~ + , VLr JkLPfl a " ) a "~TP'C~amTA.P? +a+ ±~(am)*a a ~ zJe~mZSflP)' p
(2.39)
pcLm
(A~p A m Aap + Zma A~p),
C : Cm =
(2.40)
VP
where the matrix z,,a'(~P)is given by eq. (2.30). These equations lead to the same commutation relations as eqs. (2.4), (2.5) and (2.8), because they may be obtained from them by means of the canonical transformation which replaces the operators Aa,. by (Aam-½X7 ~'(a'm')A+ ~ if, consistently, high-order terms are neglected. For / ,a'm'~ma ~-~'m'.,', instance, cubic terms in the boson operators multiplied by Z ~ ) should be regarded as X~a fifth-order terms, since (a.) -- - (O[A,,ABm[O), and therefore they should be neglected. I f we denote by: A B C . . . : the normal product of the operators A, B, C , . . . with respect to the vacuum ]0) of the operators S,, so that :s,s: : = sJs,,
:S+ Sr: = S,+ S,,
etc.,
then eq. (2.40) may be written in the following form, which agrees with eq. (2.36), as far as terms linear in the operators Sr, S~+ , are concerned: +
=
1 -r + {~(O[A~,,AB,[O)Apm
A,+m-pn
+½(OJA;mAp, IO)A+ +(O]A;mA2IO)Ap,+½: Ac,,A~mAt~,,. + + .}.
(2.41)
We have therefore justified eqs. (2.38) to (2.40) - they lead to correct commutation relations and to expectation values (0It + G c+ ca[0) possessing the correct symmetries. We shall discuss now the physical significance of the modifications we had to introduce in the Beliaev-Zelevinsky expansion in order to guarantee the correct symmetries of the expectation value (01(c+ G)(c+, ca)J0 ). Those modifications mean simply that the vector [0) cannot be an arbitrary vector of the many-boson Hilbert space. Obviously, a vector ix) belonging to the many-boson Hilbert space may be used to represent a + + state of our many-fermion system only if the vector (% G)(G ca)Ix) obtained by acting on rx) with the operator (e + ca)(c+, ca) is anti-symmetric under the interchanges m ~ n or a ~-+ ft. (It is assumed that the operators (c+ G) and (c + ca) have been expressed as power series in the boson operators A~p.) The acceptable vectors [x) span a subspace of the Hilbert space. For that reason, a canonical transformation taking us from (the ket corresponding to) the Hartree-Fock ground state to the A . . . --dt(r)A+'~ vacuum 10) of the boson operators S, = z.,. . .(,/t(r) . . . -.. --a,,, is not strictly acceptable as it stands. It must be corrected for the violation of the Pauli principle, for instance, by means of an auxiliary canonical transformation. That is the role played by the additional terms we have introduced in the Beliaev-Zelevinsky expansion. It should, therefore, be emphasized that the correction terms we have introduced are not
596
J. DA P R O V I D E N C I A
correcting the Beliaev-Zelevinsky expansion, which does not have to be corrected. They are simply correcting our canonical transformation for the violation of the Pauli principle. 3. Extended RPA
The canonical transformation
(3.1)
S, = ~_, fil/(') A --~I,(')A x/t" = m -~otm "r m ~ t - - ~ + t t n "~ / ~tm
takes us from the Hartree-Fock ground state [Cbo) to the vacuum of the bosons S,, which is given approximately by 10) = {1+½ Z
,t,(o*,t,(r)tA+ n+ tF otrn tlJ nfl \ Za~m "qL fln
la+a+~"~
x
- - 2"~'L~tn " ° - p r n J J I "x" O / "
(3.2)
{{mflnr + The term ( - ~ A1~ , + Apm ) arises from the canonical transformation we have introduced into the Beliaev-Zelevinsky expansion to allow for the Pauli principle. In appendix 3, we discuss the connection between eq. (3.2) and the corresponding RPA result. Actually we have to generalize our canonical transformation to read Sr
=
--
a ( ° + ~ (~//') A , m - -
~/1(")A + ]
(3.3)
{zm
where a (° is a constant. The vacuum 10) becomes, therefore A+ __XA + i0) ~ {1+ Z " jb(r)*~(r)A+ , m " ",m -t-1 ~ 'l'(')*'i'(OfA+ V,m V,0V'~,,"p, 2"'~,aP-,,)}Jq~o>" ctmr
(3.4)
amflnr
This generalization of eqs. (3.1) and (3.2) is necessary in order to eliminate the interaction between the vacuum and one-phonon states. Such an interaction is present even with a Hartree-Fock basis due to the existence of ground state correlations. However, the a (°, like the ~m~b (r) , are small quantities. We may ask now how are the amplitudes a <') and ~bl~> to be determined. That is equivalent to asking which vector of the many-boson Hilbert space best represents, within an i n d e p e n d e n t - b o s o n a p p r o x i m a t i o n , the actual ground state of our system. The answer to this question is given by the variational principle. The equations which determine the amplitudes a (r) and T~/~!~. ~ J ) are to be obtained by requiring that the expectation value (01HI0) of the Hamiltonian H be stationary with respect to small variations of the parameters a (° and ~ ) . It is convenient at this point to introduce a notation for many-boson wave functions. We let [rs . . . t) = S,+ S + . . . S~I0). (3.5) Since the most general infinitesimal variation of the phonon vacuum induced by small variations of the parameters a t') and 0}~) is of the form
10) = ( Z,
+ +½ Y, r$
= 2 e, lr)+½ ~ r
rs
e,~lrs),
(3.6)
RPA EXTENSION
597
where er and ers are infinitesimals, the condition fi(0fHI0) : 0
(3.7)
(rlHI0) = 0,
(3.8)
(rslH[O) = 0.
(3.9)
leads to the equations
By considering a canonical transformation a m o n g the boson operators which does not mix boson creation operators with boson annihilation operators and therefore does not change the vacuum [0), it is possible to require also
(,-Imls) = co;%s.
(3.1o)
The quantities a (~), ~,j ~I,[<) are determined by eqs. (3.8)-(3.10), which play in our problem a role completely analogous to the role of the H a r t r e e - F o c k equations in the problem of selecting the Slater determinant (independent-particle wave function) which " b e s t " approximates the ground state wave function of a system of fermions. It is now convenient to introduce the expansion of the Hamiltonian H in powers of the boson operators H = h(°)+ Z (h~1)S+ +h~l)*S,) r
+
(2) +S~+h,,(2). S,, Ss) --1 ~,/h(2)~,+S: +2h,,sS, 2! ~ "-'~ -"
+ ....
(3.11)
where h (°) = (0[H[0),
(3.12)
h~') = (01IS, H]I0),
(3.13)
(2)
(3.14)
'rs = (0liSt, [Ss, H]]/O), h(2) = (0[IS,, [H, S+]]I0), r,s
....
(3.15)
In terms of the quantities h~l), #2) and ]/(2) eqs. (3.8)-(3.10) m a y be written • -rs r, s h~ ') = 0,
(3.16)
h(2) = 0, rs
(3.17)
h~2~) = 09,6,~,
(3.18)
where cot = a ~ - h (°). It will be shown that eqs. (3.17) and (3.18) provide an extension o f the RPA, and that eq. (3.16) extends the H a r t r e e - F o c k equations by taking into account corrections due to the ground state correlations.
598
s. DA PROVIDENC1A
In order to calculate the quantities h~ ~), h (2) h (2) etc., we write the Hamiltonian H explicitly in the form
H
~t,.jc+cj+½Z ij
Vik, jlCi + Ck+ ClCj
(3.19)
ijkl
and assume a H a r t r e e - F o c k basis so that
ti, j-{- ~
Vla, jot = e i 6 i j ,
(3.20)
~ik, jl = Vik, jl -- Vik, l j "
(3.21)
o~
where
It is convenient at this point to introduce a notation for the expectation values (01Aaml0), (OIA,,,Aa.IO), (0[A]mAa,[0) etc., which are small quantities, according to the assumption that 10) is not too poorly approximated by 1~o>. We let
t/,.,a = (01Aa.,10), t/a.m = (0[A+ml0),
(3.22)
~/a.,;.a = (0IA~,.Aa.10),
A~-.10),
tlma; #n = (OlA+~m
qma;nll = (OIA+~,,,AanIO),
ha.; a. = (01A~-. Aaml0).
(3.23)
Notice that
* ~/am; n#
=
*
?]nfl; a m
(3.24) =
?]/~n; m a ,
* r/ma; "tJ = t/~;,.a = r/an;~m"
(3.25) (3.26)
In the last section, the quantities t/a, a and t/,,,~ had been defined. With the definition of t/a, ,, and t/ . . . . the matrix t/i, j became completely defined. U p to first order in t/i, j, we can write (0[c+ c,[0) --- Oi6ij + tl,, j. (3.27) The non-defined elements of t/ij;k~ are t/a~; rp = 0). With the help of eq. (3.3), one obtains
set
equal to zero
S? [,/,(r) \ ' r a m U ~(,). _ " r .~.(r)*_(r)~ t/Jma u j,
q a , m -~" / , r
tla,,; na = ~ ~',m't'(r)*'t'(')Vna'
(for instance,
(3.28) (3.29)
r
tlm~,;"a = ~ V,,a't'('l*'t'(OV,,~" r
(3.30)
599
RPA EXTENSION
Also, from eqs. (2.26) and (2.27) it follows that rl*,p = - c Ilp.;pB 9
(3.31)
ul m, n = 29,.i;.i.
