The Many-Body Problem Jastrow Correlations Versus Brueckner Theory: Proceedings of the Third Topical School Held in Granada (Spain) September 22–27, 1980

372 Pages·1981·14.159 MB·English

by  S. Rosati, S. Fantoni (auth.), R. Guardiola, J. Ros (eds.)

CORRELATIONS IQ~INFINITE SYSTEMS
.S Roeati
Istituto di Fisica dell'UniversitY, Piss, Italy
Istituto Nazionale di Fisica Nucleate, Sezione di Piss, Italy
÷
.S Fantoni
Institut for Theoretische Physik
Universit~t zu KOln, Cologne, W.-Germany
Abstract. An extensive analysis of strongly-interacting infinite Fermi
systems is presented within the framework of the variational theory ba-
sed on trial correlated wave ?unctions. The Fermi hypernetted chain
(FHNC) procedures for calculating the energy expectation value and other
quantities of physical interest are derived in correspondence to the
following two cases: )a{ polarized Fermi systems described by Jastrow-
Slater wave ?unctions with state-independent correlation faotors,(b)un-
polarized Fermi systems described by Jastrow-type wave ?unctions, with
correlation factors depending on the spin-isoepin state of the partic-
les. Numerical results are presented for a simplified model of nuclear
matter, for liquid 3He and for the hard-spheres Fermi system. The cal-
culation of the surface energy coefficient for nuclear matter is also
discussed.
÷
Permanent address: Istituto di Fisica deIl'Universit~, Piss, Italy
.I Introduction
In the present paper some problems related to the variational approach
to the study of strongly interacting Fermi systems are discussed. Basi-
cally, the analysis is limited to the case of infinite, homogeneous and
translationally invariant systems, but some results for finite systems,
obtainable in rather Simple ways, are also presented. The approach is
non relativistic, and the particles which constitute the system are sup-
posed to interact by means of two-body forces. The Hamiltonian is then
written in the form
A A
H=T+V= t(i) + ~ V(i,j) , (1.1)
i=I i<j:1
where t(i) is the kinetic energy operator and A is the number of par-
ticles. In the calculations, A and the volume 0 of the cubic box contai-
ning the system are taken to be finite but large enough to finally con-
eider the thermodynamic limit: A and £ are left to go to infinity, but
the density ~ is kept fixed.
The wave function @ for the system is written as
@:F¢=F(1 ..... A)@(1 ..... A} . (1.2)
The model function ¢ must adequately describe the ground state of the
system in absence of strong interparticle correlations. ¢ is therefore
chosen as the Slater determinant of the Fermi sea
~(1 ..... A)=detlCBz(i)=. (8)s61... 8A¢61(I)...¢6 A(A) , i.3)
where (6) stands for 18 ..... 6 A and s8 .8 is the permutation symbol,
The free-particle wave function ~6 (i)li'n" wh~chthe argument refers to the
particle . and the subscrzpt • to the si tate, has the form
BS ~.iB~i(pxe2/I-~)i()is(pq=)i( )i , )4.1(
l
where the spin (-isospin) functions qp(i), p=1 ..... v, constitute a com-
plete orthonormalized set. The number v is called the multiplicity of
the system. The allowed momenta ~. satisfy the usual periodicity con-
1
ditions on the walls of the box,
ki,x=ni,x~/a, ki,y = ni,y~/a , ki,z=ki,z~/a (ni, x,ni,y ,ni,z ~0) , I( • )5
where a is the side of the box.
The correlation factor F, in eq. (1.2), is an operator which takes care
of the correlations induced by the strong interparticle interactions.
It is require d to satisfy the cluster property, i.e. when a subset
i I ..... i r of particles is far yaw'a from all the remaining ones, the
following relation holds:
F(I ..... )A ÷Fr(i I ..... ir)FA,r(ir+ 1 ..... i )A . (I.6)
Interesting applications are possible if the system is allowed to pos-
ses different populations in the v single-particle spin states. The
problem of evaluating the mean value of the total energy for those sys-
tems has been treated by ROSATI and FANTONI )I( in the case of a state
-independent Jastrow choice )2( for the correlation factor F, namely
A
=F ~ fir..) . (1.7)
i>j= I jI
In section 2 the problem is discussed in details, and the cluster ex-
pansion for the two-body distribution function is given and summed up
in the frame of the Fermi hypernetted chain (FHNC) theory. Section 3
presents some numerical results obtained for three specific polarized
Fermi systems, namely a semirealistic model of non-symmetrical nuclear
matter, the polarized 3He and the hard-spheres system.
