PHYSICAL REVIEW C 67, 044311 共2003兲
Goldstone-Brueckner perturbation theory extended in terms of mixed nonorthogonal
Slater determinants
T. Duguet*
Service de Physique Théorique, CEA Saclay, F-91191 Gif sur Yvette Cedex, France
共Received 17 October 2002; published 29 April 2003兲
The Goldstone-Brueckner perturbation theory is extended to incorporate in a simple way correlations associated with large-amplitude collective motions in nuclei. The new energy expansion making use of nonorthogonal vacua still allows to remove the divergences originating from the hard core of the bare interaction. This is
done through the definition of a new Brueckner matrix summing generalized Brueckner ladders. At the lowest
order, this formalism motivates variational calculations beyond the mean field, such as the generator coordinate
method 共GCM兲 and the projected mean-field method, from a perturbative point of view for the first time. Going
to higher orders amounts to incorporate diabatic effects in the GCM and to extend the projection technique
from product states to well-defined correlated states.
DOI: 10.1103/PhysRevC.67.044311
PACS number共s兲: 21.60.⫺n, 21.10.Dr, 21.10.Re, 21.30.⫺x
I. INTRODUCTION
The mean-field approximation relies on a particular
choice of the trial state of the system, namely, a Slater
determinant1 兩 ⌽ ␣0 典 共see Ref. 关1兴兲 describing the system as N
independent particles. The mean-field approximation can
also be viewed as the zero-order approximation of the actual
ground-state energy in a perturbative expansion written in
terms of the residual interaction. In the case of nuclear structure, the mean-field approximation fails from the outset because of the strong repulsive core of the bare nucleonnucleon interaction which leads to divergences when limiting
the perturbative expansion to any order in the interaction.
However, Brueckner 关2兴 showed that the energy expansion
can be reordered as a function of hole-line number in the
graphs, which amounts to an expansion in the density of the
system. Such a reordering takes care of two-body, shortrange correlations induced by the repulsive core of the
nucleon-nucleon interaction. It leads to an expression of the
energy in terms of a renormalized interaction G. Using it,
one can study the lowest-order approximation and define a
meaningful mean-field picture. One can then evaluate the
improvements achieved by including higher-order diagrams
since each of them is well behaved.
In many-body systems, several ways to go beyond the
mean-field approximation exist, depending upon the physical
situation of interest 关1兴. Without particularly relating it to any
perturbative expansion, one can improve the approximate
trial wave function of the system in a variational picture in
order to include correlations in the ground state and calculate
excited states. The projected mean-field method 关3,4兴 and the
generator coordinate method 共GCM兲 关5–7兴 are such examples. The correlations considered in these two methods
are those associated with large-amplitude collective motions
*Present address: Argonne National Laboratory, Physics Division,
9700 S. Cass Ave., Argonne, IL 60439. Email address:
[email protected]
1
In its generalized form taking care of static pairing correlations,
the mean-field approximation makes use of a state being a product
of independent quasiparticles instead of independent particles.
0556-2813/2003/67共4兲/044311共13兲/$20.00
that are known to be very important in low-lying nuclear
structure. This is done by using a trial state written as a
superposition of several nonorthogonal product functions referring to different mean fields:
兩 ⌿ k典 ⫽
兺␣
f ␣k 兩 ⌽ 0␣ 典 .
共1兲
Once such a trial state is given, its mean energy
E mix
k ⬅
E mix
k ⫽
具 ⌿ k兩 H 兩 ⌿ k典
,
具 ⌿ k兩 ⌿ k典
f ␤k * f ␣k 具 ⌽ 0␤ 兩 H 兩 ⌽ ␣0 典
兺
␣,␤
兺
␣,␤
f ␤k * f ␣k 具 ⌽ ␤0 兩 ⌽ 0␣ 典
共2兲
.
共3兲
can be minimized with respect to variational parameters. The
mean-field approximation is recovered if a single coefficient
f ␣k is nonzero; its self-consistent version corresponds to the
minimization of the energy with respect to individual wave
functions.
A problem is that going beyond the mean field through
Eqs. 共1兲–共3兲, it is no longer possible to connect the variational calculation to any existing diagrammatic picture. Indeed, pertubation theories developed up to now can only deal
with small-amplitude correlations, these correlations being
short ranged or long ranged 关1兴. The treatment of largeamplitude collective motions as defined by Eqs. 共1兲–共3兲
makes use of nonorthogonal vacua connected through specific infinite sums of particle-hole excitations, which can
hardly be identified in usual expansions. To make such a link
is a question of interest since the systematic character of a
perturbative expansion is a very powerful tool in order to
classify the correlations included or forgotten in a given
variational calculation. An appropriate perturbative scheme
would also be useful to identify the effective interaction that
aims at renormalizing two-body correlations in the context of
large-amplitude collective motions.
67 044311-1
©2003 The American Physical Society
PHYSICAL REVIEW C 67, 044311 共2003兲
T. DUGUET
The aim of the present paper is to extend the BruecknerGoldstone theory in order to motivate the methods embodied
by Eqs. 共1兲–共3兲 from a perturbative point of view. The
lowest-order approximation of the new expansion provides
an energy having the form as given by Eq. 共3兲, and thus
describes large-amplitude collective motions.2 Higher orders
introduce more correlations and of different nature than in
the lowest order. Namely, they correspond to the introduction
of nonadiabatic effects in the coupling of individual and collective degrees of freedom. Moreover, the perturbative
scheme allows to eliminate divergences arising from the
strong repulsive core of the interaction by summing Brueckner ladders in the context of mixed vacua. It thus provides a
microscopically defined effective interaction to be used in
Eq. 共3兲.
The present work is organized as follows: we first remind
some important features of the usual Goldstone-Brueckner
perturbation theory in Sec. II. Then we develop the new perturbative scheme in Sec. III where we define and study a
generalized Brueckner G matrix. In Secs. IV and V, the link
between the lowest-order approximation and the configurations mixings used in variational calculations is illustrated
for the GCM and the projected mean-field method. The conclusions are given in Sec. VI.
where
h 1␣ ⫽V⫺⌫ ␣
共7兲
is characterized as the residual interaction.
In what follows, we make use of the Gell-Mann-Low
adiabatic theorem 关8兴, which states that an eigenstate of H is
obtained when evolving 兩 ⌽ 0␣ 典 adiabatically from t⫽⫺⬁ to
t⫽0. Actually, no rule exists saying that this eigenstate is
necessarily the ground state of the full Hamiltonian. We will
consider that it is so 关9兴. This theorem explicitly reads as
U ⑀␣ 共 0,⫺⬁ 兲 兩 ⌽ ␣0 典
␣
兩 ⌰ 0 典 ⫽ lim ␣ ␣
␣
⑀ →0 具 ⌽ 0 兩 U ⑀ 共 0,⫺⬁ 兲 兩 ⌽ 0 典
.
共8兲
In 兩 ⌰ ␣0 典 , the label ␣ reminds from which unperturbed
state 兩 ⌽ ␣0 典 the actual ground state comes from. The adiabatic
evolution operator U ⑀␣ in the ␣ interaction representation 关9兴
is
␣
␣
U ⑀␣ 共 t,t 0 兲 ⫽e ih 0 t/ប U ⑀ 共 t,t 0 兲 e ⫺ih 0 t 0 /ប
␣
t
␣
⫽e ih 0 t/ប e ⫺i 兰 t 0H( ⑀ , ␶ )d ␶ /ប e ⫺ih 0 t 0 /ប ,
共9兲
II. STANDARD GOLDSTONE-BRUECKNER THEORY
A. Ground-state expansion
Let us start with the Hamiltonian of the N interacting
nucleons. It consists of a kinetic energy term and a two-body
force V that includes a realistic nucleon-nucleon interaction
and the Coulomb interaction:
共4兲
H⫽t⫹V.