(3.32)
We may now calculate the quantities h!“, I$:‘, h!Ti etc., with the help of expansions (2.38), (2.39) and (2.41). The calculation is straightforward but somewhat lengthy. Making also use of the Hartree-Fock equations [eq. (3.20)], one obtains A!” = C $I;‘[(& -@(ik)%, ik
i + 7 fi/cj, i[Vl, j
+ F ( - Vi o/c, Ih yljk: Ii - “kfilj, ik Vjk;
kl)l,
(3.33)
where vi = l-28,,
(3.34)
sothatv,= -l,v,= 1. One also obtains (3.35) (3.36) where yfil,kj
=
-(&i-k&l
-Ej-Ek)HKii)
+ o(kz))qkl; i j
+ C {Cm vidij8g1, hk- VkSlkDgi, hj)Vh, g-(Dig, jhYgh; lk + 4g, khflgh; ijP(ikMG) 8h -Dih,kg~jg;lh-ogl,hj~gi;hk
+(‘il, ght?kg; hj+Ogh, kjqgi;Ih
+‘g~,kh~gi;hjI‘Dih,gjqkg;Ih)VgVh-#gi, jkqgh;Zh+61g, jkqgh;ih + ‘li, gkqjh; gh+ ‘li, jgqkh; gh)) -“fTh {sij(OIJ, ghqfh; gk + 6g.f, kh~fh; 19)
+‘klfagf, jhqfh: ig + Oif, ghqfh; gj>>.
(3.37)
The condition that (O]H]O) should be a minimum has provided us with eqs. (3.16), (3.17) and (3.18). In order to find which results arise from those equations, we transform them in the following way:
T (t@*h;f’- l@h;“;) = - co,fjf;‘. With the help of eqs. (2.14), (3.33), (3.35) and (3.36) one finally obtains
600
J. DA PROVIDENCIA
(ek--ei)O(ik)rlk, i+ ~ Ojk, zirh, j+ ~ (--Vi~ki,,hrljh;z~--Vk13,i,~hrljh;k, ) = 0, jl
(3.38)
jib
((.Or'q- 8i-- 8 k ) ~ ) = O ( i k ) ~ (~il, k j ' ~ - ~ i l ,
kj]Wjl'~'l'('>"
(3.39)
jl
With regard to eq. (3.38), we may remark that if we neglect the ground state correlations, that is, if we set ~i,j = r/ij';kl = 0, then it reduces to the identity 0 = 0. This happens because we have assumed a Hartree-Fock basis. If we had not done so, then the Hartree-Fock equations would result in that limit. As it stands, eq. (3.38) provides an extension of the Hartree-Fock equations, since it enables the calculation of corrections to the Hartree-Fock density matrix, namely the quantities r/~,~, (01c,,+c~10), once the I quantities 17ij;ki are known. With regard to eq. (3.38), we may remark that it has just the aspect of the RPA equations. However there is a difference; eqs. (3.38) and (3.39) are to be solved self-consistently, since the particlehole interaction appearing in eq. (3.39) contains the correction term #fit, kj which is a function of the quantities ~/,, ,, and r/u, k~. The quantities r/,, mare given by eq. (3.38) once the quantities ~/u;k t are known, and, in turn, the quantities ~/U;kt are expressed in terms of the amplitudes ~}~) yielded by eq. (3,39). It is difficult to assess the importance of the correction Y~zl, kj in eq. (3.39) without performing an actual calculation. However, it seems that this correction tends to reduce the importance of the backward-going graphs. Indeed we have, for instance, gh =
e
.... ~ - ( ~ . . + ~ . - ~ - ~ )
~,',~>*,',~') W~m 'Fmfl
~ (¢~ ,l,(S)*,l,(s) ~ ~ ,l,(S)*,l,(s)~ \ tJmg, flh tY gh tp'n~t T Idng) ~th t)Ugh t~ rttfl.} ghs
(s)* s
(s)
~ ~
,/,(s)q
gh
gh
~_~ f " 'l'(S)*'l'(s) " 'l'(S)*'l'(S)l = 0, I.~s W~n Wm# ~ t~"s ~t'rtn Wm,6J 8
where we used eq. (3.39) with the quantity Y~it, k:" removed. We see, therefore, that there are some terms in ~ . . . . a (or ~#'~a, ~,) which tend to cancel ~. . . . a (or ~a, ,,,) in eq. (3.39). Finally, it is interesting to compare eq. (3.37) with eq. (3.7) of ref. 2) (where, actually, a term ½[1 + O(gh)O(fd)]~oy ' dh is missing in the right-hand side). There is agreement between both results, although they are based on different approaches, since the Green-function method has been used in ref. 2), It may be seen that the modified form we have given to the Beliaev-Zelevinsky expansion is essential for achieving this agreement. For completeness we give now the leading terms of the expansion of a transition operator M = E ml, jc~cj (3.40) ij
RPA EXTENSION
601
in powers of boson operators. With the help of eqs. (2.38), (2.39) and (2.41) one easily obtains M : m(°)+ Z (m,.(1)S, + "JCt3~ r(1)* S r ) - ~ - . . . . (3.41) r
where m (°) = (0IM[0) is a constant and DI(1)
=
Z ik
'l'(r)Ym 'elk t a.,T± Z (rnk, orlo, iO(ig) O
- ~k,o too,, O(g k))O(ik) - Z (ma, o rh0;ka + }ma,~ rho; go + ½mk, a q~o: ao)}. (3.42) gh
Recent calculations 3,8), based on formulae which although derived by different arguments are analogous to this one, have shown that the ground state correlations tend to appreciably reduce transition rates, thus removing, at least in part, some wellknown discrepancies between theory and experiment. If we compare eq. (3.42) [or eq. (2.41)] with eq. (3.8) of ref. 2), we find that there is agreement between those results, and again we may verify that this agreement requires the particular version of the Beliaev-Zelevinsky expansion we have adopted.
4. The boson-boson interaction and the higher RPA The expansion of the Hamiltonian H should contain cubic and higher-order terms in the boson operators, which, however, have not been represented explicitly in eq. (3.11). If we include such terms and make use of eqs. (3.16)-(3.18), the Hamiltonian H becomes
~o, Sr+S~
H = h(°)+ r
I-h(3)K, + ~ .~ + ~, .~, + +3h,.~.,Sr ( 3 ) + + S~ S,+3ht~.,. ( 3 ) *S,. + S~S,+h,~, (3)* S,.SsSt]+ . . . . (4.1) + ~.~ rs~t L-,'st~,"
The new terms lead, of course, to phonon-phonon correlations. These correlations may be described with the help of the time-dependent variational principle, in the same way that the particle-hole correlations are conveniently described by the timedependent Hartree-Fock equations. We consider, therefore, a state vector of the form ]TJ(t)) = (exp iF)]0),
(4.2)
where F is a time-dependent Hermitian operator such that
F = ~ (f,(X)S,+ +f(1)*S,)+½ Z (f(2)S~+S+ +2f~(.2)S+S~+Z~2)*S, Ss) • r
(4.3)
rs
The time evolution of the operator F is determined with the help of the time-dependent variational principle -- i[(5 ~l (u) - (~'16 7/)1 + 6(~IHI ~u) = 0.
(4.4)
602
J. DA PROVID~NCIA
The terms of first order in F do not contribute in this equation because of eqs. (3.16) and (3.17). The terms of second order lead to
-i(Ol[6F, F]0)+½f(01[F, [H, r]]10) --- 0.
(4.5)
From this equation we finally obtain, in analogy with the time-dependent HartreeFock equations for small amplitude vibrations
_
if,.(,)+corf,.(1)+½ ~ V',,s,,~, :~,(3) ¢C2)
h~3):(2)* -,s,:s, :~
= 0,
(4.6)
St
_if~2)+(eo~+cos)f~2)+ X" it,(3) ¢~1)_h(3):(1)*'~ = 0 Z~ k'~rs, t d t "'rstJ t ]
(4.7)
t
plus complex conjugates. These equations are solved with the ansatz
f ) l ) = X,e-le, + y , ei~, f~2)
(4.8)
, * iEt Xrse-iEt + ~x,se ,
(4.9)
which leads to
EXr = ¢OrXr+½ EXrs
=
(a)
Z (hr, stgst-St
(ojr÷OJs)Xrs_}
hr~t ( 3 ) y ,s,),
- ~,"' (~h\ ( 3rs, ) tX t_ I
h,st (3) Yt),
-EY~ = cor Y,+½ V ¢h(3) v - - h"'rst t 3 ) * Y" ~ s t ]"~, /, \'~st, r "st
(4.10) (4.11) (4.12)
st
-EY,~ -- (o~,+ ~o~)r,s+ Z (h~3),Y,-ht3)*Y ~ . . , s~,:. ,
(4.13)
t
Following the Thouless 5) treatment of the RPA theory, we may write these equations in the form
B*
A* k Y 'v)] = E~ k _ r ( , ) ] ,
(4.14)
where X (~) and Y(~) represent the column matrices of the quantities y co, X~{) and y~v), y C]). The quantities A and B are appropriate square matrices. For instance
A,, s = ~Or6rs, A, , st = hi3) ,,st,
etc.
Assuming the eigenvalues E v are real and non-zero, we normalize the eigensolutions of eq. (4.14) so that (XC~),_ y(~),) [XC~)~ \y¢,,)] = 6 , ~ -IE~l -. (4.15) E, If we represent by 10} and by I/z}, respectively, the ground state wave function and the wave function of the excited state with energy E, (in the RPA applied to the phonon-phonon interaction), then to the quantities X}"), X}~), Y}") and Y}~), the following interpretation may be given 5):
RPA EXTENSION
603
{~IS)I0} = ~'(")*
{~,ls)s: IO} =
x~2 )*,
{plSrlO} = - Y~")*,
{IXlSrS,,iO} = - Y/f)*.
(4.16)
The ground state [0} differs from the ground state 10) essentially in that 10} contains boson-boson correlations. We consider now an operator
,.,,. +m~t)*Sr) M = m(°)+ ~ (,,,, ~.(~)c+ r
"-~½ Z (...(2)C,+C,+ (2) + (2)* \ f l i t s o r o S +2mr ,S, Ss+mrs SrSs),
(4.17)
rs
and introduce the column matrices
m*
m = ~,.~)/,
=
[rn~l)*] kmr,(2)* ]
(4.18)
"
The matrix element {plMI0} may be written {/~,MI0} = (X(~)t- Y(")') ( m ) m*
(4.19)
"
The well-known theorem discovered by Thouless s) in the time-dependent HartreeFock treatment of small amplitude vibrations remains valid here [{0IM[/t}[2E. g(E~, > O)
= Z Im~l)/~°.+½ Y. Im~)12(°).+'°.) r
rs
+21" Z k( f~l l(r1 ) * LIt,~ ( 3 ) st f.~(2)__ flst T
. ( 2 ) * Lflst~ ( 3 ) r l..i l t(I)
f[lst
rst
_ ~/T/(1)* m ( 2 ) * / a ( 3 ) z
r
"'~st
"~rst
X/a(3)*v~(1)w(2)~ --
2 I~rst
liar
Htst
]
= ½(01[M, [H, M]][0).