A state-independent correlation factor, as specified by eq. (1.7), ap-
pears to be appropriate for systems of particles interacting through
central forces at not to high densities (see, for example, the review
of KRALC (3)). However, to obtain very accurate results, some improve-
ments are necessary. First of all, at high densities, effective three-
body correlations become important: the generalization of eq. (1.7) to
include effective three-body correlation factors is obvious and the
corresponding calculations, which do not present particular adjunctive
difficulties, can be found in ref. (4). On the other side, serious pro-
blems are encountered if the correlation factors are taken to be state-
dependent, i.e. they contain some dependence on the spin operators, the
velocities, the angular momenta, and so on, of the particles (see, for
example, ref. (5)). The correlation factors considered in section 4 in-
blude a rather simple spin dependence which does not give troubles for
the calculation of all the cluster terms contributing to the two-body
distribution functionj the corresponding wave function reveals to be of
some interest for a satisfactory description of certain configurations
of a Fermi system.
Section 5 is devoted to some problems of interest in the variational
theory for Fermy systems. First of all, the importance of a two-body
correlation ~actor with a structure, ~or both small and medium inter-
particle separations, is stressed. The calculation o~ the momentum dis
tribution is then briefly discussed. Finally the results ~or the sur-
face energy in e Fermi system are given.
5
2. FHNC approach to polarized Fermi s~sflems
2.1. Cluster expansion. Let us indicate with x (K), K=I,...,v, the per-
centage of particles in the single-particle spin state K. The density
of particles in the state K is related to the total density P=A/Q by
the relation p(K)=x(K)p. The allowed values for the momenta of the par-
ticles in the state K are contained in the Fermi sphere of radius
k~K)=(6~2p(K)) 1/3
, (K=I ..... v) . (2.1)
The two-body distribution function g(I'J)(1,2}, with particles I and 2
in the spin states I and ,3 is defined as
(I,3) A(A-1) ~-
g (1,2) = @#~(I)(I) #(3)(2)~dx3...d~A, (2.2)
~p2 (~T
where x i represents both position ~. and spin ~ariables of the particle
1
i, so that summation over ~i' implies both integral ever all the space
and sum over the spin varxables.~r=~ . ~ ~dXl...dXA÷ ÷ , and ~(K) (i) is the
projection operator on the state qK for the particle i. The particles
(indices) I and 2 will be called as "external" particles (indices),
the remaining ones as "internal". In the limit of an infinite and
translationally invariant system, the distribution function depends on-
ly on the distance of the particles 1 and 2. With the choice of W given
by eqs. (1.2) and (1.7), we can write
(I,J)
r12) ~#-~ (1) 2)¢i>j=I~ (r.lj)dx3...dXA .2.3)
The procedure adopted here to derive the cluster expansion for the di-
stribution function, differs in some respects from the one of ref. )I(
and is quite similar to that one utilized for unpolarized systems in
ref. }6( (see also refs. )7( and (8)}. First of all, we have to obtain
the expansion of the squared Slater determinant AA=¢ ~ ¢. The model func-
tion ~ specified in eq. (1.3) can be written by performing the antlsymme
trization with respect to the particles or to the states, so that
$~(1 ..... A)= £ .~A~ ~ * (~A) =
(~) ~1'' (~I)'''~A
(2.4)
=E61...6A {~)e 1'''~A¢B1(~1)'''~BA (aA) '
If the final expression is used together with eq. (1.3), we obtain
AA(1 ..... A) = ~ s ~)¢;Ie1)'''~6ACmA)¢61(1)'''¢BA A) ' (2.5
~) ~I"''~A
Due to the antisymmetrization with respect to (~), the summation on
18 ..... B A can be extended to include also equal values, so that we £et
AA(1 ..... A)= detlP(i,3)l , (2.6
where
A
2.7
P(i,J) = !l¢8(i)¢B(J) •
8
The p-particle subdeterminants Ap defined as
A (1 ..... pl=detlp(i,j) l, i,jg1,p , (2.8)
P
satisfy the relations (see refs. (6,8))
fAp+ld~p+i=A-p)&p 2.9
& ~0, p>A (2.10)
P
Let us examine the structure of the terms which contribute to 6p defi-
ned by eq. (2.8). Any permutation of p particles can be seen as pro-
duct of independent cyclic permutations of separate subsets of these
particles. The parity of a cyclic permutation involving q particles is
(_)q-1. We have, therefore, the following rules:
The terms of Ap are products of closed loops of factors p(i,j) involving
the p particles in separate loops; to every loop of q p-factors (or par-
ticles) is associated a factor (_)q-1. In particular, if a particle i is
not exchanged with other particles, a factor p(i;i),which reduces to the
density p when the spin traces are calculated, is associated with that
particle.