We define a one-body Hamiltonian h ␣0 potentially breaking symmetries of H. Its ground state and excited states are
denoted by 兩 ⌽ 0␣ 典 and 兩 ⌽ ␣i 典 , respectively. We make the additional hypothesis that 兩 ⌽ ␣0 典 is nondegenerate. One specifies
this mean field through
h 0␣ ⫽t⫹⌫ ␣ ⫽
兺n ⑀ ␣ ␣ †n ␣ n ,
n
h 0␣ 兩 ⌽ ␣i 典 ⫽E ␣i 兩 ⌽ i␣ 典 ,
共5兲
共6兲
where the 兵 ␣ 其 single-particle basis is chosen as diagonalizing h ␣0 . For our purpose, we assume the index i to be discrete and the E ␣i to be ordered with increasing value. The
exact Hamiltonian can be related to h 0␣ through
H⫽h ␣0 ⫹h ␣1 ,
2
We concentrate on perturbation theories written in a Hilbert
space with definite particle number. In other words, only normal
contractions 具 c † c 典 are nonzero. Such theories aim at treating the
particle-hole channel of the interaction.
where U ⑀ is the evolution operator in the Schrödinger representation. The derivation of Eq. 共8兲 relies on two fundamental properties of quantum mechanics 关9,10兴.
共1兲 Making an eigenstate of H(t 0 ) to evolve adiabatically
from t⫽t 0 to t⫽t 1 through the evolution operator of the
system, drives the system to an eigenstate of H(t 1 ).
共2兲 It is possible to find an adiabatic time-dependent
Hamiltonian H( ⑀ ,t) which reduces to the independent particle Hamiltonian h ␣0 at t 0 ⫽⫺⬁ and which equals the full
Hamiltonian H of the system at t 1 ⫽0.
In the present case, the auxiliary time-dependent Hamiltonian switching the perturbation h ␣1 adiabatically is given by
H 共 ⑀ ,t 兲 ⫽h 0␣ ⫹e ⫺ ⑀ 兩 t 兩 h ␣1 ,
共10兲
where ⑀ is a small positive parameter. At very large times,
both in the past and in the future, this Hamiltonian effectively reduces to h ␣0 , which presents a known solution,
whereas at t⫽0, H( ⑀ ,t) becomes the full Hamiltonian of the
interacting system. The limit ⑀ →0 is taken at the end, and
the results are independent of ⑀ . Note that this method does
not rely on the application of the evolution operator corresponding to the actual Hamiltonian H on a given state ‘‘at
t 0 ⫽⫺⬁.’’
The normalization of the actual ground state 兩 ⌰ ␣0 典 is
具 ⌽ 0␣ 兩 ⌰ ␣0 典 ⫽1. It is chosen through the denominator of Eq.
共8兲, which exactly cancels disconnected vacuum/vacuum diagrams in the numerator 关11兴. This is crucial as both the numerator and the denominator diverge in the limit ⑀ →0.
Performing explicitly the time integrations contained in
U ⑀␣ , Goldstone 关11兴 found a linked expansion of 兩 ⌰ ␣0 典 , allowing for a diagrammatic representation and reading as
044311-2
PHYSICAL REVIEW C 67, 044311 共2003兲
GOLDSTONE-BRUECKNER PERTURBATION THEORY . . .
αq
αp
FIG. 1. G ␣ matrix elements
summing p-p ladders with respect
to the vacuum 兩 ⌽ ␣0 典 . In any ladder, all V interactions 共except possibly the first and last兲 are drawn
between two up-going lines simulating particle states outside the
Fermi sea. The rest of the diagram
is not allowed to have any interaction between the first and the last
interaction in the ladder. Points
appearing in the present figure are
superfluous here, but will become
essential in the following. They
will be defined in Fig. 3.
V
αr
αq
α
αp
αq
G
αn
αp
V
=
αn
αm
αs
V
+
+
αm
V
αu
αv
V
αn
兩 ⌰ ␣0 典 ⫽
兺n
冉
1
h 1␣
E ␣0 ⫺h 0␣
冊
n
兩 ⌽ 0␣ 典 Linked ,
共11兲
where Linked means that 兩 ⌽ ␣0 典 does not occur as an intermediate state in the diagrams.
αm
teraction between two particles to all orders of the potential
in presence of the other nucleons.
Let us now replace each h 1␣ by G 1␣ (W ␣ )⫽G ␣ (W ␣ )⫺⌫ ␣
in Eq. 共11兲. All diagrams with two successive G ␣ interactions joined through particle states are excluded from the
sum in order to avoid double counting; the sum runs over
linked irreducible 共Irred兲 diagrams 关11兴:
B. Brueckner ladders
Following Brueckner 关2兴, each V involved in a linked diagram of Eq. 共11兲 is to be replaced by a reaction matrix G ␣
summing particle-particle (p-p) ladders in the 兵 ␣ 其 singleparticle basis. This removes the strong repulsive core of the
bare nucleon-nucleon interaction, which makes the expansion in terms of V irrelevant. The replacement of V by G ␣ is
possible because each time a V interaction occurs in the
original expansion of 兩 ⌰ 0␣ 典 , one can find the infinite set of
identical diagrams except that a succession of V interactions
connecting particle states occurs instead of the original V,
and this happens before any interaction takes place somewhere else in the graph. The corresponding diagrammatic
content is shown on Fig. 1. This matrix satisfies a selfconsistent equation of the form
G ␣ 共 W ␣ 兲 ⫽V⫹V
Q␣
W ␣ ⫺h ␣0
G ␣共 W ␣ 兲 ,
兺
␣
⑀␣ ,⑀␣ ⬎⑀F
p
兩 ␣ p ␣ p ⬘ 典具 ␣ p ␣ p ⬘ 兩 .
共13兲
p⬘
The Pauli operator Q ␣ acts in the two-particle space by
excluding occupied states in 兩 ⌽ ␣0 典 as intermediate states. The
W ␣ dependence of G ␣ denotes that the in-medium interaction of two particles depends on the energy of the others
during the interaction. The G ␣ (W ␣ ) matrix elements together
with the precise definition of W ␣ are given in Appendix A.
Iterating Eq. 共12兲 shows that G ␣ takes into account the in3
The two-body states 兩 ␣ p ␣ p ⬘ 典 are non-antisymmetrized here.
兺n
冉
1
G ␣共 W ␣ 兲
␣ 1
E ␣0 ⫺h 0
冊
n
兩 ⌽ 0␣ 典 Linked/Irred . 共14兲
All graphs are generated in this way. This is nothing but a
reordering of diagrams. Each term of the perturbative expansion is now finite and ‘‘well behaved’’ even if the original
interaction contains a strong short-range repulsion as in the
nuclear case 关12兴. The Brueckner scheme has to be seen as
the simplest way to get rid of the hard-core problem. More
elaborate treatments of the many-body problem through systematic renormalization of propagators and vertices exist, using, for instance, the self-consistent Green’s function approach 关13,14兴.
共12兲
with3
Q ␣⫽
兩 ⌰ 0␣ 典 ⫽
C. One-body potential ⌫ ␣
We do not discuss in detail the question of the most appropriate one-body potentials ⌫ ␣ to be chosen for p-p, p-h,
and h-h matrix elements 关14,15兴. Let us just recall few
points. First, one can define a fixed, purely phenomenological one-body potential ⌫ ␣ . Such a choice suffers from its
nonsystematic nature. Another choice is to define ⌫ ␣ in analogy to the Hartree-Fock potential with the bare interaction V
replaced by the G ␣ matrix. This is the Brueckner–HartreeFock 共BHF兲 approximation. It reads
具 ␣ i兩 ⌫ ␣兩 ␣ j 典 ⫽
兺
␣k ,␣k⬘
具 ␣ i ␣ k 兩 G ␣ 共 0 兲 兩 ␣ j ␣ k ⬘ 典 ␳ ␣␣ k ␣ k ⬘ , 共15兲
for h-h, p-h, and p-p matrix elements, where the one-body
density matrix ␳ ␣ associated with 兩 ⌽ ␣0 典 is defined in any
single-particle basis 兵 c i 其 by
044311-3
PHYSICAL REVIEW C 67, 044311 共2003兲
T. DUGUET
αt’
α
Γ
G
α
αt’
αp
αt
αp
αt
Γ
G
αp
αt
α
= 0
a)
b)
αt
αt
αt’
α
G
Γ
α
αp
αt’
= 0
αt’’
αt
= 0
α
αp
= 0
-
α
αt’’
αp
G
Γ
α p’
c)
α
α p’
d)
FIG. 2. Canceled diagrams in 兩 ⌰ ␣0 典 in relation with the Brueckner–Hartree-Fock definition of the one-body potential ⌫ ␣ . Dashed lines
mean that the remaining part of the summed graphs are identical.