(4.20)
This is a useful result. It may serve for showing that the well-known Thomas-ReicheKuhn sum rule 9) and the usual conservation laws of momentum, angular momentum etc. are preserved in the present approach, within the approximation considered. We give now the expression of the anharmonic terms in the boson expansion of the Hamiltonian H [see eq. (4.1)]. We have, with the help of eqs. (2.38), (2.39) and (2.41) h(3) --- ~_~ Bit kj ~ rn't
ikjlh
h(3) ,,,,
,
r's't'
n(rst) 'l'(r')'i'(s')'l'(C) ~r's't"Pki Wjh 'Fhl
^ kj Y', D('s)r't'(r')'l'(")'t'")*" ,t,(").l,(')*,l,(*') ±,t,(o*,t,(r'),l,(")~ = ~ Vu, --r's'ktl"ki ~ ' j h tPlh ~Fki tPhj Whl TtPik W j h W h l I~
ikjlh
r's"
(4.21) (4.22)
604
J. DA PROVID~NCIA
where .P(r~'",,+,...')t'is equal to 1 if r ' s ' . t ' is . any . permutation . . of the . indices r s t and is zero otherwise. In eq. (4.1) we have not included quartic terms in the boson operators for simplicity reasons only. However, the inclusion of such terms would lead to straightforward generalizations of eqs. (4.6), (4.7), (4.10), (4.11) etc., but, of course, eq. (4.20) would be preserved. The terms of the Hamiltonian expansion containing four A~m have not been neglected altogether, since they contribute to the expressions of h~2) and h~! through the quantity ~/UU.kj. The operator M given by eq. (4.17) may be regarded as the expansion in boson operators of the transition operator M given by eq. (3.40). Then, according to eqs. (2.38), (2.39) and (2.41), m~1) is given by eq. (3.42). We also have m~) = Z ( -- ~i)/T/k, j(Ojl(,, ~Jikc+)J- +f'(s)+]'(r)'l 't" j i ' f ik I ,
(4.23)
ijk
rn r+2) ,,~ v r ,, ~ , , t,t,(~).t,(s~* ±,l,(+~*,t,~,h , s ~ / ~ k - - v i ] H t k , j \ t F j i Wki T~k'ij ttJlk ].
(4.24)
ijk
The boson expansion of a transition operator M contains also cubic terms in the operators S~. However, such terms do not contribute either to {/~IM[0} or to (01 [M, [H, M ]]10) in the approximation we are considering, so that eq. (4.20) remains valid when M denotes the boson expansion of an arbitrary transition operator as given by eq. (3.40). It may also be verified that in the Beliaev-Zelevinsky expansion [see eqs. (2.38)-(2.40)] enough terms have been included to insure that the value of the quantity (01[M,[H, M]]10) remains the same up to first order in q~,i or ~ i j ; k l , whether M is an operator in the many-fermion space or the corresponding boson expansion. Therefore, if we calculate the excited states I/~} in the higher RPA as formulated here, then compute the transition probabilities L{0BMI/~}]z, where M is any one-body operator (of the many-fermion space) and finally perform the sum ~,]{OIMI#}LZE,, this sum comes out to be equal to ½(0I[M,[H, M]]I0) up to first order in the quantities th, j or ~/~j;k~. The direct excitation of high-energy two-particle-two-hole states in ~60 proceeding via the ground state correlations of the RPA as well as the shifting of transition strength to higher energies due to the mixing of one-phonon states into high-energy two-phonon states have been calculated by Shakin and the author a). The final formulae on which that calculation was based are equivalent to eq. (4.23) and to a perturbative treatment of eqs. (4.10)-(4.13) (although they have been derived by different techniques). Both processes lead to the appearance of appreciable transition strength above the giant resonance region. Also, qualitatively, the appearance of transition strength at high energies seems to be compensated by the decrease of transition strength around the giant resonance, which is not surprising in view of eq. (4.20). Finally, we remark that the same formal results as derived here may be obtained by the Green-function method. Note the similarity between eq. (4.23) and eq. (4.5) of ref. 2). An earlier calculation 10) for 12C based on eq. (4.5) of ref. 2) already
RPA E X T E N S I O N
605
showed the appearance of an appreciable amount of transition strength at high energies due to the direct excitation of two-phonon states via the ground state correlations. Dr. B. Sorensen has brought to our attention two earlier attempts 1t) to take into account Pauli-principle corrections in the RPA. The relation of these methods to the Beliaev-Zelevinsky expansion used here has been discussed by Sorensen 12). I was very fortunate to be introduced by Professor Peierls into the problem of how to understand nuclei as a collection of nucleons and how to explain their properties in terms of the forces acting between the constituent particles. To me and to all those who work in that field, he is a constant source of stimulus and inspiration. On the occasion of his 60th birthday I wish, therefore, to dedicate this paper to Professor Peierls, as a token of deep respect and gratitude. I would also like to thank Professor Feshbach and the Laboratory for Nuclear Science for their warm hospitality. Further, ! would like to thank Professor Almeida Santos for his encouragement and for facilities provided at the Physics Laboratory of the University of Coimbra, where this work was initiated. Special thanks are due to J. Weneser, F. Villars, A. K. Kerman, C. Shakin, L. S. Kisslinger and J. M. Araujo for useful and interesting conversations. A leave of absence extended to me by the Faculty of Science of the University of Coimbra and a grant from Comiss~o de Estudos de Energia Nuclear, Instituto de Alta Cultura, are gratefully acknowledged.
Appendix 1 In this appendix, we derive expressions relating the expectation values (01c~+ %[0) and (01c~+ cpl0) to the transition amplitudes (rlc~+ c,,[0) = (OlS, c~+ cm]O), where 10) is the vacuum of the operators S,. We may calculate (01c,+ %[0) by introducing a complete set of states Ix) such that ~xlX)(Xl = 1, so that we get
(01c~+c,~l°) -= E (°lc.+ Ix)(xlcmlO). x
Since the states Ix) are of the one-phonon-one-hole type, we write
(0[c.+c,,I 0) = ~
(Olc+~%lr)(rlc~-cmlO) =
r},
A..a ~ "rn~,l[l(r)*l[l(r)'rm~,,
(A.1)
ry
where use has been made ofeq. (2.17). In a similar way we obtain
(01c~+cal0) - ~ a = - (01ca c~+ 10) -_ _ ~ (Olcac~lr)(rlcqc+~ lO) rq
= _ ~ (Olc~calr)(rlc+~Cql O) rq
~" ,i,~r)*,1,(,) rq
(A.2)
606
J. DA PROVIDENCIA
However, the kets c~ S,+ 10) and c~ S~+ 10) are not exactly o r t h o n o r m a l , so we have to investigate m o r e carefully the assumptions involved in these derivations. In order to calculate (01c~+ cml0), we observe that the ket c,10) is not zero due to the ground state correlations but m a y be written as a superposition of kets c~- S + 10). There is no c o m p o n e n t along csl0 ) if we assume that (01cf %10) = 0. We write, therefore cmlO) = Z z~s~ss, (') + 10). St
In order to calculate --ms, 7(t) we multiply this expression on the left by (01S~c~+. We get
(OlS, c;c~lO) = Z z~s(ols~cf ") + csS,+ 10).
(A.3)
St
We also have after some straightforward algebra
(OlS~c; csS?lO) -- (01rEst, C; Cs], S? ]lO)+(Olc; cslO)~,~ ~,~ lO~c+cs~a~,~,w~ ~ v tr..(s),,.(.. ~ ' s , ~ ' ~ , ±- ~ ' ,/.(...(,)*~.. ,f~',..
v -~ . rA.,~
P ,I,(S),h(t) * M a k i n g the a p p r o x i m a t i o n (0lc ~- cs]0) = (3es and neglecting the term /_,pwppWp7 , we m a y write (°lS, c; c,S?i°) ~- ~,~sf - ~. '~'(''('* (A.5) ~Fsp ,F #p • P
With the help of eqs. (2.22) and (2.30), we finally have in leading order
•j(s)Rift
~'~ ~l~(s)dt(t) dz(t) * = / _ , tFSp '~t'ras~t" f p ~'P
7(s) ~-~ 7 ( 0 d,(s)dt(t) * Z~mf - - / , Z~m~,,~p'sp~l" f p , SP
so that Zm
,/,(t)
(°lc~+c~l°) = ~ z~(OIc+~csS?l O) :
~ "rms\'rnsl]l(t)(l[l(t)* - - Z W6pjt(t)*dl(S~ll(S)*~tp f? ~,,~ yt Jps
st
(A.6) ).
(n.7)
In a similar way we get
(01Cc~10)- ~=f =
-(01cf c~+10) ~,q~,qf qt
-
~2 ~f J p ~ff f p "~ q 6 J"
(A.8)
Jps
Both quantities (OIcm + c,,lO) and (OIcfc+ IO) are of order (O(m~))2. It seems a p p r o p r i a t e to simplify eqs. (A.7) and (A.8) by neglecting in the right-hand sides the sums over the indices 6, p, s. Although also of the order h/'(r)~ \ 7- nlot / 2 those sums should be small as c o m p a r e d with the first terms due to incoherence effects. We recover in this way eqs. (A.1) and (A.2).
607
RPA EXTENSION
Appendix 2 Starting from the expression c,+c, z c [(+~++~)&+ +($:;*+x:;*)$J, (A.9) I we shall try to determine the correction coefficients xci and xii, which we had to introduce in order to take into account the ground state correlations. We consider the following expectation values: (Olccs’cfl~
GkJI0) = Lm c; cm - &7(w,‘d-o~
(A.lO)
(w+, 2 c~c,]lO) = 0.