For the in?inite polarized system, we are interested ,ni one £ets
A v KA
p(i,j)= ¢~(i)¢~(j) = n~i)nKj)a -1 exp( ikBK. + ÷ rj-ri)= +
a=1 K=I 6K=I
(2.11)
v oCK) 1 ~K)
= ~ nKi)nK J) (k rij),
K=I
where
l(k~K)y)= 1 dR expi~.~) 2.12)
(2~)~p '}K f k ~k p
The function defined by the latter equation is called the statistical
correlation factor and it satisfies the two following relations
d;1Ck~K)y):;d;12k~ K)y)=I/p (K) , (2,13)
;dglCkF{K)y)lCk }K I; 1
(2.14
The explicit analytical expression of the statistical factor is
iCz)=3z-3(slnz -zcos z } . C2.15)
Let us now consider the quantity
OcJ c2)A
X C2.16
~ P
where the spin sums are evaluated for all particles, except I and 2. The
various terms which contribute to thls quantity can be easily evaluated
using the expression obtained for & . The operator ~(I)c1) ~(J)C2) is
P
associated to each term of A and then the spin sums are calculated.
P
The following statistical loop rules are satisfied:
If aloop of p(i,j) factors does not include particles I and 2, after
-
spin summations,v different loops are obtained. The first is construc-
ted with the statistical factor x(1)l(k~l)rij),-~ the second with
x(2)itk(2)r " F ij), and so on. To every loop is associated a factor (_)q-1
where q is the number of statistical correlations.
If particle I is involved with other q-1 particles, but not particle
-
2, in a loop of p(i,j) factors, after spin summation, a loop of fac-
tors x(I)l(k(I)r ) involving the same q particles, with an adjunc-
F i'
tive factor (_)q-l, is obtained.
If both the particles I and 2 are involved in the same loop of p(i,j)
-
factors, after spin summation, two chains of statistical factors,
having the particles I and 2 as extremitles, result. One chain is con-
structed with x(I)l(k~I)rij)- and the other chain with x(J) l(k~J)rij).-
Let us sxpllcitly calculate the two-body distribution function in the
case of a non interacting polarized Fermi gas. This is obtained 9rom
eq. (2.3 with the positions ~I)= ~(3)=I and f(r..)=1,
JI
g2FCr I 2 ) A(A-I) A ÷ d= ÷ A" c2.17)
where the spin summations are mow extended to ell the particles. By
using eq. (2.9) and ~=A!, we get
= AA-I) & =1 !
g2F(r12) E o~ r (A-2)I 2 2p T£(I'1)p(2'2)-p(1"2)p(2'1)
v ~K) 2
= 1- x(K)l(k r12) o 2.18)
K=I
For an unpolarized system x(K)=l/v and kF(K)=k F for K=I ..... v, and the
distribution function reduces to the well known expression
I12 (2.19)
g2F(r12)=l"~ (kFr12).