␳ c␣s c r ⫽ 具 c s 兩 ␳ˆ ␣ 兩 c r 典 ⫽
具 ⌽ 0␣ 兩 c r† c s 兩 ⌽ ␣0 典
具 ⌽ ␣0 兩 ⌽ 0␣ 典
D. Energy expansion
,
共16兲
and is diagonal in the 兵 ␣ k 其 basis in such a way that ␳ ␣␣ ␣
k k⬘
⫽ ␦ kh ␦ k ⬘ h , where h denotes hole states.
In Eq. 共15兲, G ␣ is taken on the energy shell (W ␣ ⫽0).
Such a definition allows to cancel a large number of diagrams in the wave function. This is schematically shown in
Fig. 2. Figures 2共a兲 and 2共b兲 correspond to insertions on
vertices. They involve a G ␣ interaction with passive unexcited states and a ⌫ ␣ p-h interaction. Figure 2共c兲 embodies
insertions on hole lines. These three groups of diagrams have
been shown to cancel; thanks to a generalized time ordering
关16,17兴, although some of the G ␣ interactions are originally
not on the energy shell (W ␣ ⫽0). On the contrary, insertions
on particle lines cannot be systematically canceled by any
definitions of 具 ␣ p 兩 ⌫ ␣ 兩 ␣ p ⬘ 典 as depicted in Fig. 2共d兲. The two
parts of these diagrams have to be calculated explicitly when
appearing. The choice of the p-p matrix elements of ⌫ ␣ has
been of considerable controversy but we will not go into
more details here and refer to Refs. 关18 –22兴.
Finally, let us say that the BHF choice for ⌫ ␣ is not the
most appropriate one from the quantitative point of view
关15,23,24兴. A definition including higher-order diagrams is
known to take care of the rearrangement potential 关25兴 that
significantly improves density distributions in heavy nuclei
as well as the agreement between theoretical single-particle
and separation energies 关24,26兴. Indeed, higher-order graphs
are canceled with such a choice in a way that the zero-order
graph is a better approximation for observables such as the
one-body density matrix determining one-body observables.
The Gell-Mann-Low expression for the energy E 0 ⫽E ␣0
⫹⌬E ␣0 of the interacting ground state 兩 ⌰ ␣0 典 is obtained
through the scalar product of H 兩 ⌰ 0␣ 典 with the unperturbed
bra 具 ⌽ ␣0 兩 , and reads as
⌬E ␣0 ⫽ lim
⑀ →0
具 ⌽ ␣0 兩 h ␣1 U ⑀␣ 共 0,⫺⬁ 兲 兩 ⌽ ␣0 典
具 ⌽ ␣0 兩 U ⑀␣ 共 0,⫺⬁ 兲 兩 ⌽ ␣0 典
共17兲
.
The Goldstone-Brueckner linked/irreducible expansion
deduced from it is
⌬E 0␣ ⫽
兺n 具 ⌽ ␣0 兩 G 1␣共 W ␣ 兲
⫻
冉
1
G ␣1 共 W ␣ 兲
E 0␣ ⫺h ␣0
冊
n
兩 ⌽ 0␣ 典 Linked/Irred , 共18兲
where the sum runs now over all connected graphs with no
external lines. Finally, a meaningful mean-field approximation in nuclear-structure is deduced by considering the lowest
order in terms of G ␣ in Eq. 共18兲. This is the Brueckner–
Hartree-Fock approximation:
044311-4
E n⫽0
⫽E ␣0 ⫹ 具 ⌽ ␣0 兩 G 1␣ 共 0 兲 兩 ⌽ ␣0 典
0
⫽ 具 ⌽ 0␣ 兩 t⫹G ␣ 共 0 兲 兩 ⌽ 0␣ 典
⫽
t ␣ ␣ ␳ ␣␣ ␣
兺
␣
h h
h
⫹
1
2
兺
␣h␣h⬘
h h
Ḡ ␣␣
h␣ h⬘␣ h␣ h⬘
共 0 兲 ␳ ␣␣ h ␣ h ␳ ␣␣ h
⬘␣ h⬘
,
共19兲
PHYSICAL REVIEW C 67, 044311 共2003兲
GOLDSTONE-BRUECKNER PERTURBATION THEORY . . .
where Ḡ ␣␣ ␣ ␣ ␣ (0) are antisymmetrized matrix elements
m p n q
taken on the energy shell. One can effectively see that Eq.
共19兲 formally has the form of a mean-field energy for the
effective ‘‘Hamiltonian’’ t⫹G ␣ (0).
We will not go into more details concerning the standard
Brueckner theory and properties of the Brueckner G matrix.
For extensive presentations, we refer to Refs. 关2,11,15,27兴.
Let us just mention that the renormalization of short-range
correlations associated with the exchange of ␻ vector mesons is not the only significant origin of in-medium effects
and corresponding density dependence. For instance, the exchange of pions should be important. In fact, the Brueckner
summation originally introduced as the simplest way to remove the hard-core problem, provides in-medium effects not
only from the short-range repulsive part, but also from the
long-range part, i.e., from the tensor part of the bare force
关28,29兴. This is due to the dependence of the tensor-force
contribution to the energy on the Pauli operator Q ␣ . This
suggests that a part of the in-medium effects generated by
pions is taken into account through the ladder summation
共the ladders cover the whole energy range from low to high
excitation energies兲. Evidently, such a perturbation theory
based on nucleon-nucleon potentials 共possibly including
three-body ones兲 neglects the dynamical mesonic degrees of
freedom. One should go back to richer treatments of the
many-body problem including these degrees of freedom explicitly in order to evaluate quantitatively their full inmedium effects 关30兴.
A. Ground-state expansion
Let us choose a set of mean fields, corresponding residual
interactions, associated nondegenerate ground state and excited states 兵 h ␣0 ,h 1␣ , 兩 ⌽ 0␣ 典 , 兩 ⌽ ␣i 典 其 . For instance, 兵 h 0␣ 其 may
correspond to a set of identical deformed mean fields with
different intrinsic orientations. As their deformation corresponds to the minimum of the associated BHF energy, each
of them has a minimal density of individual states around the
Fermi energy. This makes any p-h excitation to cost some
energy and each ground state 兩 ⌽ 0␣ 典 to be nondegenerate. A
second case of interest consists in defining 兵 h ␣0 其 through constrained mean-field calculations along a collective deformation path. In this case, single-particle level crossings may
occur at the Fermi level for some values of the constrained
deformation ␣ . This makes the corresponding 兩 ⌽ ␣0 典 degenerate. Thus, one has to pick up a discrete number of ␣ values
along the deformation path for which this does not occur.
The rationale of the present derivation is to make each of
these vacua evolve independently from t⫽⫺⬁ to t⫽0 as
processed in Sec. II. The normalization 具 ⌽ ␣0 兩 ⌰ 0␣ 典 ⫽1 eliminating disconnected vacuum/vacuum diagrams is chosen for
each ␣ . In this way, the 兩 ⌰ 0␣ 典 simply differ through their
norm. For each ␣ , the standard Brueckner scheme is performed in order to get the ground-state wave function under
the form given by Eq. 共14兲. Finally, we perform an arbitrary
linear combination of the states 兩 ⌰ ␣0 典 :
兩 ⌰ 0典 ⫽
III. EXTENDED PERTURBATION THEORY
The preceding section reminded the Brueckner-Goldstone
expansion referring to a given mean-field unperturbed state
兩 ⌽ ␣0 典 . As said in the Introduction, variational calculations
relying on the mixing of nonorthogonal product states cannot
be motivated by such a perturbative scheme. We thus
develop a perturbation theory based on a superposition
of nonorthogonal Slater determinants. A natural idea would
be to evolve a linear combination as given by Eq. 共1兲 from
t⫽⫺⬁ to t⫽0. To do so, one has to find a one-body Hamiltonian h mix
0 , whose ground state is the mixing of nonorthogonal Slater determinants, and define an auxiliary Hamiltonian of the form
⫺⑀兩t兩
H 共 ⑀ ,t 兲 ⫽h mix
共 H⫺h mix
0 ⫹e
0 兲.