(A.ll)
From eq. (A.9) we obtain, if we keep only terms linear in the correction coefficients, (OlCc,‘c,
3 4 czllo) = &, 4lw -I- c C($2 x2* - d2*xttr,‘> + (2 $g* - x!2*$~)1 r = 6,&q 6,, + $$* + x:cB,“‘,
(OlCCcg 2 &,-JO) = c [(~6~x~*-~~~*xJT,‘)+(x~~~,~*-x~~*~j;,’)] r
(A.12) = XT’* -x$Jj*, (A.13)
where “&m, = & (&)&;* -&)*@).
(A.14)
In order to satisfy eqs. (A.lO) and (A. 1 I), it is enough to set where rln,nz = !w,‘cnlo) and to require Xc’ = X;;,B,“‘.
(A.16)
The matrix xlps”” remains, therefore, arbitrary, as long as it is symmetric. We have given to the matrix xLcB,“’ an hermitian form since a possible anti-hermitian component may always be removed by means of a canonical transformation of the type S, -+ s,- 2 c,,.S,, s with E,,, = -E,,,. Such a canonical transformation leaves the phonon vacuum invariant. With the help of the completeness relations, eq. (A.14) may be inverted and leads to “&’ = c (x;y)$rL + xjqm)*$E($ (A.17) anI The coeficients $2 and xz: i. n eq. (A.9) are now partially determined, but we have still at our disposal the quantity x$“‘).
608
J. DA PROVIDENCIA
Appendix 3 In this appendix we wish to establish the connection between the wave function given by eq. (3.2) and the ground state wave function t 3) in the RPA approximation. We begin by transforming eq. (3.2) in the following way: 1o) -=- { 1 + ½ y~ 'e'm~ . . ( . ~ , ~Wan .,.)*r ctrnflnr
~ + za~On ~, + -LZatam
. + A,~m + Ayp)Afln + - ½A+m( E Aap + Ayn + A~p)]}l~o). ' 2\ Z A~p 7P YP
Then, with the help of eq. (2.40) and within the spirit of our approximations, we obtain ~]mct~lfln (A.18) 10)~{l+½E ~') ")* CmC~tC" + + Cfl)]¢~O) , ttm~nr
thus recovering the RPA result. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
G. E. Brown, Unified theory of nuclear models (North-Holland Publ. Co., Amsterdam, 1964) J. da Provid~ncia, Nuclear Physics 83 (1966) 209 J. da Provid~ncia and C. M. Shakin, Nuclear Physics, submitted S. T. Beliaev and V. G. Zelevinsky, Nuclear Physics 39 (1962) 582 D. J. Thouless, Nuclear Physics 22 (1961) 78 M. C. Mihailovi6 and M. Rosina, Ann. of Phys., to be published N. Fukuda, F. Iwamoto and K. Sawada, Phys. Rev. 135 (1964) 705 C. M. Shakin and J. da Provid~ncia, Nuclear Physics, submitted R. G. Sachs and N. Austern, Phys. Rev. 81 (1951) 705 J. da Provid~ncia, Nuclear Physics A90 (1967) 597 K. Hara, Progr. Theor. Phys. 32 (1964) 88; K. Ikeda et al., Progr. Theor. Phys. 33 (1965) 22 12) B. Sorensen, Nuclear Physics A97 (1967) 1 13) E. A. Sanderson, Phys. Lett. 19 (1965) 141; J. da Provid~ncia, Phys. Lett. 21 (1966) 668
[
Nuclear Physics A108 (1968) 589--508; (~) North-HollandPublishiny Co., Amsterdam
I
Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
AN E X T E N S I O N OF THE R A N D O M P H A S E A P P R O X I M A T I O N J. DA PROVID~NCIA t Laboratory for Nuclear Science and Department oJ Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts tt Received 14 August 1967 Abstract: A theoretical scheme appropriate for generalizing the Hartree-Fock and RPA theories to include ground state correlations is developed. The Beliaev-Zelevinsky expansion is used for expanding the Hamiltonian in boson operators. The ground state wave function is defined as the independent boson wave function which minimizes the expectation value of the Hamiltonian. It follows that the ground state cannot mix with one- or two-boson wave functions, and this condition yields immediately equations generalizing the Hartree-Fock and the RPA equations. By consideringtheoscillations of slightly modified independent-bosonwave functions around the equilibrium position, the boson-boson correlations due to anharmonic effects are introduced. Finally, it is shown that the present approach possesses formal properties similar to those of the RPA theory.
1. Introduction
The r a n d o m phase a p p r o x i m a t i o n ( R P A ) has had a great deal of success in describing the b e h a v i o u r of spherical nuclei. W i t h i n its framework, m a n y nuclear properties have been explained t). However, besides the fact that the R P A fails to a c c o u n t for several experimental results, the p r o b l e m of defining a n d investigating the next order to the R P A remains a n interesting one in theoretical nuclear physics. Several approaches are possible in tackling that problem. I n a previous paper 2), the G r e e n - f u n c t i o n formalism has been used for extending the RPA. Corrections to R P A t r a n s i t i o n rates have also been calculated 3) with the a s s u m p t i o n that the twoparticle, two-hole correlation structure of the nuclear g r o u n d state is given by the R P A result for the g r o u n d state wave function. I n the present paper, we use the Beliaev-Zelevinsky m e t h o d for expanding operators defined in the space of m a n y - f e r m i o n wave functions in ideal b o s o n operators 4). We define the g r o u n d state wave f u n c t i o n as the i n d e p e n d e n t b o s o n wave f u n c t i o n which minimizes the expectation value of the H a m i l t o n i a n . This leads to the cond i t i o n that the g r o u n d state should n o t mix with one- or t w o - b o s o n states, which yields equations generalizing the H a r t r e e - F o c k a n d the R P A equations. By considering the oscillations of slightly distorted i n d e p e n d e n t - b o s o n wave functions a r o u n d the e q u i l i b r i u m position, the b o s o n - b o s o n correlations due to a n h a r m o n i c effects are t Present Address: Laboratorio de Fisica, Universidade de Coimbra, Coimbra, Portugal. tt This work is supported in part through funds provided by the Atomic Energy Commission under Contract AT(30-1)-2098. 589
590
s. DA PROVIDf~NCIA
introduced. Following the analogy with the Thouless s) treatment of small amplitude vibrations, it is finally shown that the present approach possesses formal properties similar to those of the RPA theory. The purpose of this paper is, therefore, to present a theoretically consistent scheme. The formal results obtained here generalize those of ref. 2), however, in spirit, this work is closer to ref. 3), where some actual numerical calculations are also presented. A very interesting preprint 6) where essentially the same problem is treated has recently appeared. In that work, the Hartree-Fock and RPA theories are extended by an ingenious variational method which yields simultaneously equations for the density matrix and transition amplitudes.
2. The Beliaev-Zelevinsky expansion We are interested in studying the behaviour of a system of fermions. It is our basic assumption that the Hartree-Fock theory provides a reasonable first approximation and a convenient starting point for a more elaborate description of that system. As usual, a single-particle state occupied in the Hartree-Fock ground state is called a hole and a non-occupied single-particle state is called a particle. We adopt the convention of denoting hole states by the Greek indices cq [3, 7, J . . . . and particle states by the Latin letters m, n, p, q , . . . . The letters 9, h, i , j , . . , may represent both particle or hole states. I f we denote by c + the creation operator for state i, the Hartree-Fock ground state of our system may be written N
I o> = 1-I
+
I>,
(2.1)
where ] ) represents the absolute vacuum. Excited states are then obtained by acting with the particle hole operators (c + c,) on I~0). However, due to the residual interaction, such states do not correspond to the normal modes of the system. In the socalled random phase approximation (RPA), the operators Cz+ c~ are treated as ideal boson operators, and the boson operators which generate the normal modes are given as linear combinations of the operators Cm+ Ca and c~,+ c m. In order to go beyond the RPA, Beliaev and Zelevinsky 4) have proposed an expansion of particle-hole operators in powers of ideal boson operators. We summarize now the derivation of the Beliaev-Zelevinsky expansion in order to present the main ideas involved. If we identify the operators c+~ c m with some ideal boson operators A,m, then from commutation relations like [ c + c = , e+ cq] = c+Cqamp,
(2.2)
I C+Cm, ca+ %] = - C#+ Cm ~ct~,
(2.3)
we arrive at the following expansions for the operators cp+ cq and cp+ %:
RPA EXTENSION +
591
+
c~ cq = ~ An, A~q, ),
(2.4)
~;c~ = ~p~- Z A~Ap,.
(2.5)
P
However, these results require also a consideration of the expectation values
<*olC;Cqla~0> -- 0,
(2.6)
<~0olc~-c~]~o> = 5¢~,
(2.7)
and the identification of the vacuum of the operators A~,, with the Hartree-Fock ground state 1~o>. Only then can the constant terms in the expansions be determined. We may now go one step further in the expansions of the operators c~+ Cr,. AS it may easily be seen, it is enough to write
c+ c,,
=
A~,,-½ E
+ A m A~p A~p
(2.8)
?P
in order to insure that the following equations are satisfied:
[c: cm, ~: cp] = c~+ ca fir,, - c,+ Cm3~#,
[C+~Cm,c-~c,] = 0.
(2.9) (2.10)
We have, for instance, if consistently we neglect quantities of second order in the correction terms
- ½Z {[A.., (A,+.A L A,~)] + [(A,~ A,. ~ ) , AL]} 7q
q
In a similar way it may also be seen that eq. (2.10) is satisfied. One might now ask if the improved accuracy in the expansion for c~ Cm requires for consistency the inclusion of new correction terms in the expansions for c~ Cq and c~ q. The answer is that it does not. We have, for instance,
7
-½ ~, [(A~,., A.,,.A..,,),
(A+ A~,q)]
= amp(A,q-- ½ E A~ra"A~,q A~m,), ~t'm"
so that eq. (2.2) remains satisfied up to first order in the correction terms; the same is true of eq. (2.3).