共20兲
for a general conHowever, it is not possible to find h mix
0
figuration mixing, such as the one defined by Eq. 共1兲. To
avoid this difficulty, we use a different approach based on the
arbitrariness of the unperturbed Hamiltonian h 0␣ , with respect to which the ground state of the system is developed in
the standard Goldstone picture. This arbitrariness, which is
usually used to optimize the unperturbed state through the
‘‘best’’ choice of ⌫ ␣ , is used here to superpose the expressions of the actual ground state obtained by starting from
several unperturbed Slater determinants 兩 ⌽ 0␣ 典 .
⫽
兺␣
f ␣ 兩 ⌰ 0␣ 典
兺␣ f ␣ 兺
n
␣
冉
1
G 1␣
E ␣0 ⫺h ␣0
冊
n␣
兩 ⌽ ␣0 典 Linked/Irred .
共21兲
As each 兩 ⌰ 0␣ 典 is the ground state of H with the eigenvalue
E 0 , 兩 ⌰ 0 典 is also the eigenstate with the same eigenenergy.
This is true whatever the coefficients f ␣ are. Actually, only
the normalization changes from Eq. 共11兲 to Eq. 共21兲. This
last equation constitutes the starting point of our extended
perturbative scheme. The linear superposition in Eq. 共21兲
seems to be redundant and trivial, but it provides richer approximate states at any order of the expansion. In particular,
the lowest order provides a superposition of nonorthogonal
product states as a well-defined approximation of the actual
ground state of the interacting system. In the following, we
will focus on the perturbative expansion of the corresponding eigenenergy.
B. Energy expansion
In the standard formulation, the starting point of the energy expansion is Eq. 共17兲, which gives the diagrammatic
development of the correction ⌬E 0␣ to the unperturbed energy E ␣0 . It is obtained through the scalar product of h ␣1 兩 ⌰ 0␣ 典
with the unperturbed bra 具 ⌽ ␣0 兩 . In our extended scheme,
however, there is no well-defined unperturbed state and unperturbed energy since each of the vacuum 兩 ⌽ ␣0 典 evolves
044311-5
PHYSICAL REVIEW C 67, 044311 共2003兲
T. DUGUET
independently from t⫽⫺⬁ to t⫽0 before the linear combination is performed. To avoid this ambiguity, we keep the
symmetry between the bra and the ket and directly express
the total energy E 0 through the mean value of H in the
ground state 兩 ⌰ 0 典 . Using Eq. 共21兲, we get
具 ⌰ 0 兩 H 兩 ⌰ 0 典 ⫽E 0 具 ⌰ 0 兩 ⌰ 0 典
⫽
兺
␣ , ␤ ,n ␣ ,n ␤
⫻
冉
f ␤* f ␣ 具 ⌽ 0␤ 兩
1
G␣
␣
␣ 1
E 0 ⫺h 0
冊
n␣
冉
兺
␣ , ␤ ,n ␣ ,n ␤
⫻
冉
1
E 0␤ ⫺h ␤0
冉
f ␤* f ␣ 具 ⌽ 0␤ 兩 G 1␤
1
G 1␣
E ␣0 ⫺h ␣0
冊
冊
n␤
关 t⫹V 兴
兩 ⌽ 0␣ 典 Linked/Irred ,
where the norm is
具 ⌰ 0兩 ⌰ 0典 ⫽
G ␤1
n␣
1
E ␤0 ⫺h 0␤
冊
共22兲
n␤
兩 ⌽ ␣0 典 Linked/Irred .
共23兲
The Goldstone diagrams corresponding to Eqs. 共22兲 and
共23兲 are irreducible in terms of the G ␤ and G ␣ matrices. The
dependence upon the energy parameters W ␤ and W ␣ is omitted for convenience. In opposition to the energy expansion in
the standard Goldstone-Brueckner theory, Eq. 共22兲 will provide explicit correlation diagrams associated to the kinetic
energy operator.
In the above equations, the diagrammatic expansion is
written as a sum over ( ␣ , ␤ ) of several types of vacuum/
vacuum diagrams 具 ⌽ 0␤ 兩 ••• 兩 ⌽ ␣0 典 involving different starting
and ending vacua. The sum runs over k 2 of these terms if k
different vacua 兩 ⌽ ␣0 典 are considered. Considering a particular
couple ( ␣ , ␤ ), let us give rules not only to calculate the
corresponding matrix elements but also to define and draw
the new associated diagrams.
First, all operators having a subscript ␤ in Eqs. 共22兲 and
共23兲 are written in the corresponding 兵 ␤ 其 single-particle basis, which is associated with dashed lines in the diagrams.
All operators having a subscript ␣ are written in the corresponding 兵 ␣ 其 single-particle basis that is associated with
solid lines in the diagrams. The Hamiltonian in the middle of
Eq. 共22兲 is chosen for symmetry reasons to be written in the
left 兵 ␤ 其 and right 兵 ␣ 其 single-particle bases:
H⫽
t␤
兺
m,n
m␣n
␤ m† ␣ n ⫹
1
4
兺
m,n,p,q
† †
V̄ ␤ m ␤ p ␣ n ␣ q ␤ m
␤ p␣ q␣ n ,
diagrams. Some of them together with basic definitions are
given in Fig. 3. For example, reading formula 共22兲 from right
to left, which is equivalent to read diagrams from bottom to
top, one can be convinced that the nucleons will first propagate from 兩 ⌽ ␣0 典 to 具 ⌽ 0␣ 兩 in 兵 ␣ 其 individual states. Propagators
involving different vacua will appear when going through H
because of the complete set of states referring to 兩 ⌽ 0␣ 典 inserted on its right and the one referring to 兩 ⌽ 0␤ 典 inserted on
its left. After that, they will propagate from 兩 ⌽ 0␤ 典 to 具 ⌽ ␤0 兩 in
兵 ␤ 其 individual states. Graphically, we choose to specify the
vacua entering the contraction in a propagator by a square for
兩 ⌽ ␤0 典 and by a circle for 兩 ⌽ ␣0 典 at the extremities of the line
共see Fig. 3兲. Finally, apart from the presence of two different
single-particle bases in the graphs, the use of the generalized
Wick theorem 关31兴 instead of the standard one 关32兴 and thus
the apparition of new propagators, other rules for the calculation of diagrams are identical to those defined in Ref. 关11兴.
Because of the nonorthogonality between particle-hole excitations built on two different vacua 兩 ⌽ ␣0 典 and 兩 ⌽ 0␤ 典 , nonzero
diagrams are more numerous than usual.
C. Generalized Brueckner ladders
We have now to treat divergences brought about by the V
interaction coming from the energy operator H in Eq. 共22兲.
Their treatment has been separated from the one of the divergences appearing in the time evolution of the wave function, which have been taken care of through the Brueckner
matrices G ␣ , because it is of different origin.
Each diagram of Eq. 共22兲 involving a V interaction can be
associated with diagrams differing only by the fact that a G ␣
interaction connects the original two-nucleons state to V
through two 兵 ␣ 其 particle states and/or that a G ␤ interaction
after V diffuses the two nucleons from 兵 ␤ 其 particle states into
the two-nucleon final state. This allows to replace V by a
G ( ␤ , ␣ ) matrix summing generalized particle-particle ladders,
where nucleons propagate in two types of particle states referring to two different nonorthogonal vacua. In this way, all
diagrams originating from Eq. 共22兲 are generated and each
graph given in terms of G ( ␤ , ␣ ) is well behaved as will be
shown in the following section. The diagram content of the
new reaction matrix is illustrated in Fig. 4.
Analytically, G ( ␤ , ␣ ) is expressed in terms of G ␣ and G ␤
through
G ( ␤ , ␣ ) 共 W ␤ ,W ␣ 兲 ⫽V⫹G ␤ 共 W ␤ 兲
共24兲
where V̄ ␤ m ␤ p ␣ n ␣ q are antisymmetrized matrix elements.
Second, one has to insert complete sets of particle-hole
excitations between these operators while keeping linked
diagrams only. These are particle-hole excitations with respect to 兩 ⌽ 0␣ 典 around operators with an ␣ subscript, and with
respect to 兩 ⌽ 0␤ 典 around operators with a ␤ subscript. In order
to describe associated graphs, several types of propagators
have to be defined depending on their localization in the
⫹V
Q␣
W ␣ ⫺h ␣0
⫹G ␤ 共 W ␤ 兲
Q␤
W ␤ ⫺h 0␤
V
G ␣共 W ␣ 兲
Q␤
V
␤
W ␤ ⫺h 0
Q␣
W ␣ ⫺h 0␣
⫽G ␤ 共 W ␤ 兲 V ⫺1 G ␣ 共 W ␣ 兲 ,
G ␣共 W ␣ 兲
共25兲
the last identity being obtained by grouping terms and using
the self-consistent equation satisfied by G ␣ (W ␣ ) and
G ␤ (W ␤ ).
044311-6
PHYSICAL REVIEW C 67, 044311 共2003兲
GOLDSTONE-BRUECKNER PERTURBATION THEORY . . .
FIG. 3. Basic definitions for drawing graphs between different vacua appearing in the generalized energy expansion 共22兲. Examples of
mixed propagators involved in these graphs are given.
FIG. 4. Diagrammatic representation of the G ( ␤ , ␣ ) matrix summing generalized particle-particle ladders.
044311-7
PHYSICAL REVIEW C 67, 044311 共2003兲
T. DUGUET
The G ( ␤ , ␣ ) matrix elements together with the definitions
of W ␤ and W ␣ are given in Appendix B. G ( ␤ , ␣ ) also satisfies
a self-consistent equation that is given in the same appendix.
Then, the energy expansion can be rewritten in terms of
G ( ␤ , ␣ ) . Equation 共22兲 becomes
具 ⌰ 0兩 H 兩 ⌰ 0典 ⫽
兺
␣ , ␤ ,n ␣ ,n ␤
⫻ 关 t⫹G
冉
f ␤* f ␣ 具 ⌽ ␤0 兩 G ␤1
(␤,␣)
兴
冉
1
E ␤0 ⫺h 0␤
1
G ␣1
E 0␣ ⫺h ␣0
冊
冊
n␤
具 ␣ r ␣ s 兩 G ␣ 共 W ␣ 兲 兩 ␣ p ␣ q 典 ⫽ 具 ␣ r ␣ s 兩 V 兩 ␺ ␣pq 共 W ␣ 兲 典 .
n␣
⫻ 兩 ⌽ ␣0 典 Linked/Irred ,
共26兲
where diagrams connecting G ␣ to G ( ␤ , ␣ ) through two 兵 ␣ 其
particle states and/or G ( ␤ , ␣ ) to G ␤ through two 兵 ␤ 其 particle
states are excluded in order to avoid double counting.
The reaction matrix G ( ␤ , ␣ ) is the effective interaction removing the hard-core problem within the framework of our
generalized expansion as shown below. A remarkable property of this effective interaction is its dependence with respect to the type of N-body matrix element in which it is
inserted in Eq. 共26兲. Thus, for k states entering the superposition defined by Eq. 共21兲, the present scheme requires the
calculation of k different G ␣ matrices referring to each
vacuum plus k 2 G ( ␤ , ␣ ) matrices referring to each type of
N-body matrix element. It may be very time consuming for
quantitative calculations.
As the aim of a Brueckner matrix is to take care of shortrange correlations that should not be very sensitive to the
single-particle basis used, the G ␣ matrices should be almost
identical for all ␣ . However, BHF calculations have shown
that this is not such a good approximation 共see Ref. 关33兴, for
instance兲. We will not go further concerning necessary approximations to implement this scheme for quantitative calculations since we stay on a formal level in the present work.
D. Pair correlated wave function and G „ ␤ , ␣ … matrix properties
In standard Brueckner theory, it is useful to illustrate the
G ␣ matrix properties through its action on an uncorrelated
two-body wave function:4
兩 ␣ p ␣ q 典 ⫽ ␣ †p ␣ †q 兩 0 典 .
共27兲
This defines a correlated two-body wave function 兩 ␺ ␣pq 典
关27兴 through the equation
V 兩 ␺ ␣pq 共 W ␣ 兲 典 ⫽G ␣ 共 W ␣ 兲 兩 ␣ p ␣ q 典 ,
共28兲
which can be rewritten using Eq. 共12兲 as
兩 ␺ ␣pq 共 W ␣ 兲 典 ⫽ 兩 ␣ p ␣ q 典 ⫹
Q␣
W ␣ ⫺h 0␣
It can be shown 关12兴 that the content of the Brueckner
summation is such that the correlated pair wave function
␺ ␣pq (W ␣ ,rជ ,Rជ ) vanishes rapidly as the separation distance rជ
between the two nucleons decreases within the range of the
repulsive core. The exclusion of the nucleons from the region
of the core in the correlated wave function makes finite twobody matrix elements of the form
V 兩 ␺ ␣pq 共 W ␣ 兲 典 .
共29兲
This is the physical reason why the standard expansion
formulas 共14兲 and 共18兲 in terms of G ␣ provide well-behaved
diagrams. Let us now study the properties of the generalized
G ( ␤ , ␣ ) matrix entering the new energy expansion. As we
have symmetrized the role of the bra and the ket in the expansion, the quantity to look at is no longer correlated twobody ket alone, but the two-body matrix element
具 ␤ r ␤ s 兩 G ( ␤ , ␣ ) 兩 ␣ p ␣ q 典 itself.
Using Eq. 共25兲 together with definition 共28兲 for the pair
correlated wave function, one can write
具 ␤ r ␤ s 兩 G ( ␤ , ␣ ) 共 W ␣ ,W ␤ 兲 兩 ␣ p ␣ q 典
⫽ 具 ␤ r ␤ s 兩 G ␤ 共 W ␤ 兲 V ⫺1 G ␣ 共 W ␣ 兲 兩 ␣ p ␣ q 典
␤
⫽ 具 ␺ rs
共 W ␤ 兲 兩 V 兩 ␺ ␣pq 共 W ␣ 兲 典 .
␣
Except for the definition of Q , the two-body states are antisymmetrized.
共31兲
This identity shows that G ( ␤ , ␣ ) is well defined to renormalize two-body, short-range correlations stemming from the
repulsive core of the interaction. It also shows how the energy expansion 共26兲 provides well-behaved diagrams. Comparing Eq. 共31兲 with Eq. 共30兲, one sees that G ( ␤ , ␣ ) renormalizes more correlations than standard Brueckner matrices
since it correlates both the two-body ket and the two-body
bra. This originates from writing the ground-state energy as
the expectation value of the Hamiltonian in the actual ground
state and resumming the ladders accordingly, instead of taking the scalar product of H 兩 ⌰ 0 典 with a vacuum and resumming the ladders, as done in the standard Brueckner theory.
Given the above rules and renormalized interactions G ␣ ,
␤
G , and G ( ␤ , ␣ ) , Eq. 共26兲 together with the norm provides a
perturbative formula for the ground-state energy. It is worth
mentioning that all the exponents (n ␤ ,n ␣ ) defining the orders of the expansion for each N-body matrix element in Eqs.
共26兲 and 共23兲 are taken equal, n ␣ ⫽n ␤ ⫽n,᭙( ␣ , ␤ ). This
means that, at a given order, identical diagrams with respect
to each unperturbed state will be dropped both in the ket and
in the bra.5 The order at which these expansions are truncated must be the same for 具 ⌰ 0 兩 H 兩 ⌰ 0 典 and 具 ⌰ 0 兩 ⌰ 0 典 .