J. DA PROVIDF-NCTA
592
It was mentioned already that in the RPA, the boson operators describing the normal modes of a system are some linear combinations of the operators c+ c,. and Cm+ ¢~ ,,., (dt(r)~ + (r) + S~ = Z ,~',m~, Cm--t~r~CmC,), (2.11) ~tm
where the amplitudes ~/,.~9 ~'tJ satisfy the following orthogonality and completeness s.v) relations: Z (')* ¢~) "~ (2.12) ij
¢,,j(r)4,~,(s)o(,j).. = o,
(2.13)
~ t,l,(O*,t,(,) ,t,(~),t,(')*'~ t.7~'ij tFkl - - W j i tFlk ) : 61k6jtO(ij),
(2.14)
r
the quantity O(ij) being defined by O(ij) = 0 ~ - Oj ,
(2.15)
with O~ = 1
if
i is a hole state,
O~ = 0
if
i is a particle state.
(2.16) (We have therefore 0, = 1, 0 m -~- O, O ( ~ m ) = - - O ( m ~ ) = 1, O(~fl) = O(mn) = 0.) Our quantities ~/, (r) correspond to the J((') rctm -- ~tm and the a/,(') "tract correspond to the y(O - - ~m in the more usual RPA notation as adopted, for instance by Thouless s). The notation we follow is due to Fukuda, Iwamoto and Sawada 7). Our assumption about the HartreeFock ground state providing a reasonable first approximation to the actual ground state of our system means that the ground state correlations are in some sense small, so that the quantities ~-z~'/'(°(or the --~m,Y(O~are on the average small as compared with the ~b(° (or the Y(')~ for a positive eigenfrequency ~o~. We shall now show that the ground state correlations require a small modification of the Beliaev-Zelevinsky expansion. With the help of eqs. (2.12)-(2.14), we may invert eq. (2.11) (and its complex conjugate) and obtain +
(tb(r).~ "+ 4- Ib(r)*,~ )l
¢mect
~
Z ~Yam--r r
C;Cm
"-~ Z
- - v'm~t --rl~
(r)* S-r ) . .(|]/.(r)~+ ... --r "~-~Jatm
(2.17)
r
Now we remark that these equations are not consistent with the expectation values of eqs. (2.9) and (2.10) with respect to the phonon vacuum. We have indeed (01EC + C m , C+ ¢fl][O) = (0[ C+ CfllO)~mn - - (01 C+ ¢mlO)6~fl ,
(2.18)
(Ol[c+c,.,
(2.19)
c~-c,]lO) = O,
and in appendix 1 the following expressions for the expectation values (OIc+ %10)
RPA EXTENSION
593
and (OIc~+ Cr~lO) with corrections due to ground state correlations included have been derived (°lc+cal °) -- a = t - Y~ ~',t't'<~)*'t'{~'V'q=, (2.20) rq
(Oi4cmlO) = Z ~~/I,<~ ~ . ~//"~* , •
(2.21 )
r7
It is shown in appendix 2 that in order to satisfy eqs. (2.18)-(2.21), one must replace eqs. (2.17) by the following equation together with the corresponding complex conjugate c r+a g = ~ [ ( ~ c t(.> (2.22) m "~- Z.(,)~c+ , m J O r 4_(,/?r)*+ -- ",'rmo~ - - Z.¢,)*~c m . J O r Jq, r
where the correction factors/,~tm " ( ' ) and Zm. "(') have the following expressions: .(r) 7.,m = ~
On ~(r)
m. = Z
[ ~,(fln)d,(r) \A~tm W'an _L~,(a,,)*,t,¢,)a ~ Am~t Wnfl],
(2.23)
[,~(tn),l,(r) ± ,,(fln)* ,l,(r) x \l.m~: '/',an V A a m Wnl3)"
(2.24)
Here, the quantities .(t.) are Hermitian matrices given by (2.25) with = -
Y, V'q~'Z'(r)*'l'(')V'qt,
(2.26)
rq
".,m ---- (OlCm+C.lO)
(2.27)
= Z '1/')*'1'(') w m 7 "Yn7 ' r7
and the quantities Z~ff) are symmetrical matrices in the following sense: Z(~,) ma
=
.(,m),
(2.28)
~n t
which otherwise remain at our disposal. Following a discussion of Shakin and the author 8), we consider now the expectation value (Ore+ c t c + G]0) in order to specify the quantities Z~"). From eq. (2.22) we have
(Ole~+cpC+m¢,lO) \Writ
T Anti
] A kT"*atmT
r "~ Z [ d ' ( r ) * ' l ' ( r ) ~ ~(r)*'l'(r)'~ "~\ W n t Worm T A n t Wotml r -~ Z Wnfl'b(r)*'lt(r)Watm.~,,(¢tm)@znlJ r
,
(2.29)
594
J. DA P R O V I D E N C I A
where small quantities like V ~(,),t,(,)* and ~/ ,r /~(r)~(,). /_.,,~,mv,# have been neglected. Now, , a m Anti we have (01c~+ c# C c~l0) = -- (01c2 c~c~+ c0 0), so that the following condition should be verified: Z "t'(r)*'t'(r) -t- ,,(~m)*
,I,(r)*d,(r) ± ,/am)*~
r
r
,,(~ra)
A convenient choice for ,~,# , which is in agreement with this condition and symmetrical in the sense of eq. (2.28), is Zt~m)
,#
= _ ~ ~',='t'(')'l'(')*~'#~.
(2.30)
r
In terms of the operators A~m = S" (dt(r)R + 4-~b(r)*R r
A + = ~ (,I,(~)R + .4-,h(')*.~ ~
(2.31)
r
it is possible to give the following simpler form to eq. (2.22): +
Cmc~
~
A + + ~V\ ~ m:,,(0.)a ~ a O+n V_L%~,)*A¢,)"
(2.32)
an
The quantities t/,,# q,, m and Z ~") have also simple forms in terms of the operators A,m, namely q.,p = - E (0lA~Aaql0), (2.33) q
ft.,,. = E (oIa+.,a,.lO),
(2.34)
7
Z(") n0
=
-(OIA#,,A,.IO).
(2.35)
These results may be combined with eqs. (2.25) and (2.32) to give %+c , ~ A + m - Z { ½ ( 0 1 A + . A~,IO)Ao,.+~(OIAt3,,Ao.IO)A.,+(OIAm, + ' + + + A,.IO)Ao. + }.
(2.36)
On
Next we look for higher-order terms in the expansions of the operators c + ci in powers of the operators S, and S +. These higher-order terms must be constructed in such a way as to preserve the commutation relations
[c~cj, C?Ck] = C+Ckfjl --Cl+Cjfik
(2.37)
up to first order in the corrections. Although eqs. (2.4), (2.5) and (2.8) are basically acceptable, a slight modification of these equations is imposed by eq. (2.36), which specifies the form of the terms linear in the operators S~ and S +. The equations which replace eqs. (2.4), (2.5) and (2.8) are
RPA EXTENSION
+ Cp Cq =
+
(~,(a")A+ A + 4-v(am)*A A~q),
1
(2.38)
7am
7 + C a C., =
595
6 ~ 7 - ~ A~pAap+~ + , VLr JkLPfl a " ) a "~TP'C~amTA.P? +a+ ±~(am)*a a ~ zJe~mZSflP)' p
(2.39)
pcLm
(A~p A m Aap + Zma A~p),
C : Cm =
(2.40)
VP
where the matrix z,,a'(~P)is given by eq. (2.30). These equations lead to the same commutation relations as eqs. (2.4), (2.5) and (2.8), because they may be obtained from them by means of the canonical transformation which replaces the operators Aa,. by (Aam-½X7 ~'(a'm')A+ ~ if, consistently, high-order terms are neglected. For / ,a'm'~ma ~-~'m'.,', instance, cubic terms in the boson operators multiplied by Z ~ ) should be regarded as X~a fifth-order terms, since (a.) -- - (O[A,,ABm[O), and therefore they should be neglected. I f we denote by: A B C . . . : the normal product of the operators A, B, C , . . . with respect to the vacuum ]0) of the operators S,, so that :s,s: : = sJs,,
:S+ Sr: = S,+ S,,
etc.,
then eq. (2.40) may be written in the following form, which agrees with eq. (2.36), as far as terms linear in the operators Sr, S~+ , are concerned: +
=
1 -r + {~(O[A~,,AB,[O)Apm
A,+m-pn
+½(OJA;mAp, IO)A+ +(O]A;mA2IO)Ap,+½: Ac,,A~mAt~,,. + + .}.
(2.41)
We have therefore justified eqs. (2.38) to (2.40) - they lead to correct commutation relations and to expectation values (0It + G c+ ca[0) possessing the correct symmetries. We shall discuss now the physical significance of the modifications we had to introduce in the Beliaev-Zelevinsky expansion in order to guarantee the correct symmetries of the expectation value (01(c+ G)(c+, ca)J0 ). Those modifications mean simply that the vector [0) cannot be an arbitrary vector of the many-boson Hilbert space. Obviously, a vector ix) belonging to the many-boson Hilbert space may be used to represent a + + state of our many-fermion system only if the vector (% G)(G ca)Ix) obtained by acting on rx) with the operator (e + ca)(c+, ca) is anti-symmetric under the interchanges m ~ n or a ~-+ ft. (It is assumed that the operators (c+ G) and (c + ca) have been expressed as power series in the boson operators A~p.) The acceptable vectors [x) span a subspace of the Hilbert space. For that reason, a canonical transformation taking us from (the ket corresponding to) the Hartree-Fock ground state to the A . . . --dt(r)A+'~ vacuum 10) of the boson operators S, = z.,. . .(,/t(r) . . . -.. --a,,, is not strictly acceptable as it stands. It must be corrected for the violation of the Pauli principle, for instance, by means of an auxiliary canonical transformation. That is the role played by the additional terms we have introduced in the Beliaev-Zelevinsky expansion. It should, therefore, be emphasized that the correction terms we have introduced are not
596
J. DA P R O V I D E N C I A
correcting the Beliaev-Zelevinsky expansion, which does not have to be corrected. They are simply correcting our canonical transformation for the violation of the Pauli principle. 3. Extended RPA
The canonical transformation
(3.1)
S, = ~_, fil/(') A --~I,(')A x/t" = m -~otm "r m ~ t - - ~ + t t n "~ / ~tm
takes us from the Hartree-Fock ground state [Cbo) to the vacuum of the bosons S,, which is given approximately by 10) = {1+½ Z
,t,(o*,t,(r)tA+ n+ tF otrn tlJ nfl \ Za~m "qL fln
la+a+~"~
x
- - 2"~'L~tn " ° - p r n J J I "x" O / "
(3.2)
{{mflnr + The term ( - ~ A1~ , + Apm ) arises from the canonical transformation we have introduced into the Beliaev-Zelevinsky expansion to allow for the Pauli principle. In appendix 3, we discuss the connection between eq. (3.2) and the corresponding RPA result. Actually we have to generalize our canonical transformation to read Sr
=
--
a ( ° + ~ (~//') A , m - -
~/1(")A + ]
(3.3)
{zm
where a (° is a constant. The vacuum 10) becomes, therefore A+ __XA + i0) ~ {1+ Z " jb(r)*~(r)A+ , m " ",m -t-1 ~ 'l'(')*'i'(OfA+ V,m V,0V'~,,"p, 2"'~,aP-,,)}Jq~o>" ctmr
(3.4)
amflnr
This generalization of eqs. (3.1) and (3.2) is necessary in order to eliminate the interaction between the vacuum and one-phonon states. Such an interaction is present even with a Hartree-Fock basis due to the existence of ground state correlations. However, the a (°, like the ~m~b (r) , are small quantities. We may ask now how are the amplitudes a <') and ~bl~> to be determined. That is equivalent to asking which vector of the many-boson Hilbert space best represents, within an i n d e p e n d e n t - b o s o n a p p r o x i m a t i o n , the actual ground state of our system. The answer to this question is given by the variational principle. The equations which determine the amplitudes a (r) and T~/~!~. ~ J ) are to be obtained by requiring that the expectation value (01HI0) of the Hamiltonian H be stationary with respect to small variations of the parameters a (° and ~ ) . It is convenient at this point to introduce a notation for many-boson wave functions. We let [rs . . . t) = S,+ S + . . . S~I0). (3.5) Since the most general infinitesimal variation of the phonon vacuum induced by small variations of the parameters a t') and 0}~) is of the form
10) = ( Z,
+ +½ Y, r$
= 2 e, lr)+½ ~ r
rs
e,~lrs),
(3.6)
RPA EXTENSION
597
where er and ers are infinitesimals, the condition fi(0fHI0) : 0
(3.7)
(rlHI0) = 0,
(3.8)
(rslH[O) = 0.
(3.9)
leads to the equations
By considering a canonical transformation a m o n g the boson operators which does not mix boson creation operators with boson annihilation operators and therefore does not change the vacuum [0), it is possible to require also
(,-Imls) = co;%s.
(3.1o)
The quantities a (~), ~,j ~I,[<) are determined by eqs. (3.8)-(3.10), which play in our problem a role completely analogous to the role of the H a r t r e e - F o c k equations in the problem of selecting the Slater determinant (independent-particle wave function) which " b e s t " approximates the ground state wave function of a system of fermions. It is now convenient to introduce the expansion of the Hamiltonian H in powers of the boson operators H = h(°)+ Z (h~1)S+ +h~l)*S,) r
+
(2) +S~+h,,(2). S,, Ss) --1 ~,/h(2)~,+S: +2h,,sS, 2! ~ "-'~ -"
+ ....
(3.11)
where h (°) = (0[H[0),
(3.12)
h~') = (01IS, H]I0),
(3.13)
(2)
(3.14)
'rs = (0liSt, [Ss, H]]/O), h(2) = (0[IS,, [H, S+]]I0), r,s
....
(3.15)
In terms of the quantities h~l), #2) and ]/(2) eqs. (3.8)-(3.10) m a y be written • -rs r, s h~ ') = 0,
(3.16)
h(2) = 0, rs
(3.17)
h~2~) = 09,6,~,
(3.18)
where cot = a ~ - h (°). It will be shown that eqs. (3.17) and (3.18) provide an extension o f the RPA, and that eq. (3.16) extends the H a r t r e e - F o c k equations by taking into account corrections due to the ground state correlations.
598
s. DA PROVIDENC1A
In order to calculate the quantities h~ ~), h (2) h (2) etc., we write the Hamiltonian H explicitly in the form
H
~t,.jc+cj+½Z ij
Vik, jlCi + Ck+ ClCj
(3.19)
ijkl
and assume a H a r t r e e - F o c k basis so that
ti, j-{- ~
Vla, jot = e i 6 i j ,
(3.20)
~ik, jl = Vik, jl -- Vik, l j "
(3.21)
o~
where
It is convenient at this point to introduce a notation for the expectation values (01Aaml0), (OIA,,,Aa.IO), (0[A]mAa,[0) etc., which are small quantities, according to the assumption that 10) is not too poorly approximated by 1~o>. We let
t/,.,a = (01Aa.,10), t/a.m = (0[A+ml0),
(3.22)
~/a.,;.a = (0IA~,.Aa.10),
A~-.10),
tlma; #n = (OlA+~m
qma;nll = (OIA+~,,,AanIO),
ha.; a. = (01A~-. Aaml0).
(3.23)
Notice that
* ~/am; n#
=
*
?]nfl; a m
(3.24) =
?]/~n; m a ,
* r/ma; "tJ = t/~;,.a = r/an;~m"
(3.25) (3.26)
In the last section, the quantities t/a, a and t/,,,~ had been defined. With the definition of t/a, ,, and t/ . . . . the matrix t/i, j became completely defined. U p to first order in t/i, j, we can write (0[c+ c,[0) --- Oi6ij + tl,, j. (3.27) The non-defined elements of t/ij;k~ are t/a~; rp = 0). With the help of eq. (3.3), one obtains
set
equal to zero
S? [,/,(r) \ ' r a m U ~(,). _ " r .~.(r)*_(r)~ t/Jma u j,
q a , m -~" / , r
tla,,; na = ~ ~',m't'(r)*'t'(')Vna'
(for instance,
(3.28) (3.29)
r
tlm~,;"a = ~ V,,a't'('l*'t'(OV,,~" r
(3.30)
599
RPA EXTENSION
Also, from eqs. (2.26) and (2.27) it follows that rl*,p = - c Ilp.;pB 9
(3.31)
ul m, n = 29,.i;.i.
(3.32)
We may now calculate the quantities h!“, I$:‘, h!Ti etc., with the help of expansions (2.38), (2.39) and (2.41). The calculation is straightforward but somewhat lengthy. Making also use of the Hartree-Fock equations [eq. (3.20)], one obtains A!” = C $I;‘[(& -@(ik)%, ik
i + 7 fi/cj, i[Vl, j
+ F ( - Vi o/c, Ih yljk: Ii - “kfilj, ik Vjk;
kl)l,
(3.33)
where vi = l-28,,
(3.34)
sothatv,= -l,v,= 1. One also obtains (3.35) (3.36) where yfil,kj
=
-(&i-k&l
-Ej-Ek)HKii)
+ o(kz))qkl; i j
+ C {Cm vidij8g1, hk- VkSlkDgi, hj)Vh, g-(Dig, jhYgh; lk + 4g, khflgh; ijP(ikMG) 8h -Dih,kg~jg;lh-ogl,hj~gi;hk
+(‘il, ght?kg; hj+Ogh, kjqgi;Ih
+‘g~,kh~gi;hjI‘Dih,gjqkg;Ih)VgVh-#gi, jkqgh;Zh+61g, jkqgh;ih + ‘li, gkqjh; gh+ ‘li, jgqkh; gh)) -“fTh {sij(OIJ, ghqfh; gk + 6g.f, kh~fh; 19)
+‘klfagf, jhqfh: ig + Oif, ghqfh; gj>>.
(3.37)
The condition that (O]H]O) should be a minimum has provided us with eqs. (3.16), (3.17) and (3.18). In order to find which results arise from those equations, we transform them in the following way:
T (t@*h;f’- l@h;“;) = - co,fjf;‘. With the help of eqs. (2.14), (3.33), (3.35) and (3.36) one finally obtains
600
J. DA PROVIDENCIA
(ek--ei)O(ik)rlk, i+ ~ Ojk, zirh, j+ ~ (--Vi~ki,,hrljh;z~--Vk13,i,~hrljh;k, ) = 0, jl
(3.38)
jib
((.Or'q- 8i-- 8 k ) ~ ) = O ( i k ) ~ (~il, k j ' ~ - ~ i l ,
kj]Wjl'~'l'('>"
(3.39)
jl
With regard to eq. (3.38), we may remark that if we neglect the ground state correlations, that is, if we set ~i,j = r/ij';kl = 0, then it reduces to the identity 0 = 0. This happens because we have assumed a Hartree-Fock basis. If we had not done so, then the Hartree-Fock equations would result in that limit. As it stands, eq. (3.38) provides an extension of the Hartree-Fock equations, since it enables the calculation of corrections to the Hartree-Fock density matrix, namely the quantities r/~,~, (01c,,+c~10), once the I quantities 17ij;ki are known. With regard to eq. (3.38), we may remark that it has just the aspect of the RPA equations. However there is a difference; eqs. (3.38) and (3.39) are to be solved self-consistently, since the particlehole interaction appearing in eq. (3.39) contains the correction term #fit, kj which is a function of the quantities ~/,, ,, and r/u, k~. The quantities r/,, mare given by eq. (3.38) once the quantities ~/u;k t are known, and, in turn, the quantities ~/U;kt are expressed in terms of the amplitudes ~}~) yielded by eq. (3,39). It is difficult to assess the importance of the correction Y~zl, kj in eq. (3.39) without performing an actual calculation. However, it seems that this correction tends to reduce the importance of the backward-going graphs. Indeed we have, for instance, gh =
e
.... ~ - ( ~ . . + ~ . - ~ - ~ )
~,',~>*,',~') W~m 'Fmfl
~ (¢~ ,l,(S)*,l,(s) ~ ~ ,l,(S)*,l,(s)~ \ tJmg, flh tY gh tp'n~t T Idng) ~th t)Ugh t~ rttfl.} ghs
(s)* s
(s)
~ ~
,/,(s)q
gh
gh
~_~ f " 'l'(S)*'l'(s) " 'l'(S)*'l'(S)l = 0, I.~s W~n Wm# ~ t~"s ~t'rtn Wm,6J 8
where we used eq. (3.39) with the quantity Y~it, k:" removed. We see, therefore, that there are some terms in ~ . . . . a (or ~#'~a, ~,) which tend to cancel ~. . . . a (or ~a, ,,,) in eq. (3.39). Finally, it is interesting to compare eq. (3.37) with eq. (3.7) of ref. 2) (where, actually, a term ½[1 + O(gh)O(fd)]~oy ' dh is missing in the right-hand side). There is agreement between both results, although they are based on different approaches, since the Green-function method has been used in ref. 2), It may be seen that the modified form we have given to the Beliaev-Zelevinsky expansion is essential for achieving this agreement. For completeness we give now the leading terms of the expansion of a transition operator M = E ml, jc~cj (3.40) ij
RPA EXTENSION
601
in powers of boson operators. With the help of eqs. (2.38), (2.39) and (2.41) one easily obtains M : m(°)+ Z (m,.(1)S, + "JCt3~ r(1)* S r ) - ~ - . . . . (3.41) r
where m (°) = (0IM[0) is a constant and DI(1)
=
Z ik
'l'(r)Ym 'elk t a.,T± Z (rnk, orlo, iO(ig) O
- ~k,o too,, O(g k))O(ik) - Z (ma, o rh0;ka + }ma,~ rho; go + ½mk, a q~o: ao)}. (3.42) gh
Recent calculations 3,8), based on formulae which although derived by different arguments are analogous to this one, have shown that the ground state correlations tend to appreciably reduce transition rates, thus removing, at least in part, some wellknown discrepancies between theory and experiment. If we compare eq. (3.42) [or eq. (2.41)] with eq. (3.8) of ref. 2), we find that there is agreement between those results, and again we may verify that this agreement requires the particular version of the Beliaev-Zelevinsky expansion we have adopted.