5
4
共30兲
Once the hard-core problem is solved, the meaningful order in
energy expansion 共26兲 is defined as the number of explicit G ␣ (G ␤ )
interactions originating from the ket 共the bra兲.
044311-8
PHYSICAL REVIEW C 67, 044311 共2003兲
GOLDSTONE-BRUECKNER PERTURBATION THEORY . . .
(β,α)
G (0,0)
(n=0)
E
0
θ0 θ0
f *β f α
=
+
T
α,β
FIG. 5. Lowest-order approximation for the energy in terms of
the G ( ␤ , ␣ ) matrix. Square and dot symbols are defined in Fig. 3.
E. Lowest-order approximation
Let us consider the lowest-order approximation. It corresponds to retaining the n⫽0 term only in Eqs. 共23兲 and 共26兲.
The approximate energy of the interacting system becomes
⫽
E n⫽0
0
f ␤* f ␣ 具 ⌽ ␤0 兩 t⫹G ( ␤ , ␣ ) 共 0,0兲 兩 ⌽ ␣0 典
兺
␣,␤
兺
␣,␤
f ␤* f ␣ 具 ⌽ ␤0 兩 ⌽ 0␣ 典
,
共32兲
which is identical to Eq. 共3兲 except that the starting two-body
interaction V has been replaced by the G ( ␤ , ␣ ) matrix on the
energy shell (W ␣ ⫽W ␤ ⫽0). Consequently, we have
achieved the goal to provide general configuration mixing
calculations with a well-defined perturbative equivalent in
terms of an effective interaction removing the hard-core
problem. The corresponding diagrammatic picture is shown
in Fig. 5. This lowest order already contains long-range correlations through the mixing of nonorthogonal vacua. This
suggests that the presently developed perturbation theory
should be appropriate for systems being soft with respect to
some collective degree of freedom since the mixing of several mean fields is an efficient way to take the associated
correlations into account. Within this framework, the lowest
order corresponds to an adiabatic Born-Oppenheimer approximation justified when the individual degrees of freedom
relax much faster than the collective one. Going to higher
orders in the expansion aims at including diabatic effects in
the coupling of these two kinds of degrees of freedom.
A mean-field approximation is recovered by taking a
single f ␣ 0 coefficient to be different from zero. In this case,
Eq. 共32兲 contains a diagonal matrix element only and G ( ␤ , ␣ )
reduces to a single term G ( ␣ 0 , ␣ 0 ) . In comparison with Eq.
共19兲, the effective interaction appearing at the mean-field
level in the present case is different from the standard one.
As shown in Sec. III D, G ( ␣ 0 , ␣ 0 ) generates two-body matrix
elements including more correlations than G ␣ 0 does. In some
way, our scheme is not a simple extension of the standard
Goldstone formulas since we do not recover it for f ␣
⫽ ␦ ␣␣ 0 f ␣ 0 . Looking at this particular case, one could think
that the expression of the energy 共22兲 together with the
G ( ␣ 0 , ␣ 0 ) matrix lead to some double counting with respect to
the standard scheme. Equation 共22兲 being exact, this is simply a different way of resumming two-body correlations into
the interaction when expanding the whole energy instead of
some correlation energy. It may appear as an unnecessary
complication when considering a single Fermi sea, but is the
simplest way of superposing several of them to get the energy as given by Eq. 共32兲 at the lowest order in G ␤ and G ␣ .
This was our original goal.
The matrix G ( ␤ , ␣ ) depends upon the matrix element
具 ⌽ 0␤ 兩 ••• 兩 ⌽ ␣0 典 in which it is inserted. Thus, the energy at the
lowest order cannot be factorized into a form as given by Eq.
共2兲 which is built from a defined state 兩 ⌿ k 典 关see Eq. 共1兲兴.
Within the perturbative calculation renormalizing short-range
correlations, 兩 ⌿ k 典 should not be considered as the approximate wave function of the actual ground state of the system
corresponding to Eq. 共32兲, from which other observables
could be calculated together with bare operators. Actually,
this is only by analogy with a true variational calculation
making use of the starting Hamiltonian that one would use
兩 ⌿ k 典 as the corresponding approximate state of the system.
The approximate value of a given observable obtained by
using 兩 ⌿ k 典 will be ofthe same quality as for the energy 共32兲
only if short-range correlations associated with the corresponding operator are small. This is precisely the situation
which has been encountered in the GCM and the projected
mean-field method when using phenomenological interacជ ) 关7,34 –
tions depending on the mixed local density ␳ ( ␤ , ␣ ) (R
36兴. Indeed, while this interaction depends on the matrix
element 具 ⌽ 0␤ 兩 ••• 兩 ⌽ ␣0 典 , the state 兩 ⌿ k 典 has been used to calculate observables such as probability transitions.
The reason why no perturbatively well defined approximate state can be extracted from Eq. 共32兲 is related to the
fact that some diagrams are summed in G ( ␤ , ␣ ) incoherently
with respect to a well-defined order in the expansion of the
state 兩 ⌰ 0 典 . We have actually defined a perturbative expansion for the mean value of an observable 共energy兲 without
taking care of the state of the system itself. It is worth noting
that defining an approximate energy through the mean value
of H⫽t⫹V in any approximate form of 兩 ⌰ 0 典 would lead to
divergent diagrams. Indeed, V could not systematically enter
a ladder in this case. That is why ladder resummations have
to be performed in the energy before any approximation
takes place.6
The same kind of situation is encountered in usual shellmodel calculations making use of effective operators in a
limited valence space. In fact, this expresses the fact that the
nuclear mean-field energy or beyond mean-field energy in
calculations making use of phenomenological effective
forces is more to be seen as a functional 共of local densities in
general兲 than as the mean value of a two-body Hamiltonian
in a defined state. We anticipate the extension of the Skyrme
functional beyond the vicinity of a single mean field 关37兴,
which will be proposed in a forthcoming publication. This
extension relying on Eq. 共32兲, a formal link is kept with the
bare interaction. Contrary to Kohn-Sham philosophy 关38兴,
this strategy is by nature systematic and does not aim at
incorporating all the missing correlations at the considered
order.
6
The situation is similar when dealing with approximate forms of
兩 ⌰ 0 典 in E 0 ⫽E ␣0 ⫹⌬E ␣0 ⫽ 具 ⌽ ␣0 兩 H 兩 ⌰ 0 典 in the standard BruecknerGoldstone theory. Then, 兩 ⌽ 0␣ 典 is not rigorously defined as the approximate state associated with the BHF energy.
044311-9
PHYSICAL REVIEW C 67, 044311 共2003兲
T. DUGUET
The scheme developed up to now is valid whatever the
coefficients f ␣ are. While they are of no importance as long
as no approximation is done, they will influence the results at
all orders of the expansion. Let us now study two applications associated with two different choices of these coefficients.
IV. GCM
Once the set of product states is chosen along a collective
deformation path, the perturbative energy can be minimized
with respect to f ␤* coefficients at any order of the expansion.
At the lowest order, the minimization of E n⫽0
关 f ␤* , f ␣ 兴 is
0
equivalent to the eigenvalue problem:
冉具
⌽ 0␤ 兩 t⫹G ( ␤ , ␣ ) 兩 ⌽ 0␣ 典
冊冉 冊
冉
f ␣ ⫽E n⫽0
具 ⌽ ␤0 兩 ⌽ 0␣ 典
0
冊冉 冊
f␣ ,
共33兲
written in matrix form because of the dependence of the
effective ‘‘Hamiltonian’’ t⫹G ( ␤ , ␣ ) (0,0) on the matrix element 具 ⌽ ␤0 兩 ••• 兩 ⌽ 0␣ 典 .