4. The boson-boson interaction and the higher RPA The expansion of the Hamiltonian H should contain cubic and higher-order terms in the boson operators, which, however, have not been represented explicitly in eq. (3.11). If we include such terms and make use of eqs. (3.16)-(3.18), the Hamiltonian H becomes
~o, Sr+S~
H = h(°)+ r
I-h(3)K, + ~ .~ + ~, .~, + +3h,.~.,Sr ( 3 ) + + S~ S,+3ht~.,. ( 3 ) *S,. + S~S,+h,~, (3)* S,.SsSt]+ . . . . (4.1) + ~.~ rs~t L-,'st~,"
The new terms lead, of course, to phonon-phonon correlations. These correlations may be described with the help of the time-dependent variational principle, in the same way that the particle-hole correlations are conveniently described by the timedependent Hartree-Fock equations. We consider, therefore, a state vector of the form ]TJ(t)) = (exp iF)]0),
(4.2)
where F is a time-dependent Hermitian operator such that
F = ~ (f,(X)S,+ +f(1)*S,)+½ Z (f(2)S~+S+ +2f~(.2)S+S~+Z~2)*S, Ss) • r
(4.3)
rs
The time evolution of the operator F is determined with the help of the time-dependent variational principle -- i[(5 ~l (u) - (~'16 7/)1 + 6(~IHI ~u) = 0.
(4.4)
602
J. DA PROVID~NCIA
The terms of first order in F do not contribute in this equation because of eqs. (3.16) and (3.17). The terms of second order lead to
-i(Ol[6F, F]0)+½f(01[F, [H, r]]10) --- 0.
(4.5)
From this equation we finally obtain, in analogy with the time-dependent HartreeFock equations for small amplitude vibrations
_
if,.(,)+corf,.(1)+½ ~ V',,s,,~, :~,(3) ¢C2)
h~3):(2)* -,s,:s, :~
= 0,
(4.6)
St
_if~2)+(eo~+cos)f~2)+ X" it,(3) ¢~1)_h(3):(1)*'~ = 0 Z~ k'~rs, t d t "'rstJ t ]
(4.7)
t
plus complex conjugates. These equations are solved with the ansatz
f ) l ) = X,e-le, + y , ei~, f~2)
(4.8)
, * iEt Xrse-iEt + ~x,se ,
(4.9)
which leads to
EXr = ¢OrXr+½ EXrs
=
(a)
Z (hr, stgst-St
(ojr÷OJs)Xrs_}
hr~t ( 3 ) y ,s,),
- ~,"' (~h\ ( 3rs, ) tX t_ I
h,st (3) Yt),
-EY~ = cor Y,+½ V ¢h(3) v - - h"'rst t 3 ) * Y" ~ s t ]"~, /, \'~st, r "st
(4.10) (4.11) (4.12)
st
-EY,~ -- (o~,+ ~o~)r,s+ Z (h~3),Y,-ht3)*Y ~ . . , s~,:. ,
(4.13)
t
Following the Thouless 5) treatment of the RPA theory, we may write these equations in the form
B*
A* k Y 'v)] = E~ k _ r ( , ) ] ,
(4.14)
where X (~) and Y(~) represent the column matrices of the quantities y co, X~{) and y~v), y C]). The quantities A and B are appropriate square matrices. For instance
A,, s = ~Or6rs, A, , st = hi3) ,,st,
etc.
Assuming the eigenvalues E v are real and non-zero, we normalize the eigensolutions of eq. (4.14) so that (XC~),_ y(~),) [XC~)~ \y¢,,)] = 6 , ~ -IE~l -. (4.15) E, If we represent by 10} and by I/z}, respectively, the ground state wave function and the wave function of the excited state with energy E, (in the RPA applied to the phonon-phonon interaction), then to the quantities X}"), X}~), Y}") and Y}~), the following interpretation may be given 5):
RPA EXTENSION
603
{~IS)I0} = ~'(")*
{~,ls)s: IO} =
x~2 )*,
{plSrlO} = - Y~")*,
{IXlSrS,,iO} = - Y/f)*.
(4.16)
The ground state [0} differs from the ground state 10) essentially in that 10} contains boson-boson correlations. We consider now an operator
,.,,. +m~t)*Sr) M = m(°)+ ~ (,,,, ~.(~)c+ r
"-~½ Z (...(2)C,+C,+ (2) + (2)* \ f l i t s o r o S +2mr ,S, Ss+mrs SrSs),
(4.17)
rs
and introduce the column matrices
m*
m = ~,.~)/,
=
[rn~l)*] kmr,(2)* ]
(4.18)
"
The matrix element {plMI0} may be written {/~,MI0} = (X(~)t- Y(")') ( m ) m*
(4.19)
"
The well-known theorem discovered by Thouless s) in the time-dependent HartreeFock treatment of small amplitude vibrations remains valid here [{0IM[/t}[2E. g(E~, > O)
= Z Im~l)/~°.+½ Y. Im~)12(°).+'°.) r
rs
+21" Z k( f~l l(r1 ) * LIt,~ ( 3 ) st f.~(2)__ flst T
. ( 2 ) * Lflst~ ( 3 ) r l..i l t(I)
f[lst
rst
_ ~/T/(1)* m ( 2 ) * / a ( 3 ) z
r
"'~st
"~rst
X/a(3)*v~(1)w(2)~ --
2 I~rst
liar
Htst
]
= ½(01[M, [H, M]][0).
(4.20)
This is a useful result. It may serve for showing that the well-known Thomas-ReicheKuhn sum rule 9) and the usual conservation laws of momentum, angular momentum etc. are preserved in the present approach, within the approximation considered. We give now the expression of the anharmonic terms in the boson expansion of the Hamiltonian H [see eq. (4.1)]. We have, with the help of eqs. (2.38), (2.39) and (2.41) h(3) --- ~_~ Bit kj ~ rn't
ikjlh
h(3) ,,,,
,
r's't'
n(rst) 'l'(r')'i'(s')'l'(C) ~r's't"Pki Wjh 'Fhl
^ kj Y', D('s)r't'(r')'l'(")'t'")*" ,t,(").l,(')*,l,(*') ±,t,(o*,t,(r'),l,(")~ = ~ Vu, --r's'ktl"ki ~ ' j h tPlh ~Fki tPhj Whl TtPik W j h W h l I~
ikjlh
r's"
(4.21) (4.22)
604
J. DA PROVID~NCIA
where .P(r~'",,+,...')t'is equal to 1 if r ' s ' . t ' is . any . permutation . . of the . indices r s t and is zero otherwise. In eq. (4.1) we have not included quartic terms in the boson operators for simplicity reasons only. However, the inclusion of such terms would lead to straightforward generalizations of eqs. (4.6), (4.7), (4.10), (4.11) etc., but, of course, eq. (4.20) would be preserved. The terms of the Hamiltonian expansion containing four A~m have not been neglected altogether, since they contribute to the expressions of h~2) and h~! through the quantity ~/UU.kj. The operator M given by eq. (4.17) may be regarded as the expansion in boson operators of the transition operator M given by eq. (3.40). Then, according to eqs. (2.38), (2.39) and (2.41), m~1) is given by eq. (3.42). We also have m~) = Z ( -- ~i)/T/k, j(Ojl(,, ~Jikc+)J- +f'(s)+]'(r)'l 't" j i ' f ik I ,
(4.23)
ijk
rn r+2) ,,~ v r ,, ~ , , t,t,(~).t,(s~* ±,l,(+~*,t,~,h , s ~ / ~ k - - v i ] H t k , j \ t F j i Wki T~k'ij ttJlk ].
(4.24)
ijk
The boson expansion of a transition operator M contains also cubic terms in the operators S~. However, such terms do not contribute either to {/~IM[0} or to (01 [M, [H, M ]]10) in the approximation we are considering, so that eq. (4.20) remains valid when M denotes the boson expansion of an arbitrary transition operator as given by eq. (3.40). It may also be verified that in the Beliaev-Zelevinsky expansion [see eqs. (2.38)-(2.40)] enough terms have been included to insure that the value of the quantity (01[M,[H, M]]10) remains the same up to first order in q~,i or ~ i j ; k l , whether M is an operator in the many-fermion space or the corresponding boson expansion. Therefore, if we calculate the excited states I/~} in the higher RPA as formulated here, then compute the transition probabilities L{0BMI/~}]z, where M is any one-body operator (of the many-fermion space) and finally perform the sum ~,]{OIMI#}LZE,, this sum comes out to be equal to ½(0I[M,[H, M]]I0) up to first order in the quantities th, j or ~/~j;k~. The direct excitation of high-energy two-particle-two-hole states in ~60 proceeding via the ground state correlations of the RPA as well as the shifting of transition strength to higher energies due to the mixing of one-phonon states into high-energy two-phonon states have been calculated by Shakin and the author a). The final formulae on which that calculation was based are equivalent to eq. (4.23) and to a perturbative treatment of eqs. (4.10)-(4.13) (although they have been derived by different techniques). Both processes lead to the appearance of appreciable transition strength above the giant resonance region. Also, qualitatively, the appearance of transition strength at high energies seems to be compensated by the decrease of transition strength around the giant resonance, which is not surprising in view of eq. (4.20). Finally, we remark that the same formal results as derived here may be obtained by the Green-function method. Note the similarity between eq. (4.23) and eq. (4.5) of ref. 2). An earlier calculation 10) for 12C based on eq. (4.5) of ref. 2) already
RPA E X T E N S I O N
605
showed the appearance of an appreciable amount of transition strength at high energies due to the direct excitation of two-phonon states via the ground state correlations. Dr. B. Sorensen has brought to our attention two earlier attempts 1t) to take into account Pauli-principle corrections in the RPA. The relation of these methods to the Beliaev-Zelevinsky expansion used here has been discussed by Sorensen 12). I was very fortunate to be introduced by Professor Peierls into the problem of how to understand nuclei as a collection of nucleons and how to explain their properties in terms of the forces acting between the constituent particles. To me and to all those who work in that field, he is a constant source of stimulus and inspiration. On the occasion of his 60th birthday I wish, therefore, to dedicate this paper to Professor Peierls, as a token of deep respect and gratitude. I would also like to thank Professor Feshbach and the Laboratory for Nuclear Science for their warm hospitality. Further, ! would like to thank Professor Almeida Santos for his encouragement and for facilities provided at the Physics Laboratory of the University of Coimbra, where this work was initiated. Special thanks are due to J. Weneser, F. Villars, A. K. Kerman, C. Shakin, L. S. Kisslinger and J. M. Araujo for useful and interesting conversations. A leave of absence extended to me by the Faculty of Science of the University of Coimbra and a grant from Comiss~o de Estudos de Energia Nuclear, Instituto de Alta Cultura, are gratefully acknowledged.
Appendix 1 In this appendix, we derive expressions relating the expectation values (01c~+ %[0) and (01c~+ cpl0) to the transition amplitudes (rlc~+ c,,[0) = (OlS, c~+ cm]O), where 10) is the vacuum of the operators S,. We may calculate (01c,+ %[0) by introducing a complete set of states Ix) such that ~xlX)(Xl = 1, so that we get
(01c~+c,~l°) -= E (°lc.+ Ix)(xlcmlO). x
Since the states Ix) are of the one-phonon-one-hole type, we write
(0[c.+c,,I 0) = ~
(Olc+~%lr)(rlc~-cmlO) =
r},
A..a ~ "rn~,l[l(r)*l[l(r)'rm~,,
(A.1)
ry
where use has been made ofeq. (2.17). In a similar way we obtain
(01c~+cal0) - ~ a = - (01ca c~+ 10) -_ _ ~ (Olcac~lr)(rlcqc+~ lO) rq
= _ ~ (Olc~calr)(rlc+~Cql O) rq
~" ,i,~r)*,1,(,) rq
(A.2)
606
J. DA PROVIDENCIA
However, the kets c~ S,+ 10) and c~ S~+ 10) are not exactly o r t h o n o r m a l , so we have to investigate m o r e carefully the assumptions involved in these derivations. In order to calculate (01c~+ cml0), we observe that the ket c,10) is not zero due to the ground state correlations but m a y be written as a superposition of kets c~- S + 10). There is no c o m p o n e n t along csl0 ) if we assume that (01cf %10) = 0. We write, therefore cmlO) = Z z~s~ss, (') + 10). St
In order to calculate --ms, 7(t) we multiply this expression on the left by (01S~c~+. We get
(OlS, c;c~lO) = Z z~s(ols~cf ") + csS,+ 10).
(A.3)
St
We also have after some straightforward algebra
(OlS~c; csS?lO) -- (01rEst, C; Cs], S? ]lO)+(Olc; cslO)~,~ ~,~ lO~c+cs~a~,~,w~ ~ v tr..(s),,.(.. ~ ' s , ~ ' ~ , ±- ~ ' ,/.(...(,)*~.. ,f~',..
v -~ . rA.,~
P ,I,(S),h(t) * M a k i n g the a p p r o x i m a t i o n (0lc ~- cs]0) = (3es and neglecting the term /_,pwppWp7 , we m a y write (°lS, c; c,S?i°) ~- ~,~sf - ~. '~'(''('* (A.5) ~Fsp ,F #p • P
With the help of eqs. (2.22) and (2.30), we finally have in leading order
•j(s)Rift
~'~ ~l~(s)dt(t) dz(t) * = / _ , tFSp '~t'ras~t" f p ~'P
7(s) ~-~ 7 ( 0 d,(s)dt(t) * Z~mf - - / , Z~m~,,~p'sp~l" f p , SP
so that Zm
,/,(t)
(°lc~+c~l°) = ~ z~(OIc+~csS?l O) :
~ "rms\'rnsl]l(t)(l[l(t)* - - Z W6pjt(t)*dl(S~ll(S)*~tp f? ~,,~ yt Jps
st
(A.6) ).
(n.7)
In a similar way we get
(01Cc~10)- ~=f =
-(01cf c~+10) ~,q~,qf qt
-
~2 ~f J p ~ff f p "~ q 6 J"
(A.8)
Jps
Both quantities (OIcm + c,,lO) and (OIcfc+ IO) are of order (O(m~))2. It seems a p p r o p r i a t e to simplify eqs. (A.7) and (A.8) by neglecting in the right-hand sides the sums over the indices 6, p, s. Although also of the order h/'(r)~ \ 7- nlot / 2 those sums should be small as c o m p a r e d with the first terms due to incoherence effects. We recover in this way eqs. (A.1) and (A.2).
607
RPA EXTENSION
Appendix 2 Starting from the expression c,+c, z c [(+~++~)&+ +($:;*+x:;*)$J, (A.9) I we shall try to determine the correction coefficients xci and xii, which we had to introduce in order to take into account the ground state correlations. We consider the following expectation values: (Olccs’cfl~
GkJI0) = Lm c; cm - &7(w,‘d-o~
(A.lO)
(w+, 2 c~c,]lO) = 0.
(A.ll)
From eq. (A.9) we obtain, if we keep only terms linear in the correction coefficients, (OlCc,‘c,
3 4 czllo) = &, 4lw -I- c C($2 x2* - d2*xttr,‘> + (2 $g* - x!2*$~)1 r = 6,&q 6,, + $$* + x:cB,“‘,
(OlCCcg 2 &,-JO) = c [(~6~x~*-~~~*xJT,‘)+(x~~~,~*-x~~*~j;,’)] r
(A.12) = XT’* -x$Jj*, (A.13)
where “&m, = & (&)&;* -&)*@).
(A.14)
In order to satisfy eqs. (A.lO) and (A. 1 I), it is enough to set where rln,nz = !w,‘cnlo) and to require Xc’ = X;;,B,“‘.
(A.16)
The matrix xlps”” remains, therefore, arbitrary, as long as it is symmetric. We have given to the matrix xLcB,“’ an hermitian form since a possible anti-hermitian component may always be removed by means of a canonical transformation of the type S, -+ s,- 2 c,,.S,, s with E,,, = -E,,,. Such a canonical transformation leaves the phonon vacuum invariant. With the help of the completeness relations, eq. (A.14) may be inverted and leads to “&’ = c (x;y)$rL + xjqm)*$E($ (A.17) anI The coeficients $2 and xz: i. n eq. (A.9) are now partially determined, but we have still at our disposal the quantity x$“‘).
608
J. DA PROVIDENCIA
Appendix 3 In this appendix we wish to establish the connection between the wave function given by eq. (3.2) and the ground state wave function t 3) in the RPA approximation. We begin by transforming eq. (3.2) in the following way: 1o) -=- { 1 + ½ y~ 'e'm~ . . ( . ~ , ~Wan .,.)*r ctrnflnr
~ + za~On ~, + -LZatam
. + A,~m + Ayp)Afln + - ½A+m( E Aap + Ayn + A~p)]}l~o). ' 2\ Z A~p 7P YP
Then, with the help of eq. (2.40) and within the spirit of our approximations, we obtain ~]mct~lfln (A.18) 10)~{l+½E ~') ")* CmC~tC" + + Cfl)]¢~O) , ttm~nr
thus recovering the RPA result. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)
G. E. Brown, Unified theory of nuclear models (North-Holland Publ. Co., Amsterdam, 1964) J. da Provid~ncia, Nuclear Physics 83 (1966) 209 J. da Provid~ncia and C. M. Shakin, Nuclear Physics, submitted S. T. Beliaev and V. G. Zelevinsky, Nuclear Physics 39 (1962) 582 D. J. Thouless, Nuclear Physics 22 (1961) 78 M. C. Mihailovi6 and M. Rosina, Ann. of Phys., to be published N. Fukuda, F. Iwamoto and K. Sawada, Phys. Rev. 135 (1964) 705 C. M. Shakin and J. da Provid~ncia, Nuclear Physics, submitted R. G. Sachs and N. Austern, Phys. Rev. 81 (1951) 705 J. da Provid~ncia, Nuclear Physics A90 (1967) 597 K. Hara, Progr. Theor. Phys. 32 (1964) 88; K. Ikeda et al., Progr. Theor. Phys. 33 (1965) 22 12) B. Sorensen, Nuclear Physics A97 (1967) 1 13) E. A. Sanderson, Phys. Lett. 19 (1965) 141; J. da Provid~ncia, Phys. Lett. 21 (1966) 668