These are the Hill-Wheeler equations encountered in the
GCM 关5,12兴 written in the nonorthogonal basis 兵 兩 ⌽ 0␣ 典 其 in
terms of the appropriate effective interaction. This result motivates the GCM from the perturbative point of view for the
first time. Let us emphasize that we are in no way referring
to the Ritz variational principle here since, as already mentioned in the preceding section, Eq. 共32兲 is not obtained
through the expectation value of the actual Hamiltonian into
a well-defined approximate state. In particular, the corresponding energy can be lower than the actual ground-state
energy. Finally, minimizing the energy at some higher orders
in the expansion motivates the introduction of diabatic effects in the GCM 关4,39,40兴.
Within the present interpretation, the diagonalization of
the GCM matrix provides several sets of ‘‘eigenvectors’’
兵 f ␣k 其 , k⫽0,1,2, . . . , corresponding to different approximations of the ground-state energy 共32兲. Everything having
been written from the outset for the ground state, no excited
state is obtained here. The corresponding 兩 ⌿ k 典 form a set of
orthogonal states where each of them is not orthogonal to the
actual ground state 兩 ⌰ 0 典 which it approximates. Using a
simple argument, it is, however, possible to interpret the
兩 ⌿ k 典 , k⭓1, as good approximations of the excited states of
the system. Indeed, while 兩 ⌿ 0 典 has a maximum overlap with
兩 ⌰ 0 典 , 兩 ⌿ k 典 共which is orthogonal to 兩 ⌿ 0 典 ) will have a large
overlap with the eigenstate 兩 ⌰ k 典 corresponding to the same
ordering of the energies.
the n⫽0 term only is retained, one recovers the symmetry
breaking mean-field solution 兩 ⌽ 0␣ 典 . Another way to stress
this point is to remember that usual perturbative corrections
with respect to 兩 ⌽ ␣0 典 will not restore any broken symmetry
unless an infinite number of diagrams is summed7 关1兴.
In variational calculations, it is well known that such an
infinity can be taken care of through a linear superposition of
symmetry breaking states 兩 ⌽ ␣0 典 as given by Eq. 共1兲. This is
the projected mean-field method. According to the symmetry,
it corresponds to a particular choice of the unperturbed states
兩 ⌽ ␣0 典 and mixing coefficients f ␣ . Let us now show how our
extended perturbation theory motivates such a method by
restoring good quantum numbers in the standard GoldstoneBrueckner expansion. We exemplify this through the restoration of rotational invariance.
We define 兩 ⌽ 00 典 as the nondegenerate ground state of an
axially symmetric deformed one-body Hamiltonian h 00 . A set
of rotated states around the y axis orthogonal to the symmetry axis is obtained through
冉
兩 ⌽ ␣0 典 ⫽exp i
共34兲
with ␣ varying from ⫺n to n as an integer. The Hamiltonian
being invariant under this tranformation, the rotating operator R( ␣ ) commutes with it. 兩 ⌽ 00 典 being h 00 ’s nondegenerate
ground state with the energy E 00 , 兩 ⌽ ␣0 典 will be the one of the
rotated Hamiltonian h ␣0 ⫽R( ␣ )h 00 R † ( ␣ ) with the same
eigenenergy. Thus, the adiabatic evolution operator is written
in the ␣ interaction representation under the form
␣
␣
U ⑀␣ 共 t,t 0 兲 ⫽e ih 0 t/ប U ⑀ 共 t,t 0 兲 e ⫺ih 0 t 0 /ប
冉 冕
0
⫽R 共 ␣ 兲 e ih 0 t/ប R † 共 ␣ 兲 exp ⫺i
t
t0
H ␣ 共 ⑀ , ␶ 兲 d ␶ /ប
冊
0
⫻R 共 ␣ 兲 e ⫺ih 0 t 0 /ប R † 共 ␣ 兲
⫽R 共 ␣ 兲 U 0⑀ 共 t,t 0 兲 R † 共 ␣ 兲
共35兲
because the ␣ dependence of the auxiliary Hamiltonian is
H ␣ 共 ⑀ , ␶ 兲 ⫽R 共 ␣ 兲 H 0 共 ⑀ , ␶ 兲 R † 共 ␣ 兲 .
V. SYMMETRY RESTORATION
The second case of interest consists in concentrating on
the approximate state rather than on the approximate energy
of the system. The actual ground state 兩 ⌰ 0␣ 典 defined through
Eq. 共11兲 obeys the symmetries of the system whatever the
symmetries broken by the product state 兩 ⌽ ␣0 典 from which it
develops at t⫽⫺⬁. However, a problem arises when diagrams are dropped in the expansion, since then, symmetry
breaking may appear in the wave function. For instance, if
冊
␲␣
J 兩 ⌽ 00 典 ⫽R 共 ␣ 兲 兩 ⌽ 00 典 ,
n y
共36兲
Consequently, the ground state 共21兲 can be rewritten as
7
In the particular case of rotational invariance, one could start
from a spherical mean field and use Bloch-Horowitz perturbation
theory for degenerate unperturbed states 关41兴 if dealing with an
open-shell nucleus. As we are interested in relating the projected
mean-field method to a perturbative expansion, we omit this possibility and continue with deformed mean-field unperturbed states.
044311-10
PHYSICAL REVIEW C 67, 044311 共2003兲
GOLDSTONE-BRUECKNER PERTURBATION THEORY . . .
U ⑀␣ 共 0,⫺⬁ 兲 兩 ⌽ 0␣ 典
n
兩 ⌰ 0 典 ⫽ lim
兺
⑀ →0 ␣ ⫽⫺n
冋兺
冋兺
f␣
具 ⌽ ␣0 兩 U ⑀␣ 共 0,⫺⬁ 兲 兩 ⌽ ␣0 典
n
⫽
␣ ⫽⫺n
f ␣R共 ␣ 兲
n
⫽
␣ ⫽⫺n
G (0,␣ ) 共 W 0⬘ ,W 0 兲 ⫽V⫹G 0 共 W 0⬘ 兲
f ␣R共 ␣ 兲
册冋
册冋 兺 冉
lim
U 0⑀ 共 0,⫺⬁ 兲 兩 ⌽ 00 典
0
0
0
⑀ →0 具 ⌽ 0 兩 U ⑀ 共 0,⫺⬁ 兲 兩 ⌽ 0 典
n
1
G 01
E 00 ⫺h 00
冊
册
⫻
n
兩 ⌽ 00 典 Linked/Irred
册
.
2I⫹1
I
* 共 ␲ ␣ /n 兲 ,
sin共 ␲ ␣ /n 兲 d 00
2n
具 ⌰ 00 兩 H P̂ 00兩 ⌰ 00 典
具 ⌰ 00 兩 P̂ 00兩 ⌰ 00 典
共40兲
,
⫽
E n⫽0
0
The projector for M ⫽0 or for a triaxially deformed product state
兩 ⌽ 00 典 is more complicated but the calculation can be extended without major difficulty.
兺
␣ ⫽⫺n
W 0 ⫺h 00
G 0共 W 0 兲 R †共 ␣ 兲 ,
f ␣00具 ⌽ 00 兩 关 t⫹G (0,␣ ) 共 0,0兲兴 R 共 ␣ 兲 兩 ⌽ 00 典
. 共43兲
n
兺
␣ ⫽⫺n
f ␣00具 ⌽ 00 兩 R 共 ␣ 兲 兩 ⌽ 00 典
It takes the form of the projected mean-field energy on
spin I⫽0 with the appropriate effective interaction
G (0,␣ ) (0,0). Two remarks still have to be made.
In agreement with the discussion given in Sec. III E, the
correlated wave function
兩 ⌿ 0 典 ⫽ P̂ 00兩 ⌽ 00 典
共44兲
is taken as the approximate ground state of the system corresponding to the energy 共43兲.
Even if the projection of 兩 ⌰ 00 典 for I⫽0 is zero, this is not
necessarily true for an approximation of this state. For the
same reasons of orthogonality as in the GCM, it is natural to
use equations similar to Eqs. 共43兲 and 共44兲 for I⫽0 as good
approximations of the energy and the wave function of the
excited states 兩 ⌰ I 典 of the system.
VI. CONCLUSIONS
共41兲
which includes a simple sum over ␣ . In order to get the form
共26兲, the divergences associated with the V interaction coming from H have to be regularized. The generalized Brueckner matrix doing so reads as
8
W 0⬘ ⫺h 0
Q0
n
共39兲
the projection being trivial if no approximation is done on
兩 ⌰ 00 典 . For an even-even nucleus, the projection gives 0 for
I⫽0 and identity for I⫽0.
Finally, the extended perturbation theory allows to restore
broken symmetries in the standard Goldstone-Brueckner expansion by projecting the approximate state. This formulation motivates the standard projected mean-field method
from a perturbative point of view for the first time and generalizes it to any order of the expansion. Moreover, the
method allows for a coherent summation of Brueckner ladders through the use of rotated Brueckner matrices
R( ␣ )G 0 R † ( ␣ ) for each component of the approximate wave
function.
Let us now study the corresponding expansion of the en2
⫽ P̂ I0 , the
ergy. As P̂ I0 commutes with H and satisfies P̂ I0
actual ground-state energy defined by Eq. 共22兲 becomes
E 0⫽
VR 共 ␣ 兲
0
共38兲
is the projector on total spin I and projection on the z axis,
M ⫽0.8 Thus, Eq. 共37兲 becomes
兩 ⌰ 0 典 ⫽ P̂ I0 兩 ⌰ 00 典 ,
G 0 共 W 0 兲 R † 共 ␣ 兲 ⫹G 0 共 W 0⬘ 兲
and depends on the matrix element 具 ⌰ 00 兩 •••R( ␣ ) 兩 ⌰ 00 典 in
which it is inserted. Thus, the projection presents a simplification with respect to the general case since a single standard
G ␣ ⫽0 matrix and k number of G (0,␣ ) matrices have to be
calculated explicitly. Once two-body correlations are
summed in the interaction, the energy at the lowest order in
G ␣ ⫽0 is written as
n
兺 f ␣I0 R 共 ␣ 兲
␣ ⫽⫺n
Q0
V⫹VR 共 ␣ 兲
共42兲
where d IM 0 is the Wigner function for (I,M ,K⫽0) quantum
numbers 关1兴, the operator
P̂ I0 ⫽
W 0 ⫺h 00
W ⬘0 ⫺h 00
共37兲
If f ␣ satisfies 关12兴
f ␣I0 ⫽
⫻
Q0
Q0
We have derived an extended Brueckner-Goldstone perturbative expansion for the ground-state energy of an interacting system. This scheme allows to recover at the lowest
order the energy of a mixing of nonorthogonal product states
as used in variational calculations. The expansion is valid for
systems interacting through a strongly repulsive interaction
at short distances as it is written in terms of a generalized
Brueckner matrix renormalizing two-body, short-range correlations in the appropriate way.
The main achievement of the present work is thus to provide a link between the generator coordinate or the projected
044311-11
PHYSICAL REVIEW C 67, 044311 共2003兲
T. DUGUET
mean-field methods making use of 共phenomenological兲 effective interactions and a perturbative expansion of the actual ground-state energy of the system. In particular, our
method allows us to understand the usual projected meanfield method as the lowest-order approximation of a symmetry restored perturbation theory.
In addition, this work formally provides a well-defined
effective interaction removing the hard-core problem to be
used in the GCM or in the projected mean-field method. In
particular, the summation of particle-particle ladders is
adapted to the superposition of nonorthogonal vacua characterizing these two methods. The link between the presently
defined effective interaction and phenomenological interactions used in configuration mixing calculations of finite nuclei such as the Gogny or the Skyrme forces is the aim of a
forthcoming publication. A new prescription for their density
dependence beyond the mean field relying on the presently
developed perturbation theory will then be tested on quantitative configuration mixing calculations in finite nuclei. Let
us say that the feasibility of perturbative calculations using
this extended Brueckner-Goldstone theory to deal with largeamplitude collective motions may be questionable at the
present time. This is a reason why we have planned in a near
future to use it mainly as a tool to get theoretically grounded
guidelines for density-dependent phenomenological interactions to be used in beyond the mean-field calculations.
G ␣␣
m␣n␣ p␣q
共W␣兲
1
⫽V ␣ m ␣ n ␣ p ␣ q ⫹
2
V ␣ m ␣ n ␣ r ␣ s G ␣␣
兺
␣
⑀␣ ,⑀␣ ⬎⑀F
r
s
r␣s␣ p␣q
⑀ ␣ p ⫹ ⑀ ␣ q ⫺ ⑀ ␣ r ⫺ ⑀ ␣ s ⫺W ␣
In this definition, W ␣ corresponds to the total excitation
energy of the intermediate state before the G ␣ interaction. In
Eq. 共A1兲, we took non-antisymmetrized matrix elements for
V, which brings about the necessity to antisymmetrize those
of G ␣ . The other possibility is to start from the outset with
antisymmetrized matrix elements of V.
APPENDIX B: G „ ␤ , ␣ … MATRIX ELEMENTS
AND STARTING ENERGIES
The reaction matrix G ( ␤ , ␣ ) satisfies a self-consistent equation generalizing the one for the usual G ␣ Brueckner matrix:
G ( ␤ , ␣ ) 共 W ␤ ,W ␣ 兲 ⫽V⫹G ( ␤ , ␣ ) 共 W ␤ ,W ␣ 兲
⫹V
The author is very grateful to G. Ripka for fruitful discussions during the elaboration of this work and to P. Bonche
and P.-H. Heenen for a careful reading of the manuscript.
⫺V
Q␤
W ␤ ⫺h ␤0
Q␤
Q␣
W ␣ ⫺h ␣0
V
G ( ␤ , ␣ ) 共 W ␤ ,W ␣ 兲
G ( ␤ , ␣ ) 共 W ␤ ,W ␣ 兲
␤
W ␤ ⫺h 0
Q␣
W ␣ ⫺h ␣0
An explicit matrix element of G ␣ as defined by Eq. 共12兲 is
1
⫹
2
1
⫺
4
Then, an explicit matrix element is
G ␤( ␤ ,␤␣ ) ␣
m n r␣s
兺
␣
⑀␣ ,⑀␣ ⬎⑀F
r
s
t u p␣q
兺
共 W ␤ ,W ␣ 兲 V ␣ r ␣ s ␣ p ␣ q
⑀ ␣ p ⫹ ⑀ ␣ q ⫺ ⑀ ␣ r ⫺ ⑀ ␣ s ⫺W ␣
V ␤ m ␤ n ␤ t ␤ u G ␤( ␤␤, ␣ )␣
共 W ␤ ,W ␣ 兲
⑀ ␤ m ⫹ ⑀ ␤ n ⫺ ⑀ ␤ t ⫺ ⑀ ␤ u ⫺W ␤
␤
⑀␤ ,⑀␤ ⬎⑀F
t
V.
共B1兲
APPENDIX A: G ␣ MATRIX ELEMENTS
AND STARTING ENERGY
1
2
.
共A1兲
ACKNOWLEDGMENTS
G ␤( ␤ ,␤␣ ) ␣ ␣ 共 W ␤ ,W ␣ 兲 ⫽V ␤ m ␤ n ␣ p ␣ q ⫹
m n p q
共W␣兲
u
V ␤ m ␤ n ␤ t ␤ u G ␤( ␤␤, ␣ )␣
t u r␣s
兺
␤
⑀␤ ,⑀␤ ⬎⑀F
⑀ ␣ , ⑀ ␣ ⬎ ⑀ F␣
t
u
r
共 W ␤ ,W ␣ 兲 V ␣ r ␣ s ␣ p ␣ q
共 ⑀ ␤ m ⫹ ⑀ ␤ n ⫺ ⑀ ␤ t ⫺ ⑀ ␤ u ⫺W ␤ 兲共 ⑀ ␣ p ⫹ ⑀ ␣ q ⫺ ⑀ ␣ r ⫺ ⑀ ␣ s ⫺W ␣ 兲
.
共B2兲
s
In this definition, W ␣ corresponds to the total excitation energy of the intermediate state before the G ( ␤ , ␣ ) interaction while
W ␤ corresponds to the total excitation energy of the intermediate state after the G ( ␤ , ␣ ) interaction.
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