The calculation of the two-body distribution function for a correlated
system is a far more difficult problem. First of all, let us consider
the quantity F 2= A f~rij)J the function f(r) which will be considered
i>j=1
is such that fCr)~l as r÷® and, moreover, it does not differ apprecia-
bly from unity for most of the values of r. As a consequence, the func-
tion h(r)=f2(r)_1 (2,20)
is appreciably different from zero only in a small region and can be
used as an expansion parameter in evaluating the distribution function
or other interesting quantities. Let us now substitute by 1+hrij) all
the factors f2{r..) which constitute F 2 and then expand the products in
JI
terms of the function hit..). If all the terms related to a given num-
JI
bet p of particles particles I and 2 being always included) are grou-
ped together, we get the expression
A A
F2=X2 (1,21 + ~ X31,2,i 3) + Z X411,2,i ,i 4} (2 21)
i3=3 i3>i4= 3 3 . . . .
One has X211,2)=f2r12) and all the remaining Xp functions contain
f2(r12) as a factorj the explicit expressions of the various Xp func-
tions can be derived without difficulty. If expression 2.21) is sub-
stituted in eq. (2.3) for g(I'O{m12), since the quantity in eq. (2.16)
is symmetric with respect to the interchange of the particle coordina-
tes ~3 ..... A+X , all the terms Xp which differ only in the labels of their
arguments may be relabelled and summed to~ether to give
gCI'Jlcr12 =A(A-! ~' ~ (11 21& +
~Pp~ T 2 (A-plIp-2IlXp(1 .... P)d~3"'drA"
{2.22)
Integratlon and spin summation over the particles p+1 .... ,A not appea-
ring in Xp, can be readily performed using the identity (2.9) to give a
factor A-pIA :
P
g(I'Jlr121Np p 2 T~'TT • 1 2)A Xp(1 .... p)dr3..drp.2.23)
The summation over p has been extended to ~ in virtue of identiy (2.10).
The calculation can be carried on in the same way as that one employed
for unpolarized systems dna discussed in ref. C6). eW od not repeat here
all the details of the procedure but limit ourselves to exhibit the most
important feature of the calculation.
ehT quantity enclosed in square brackets in eq. [2.23) nac eb expanded
sa stated before, sa mus of statistical terms Tstat, each characterized
by a product of non-overlapping closed loops of statistical factors. A
generic term T appearing sa integrand in eq. (2.23) is the product of a
statistical term Tstatwithadynamical eno ydT n, contributing to Xp,
os that T=TstatTdy n is a function of the dynamical correlation factors
h(rij) dna the statistical correlation factors. In general T nac eb
written sa product of unlinked terms,i.e, functions without nommoc in-
dices. Let su consider all the terms which enter in eq. (2.23) dna have
a same llnked part involving a given number of particles together with
particles I dna 2. All the unlinked parts of these terms nac eb integra-
ted with respect to their arguments ,and the results demmus pu to give
the normalization constant~ . In this yaw ew ~et
C(I.')
1 r F n ~ [1,2 ..... n) id~3...d~.n,
gCI,J)Crl2)=~Zn_L2j (n-2)l (2.24)
where C[I'3)(1,.2. ... n) indicates the mus of all the linked terms con-
n
structed sa allowed products of statistical ant dynamical factors inte-
resting the particles 1,2 ..... n. All the ways f labelling the n-2 in-
ternal particles in a term lead to the same coqtribution in eq. [2.24):
the number of different labelling ways is (n-2)l! ~ divided by the symmetry
I
)9((rebmun ') ,S of the particular term considered. I Moreover, it nac eb
nwohs that all the constributions from the Ireducible [") terms can-
ceil out so that, finally, ew set ii
I
I
!
®
CI,J) ~ r CI'J] ,2]=~ ~ CI'J)C1,2, i.,p)d~3 .d; (2.25)
g Cri2)= (I . . . .
2=p 2=p i P "
I
I
(') ehT symmetry number ,S associated with a given term containing the
indices 1,2 ..... n , is defined sa the number of permutations of the
indices 3,4 ..... n which leave the term unchanged.
(") A term is called reducible when its integral, sa specified by eq.
(2.24), nac eb factorized into a product of 2 or more integrals.
01
In the latter equation ~(I'3)(I,2 .... )p indicates the sum Of ail irredu-
cible, topologically distinct terms corresponding to the external par-
ticles I and 2 in the states I and ,J respectively, and with p-2 inter-
nal particles. Each term is divided by its symmetry number S and is an
irreducible product of statistical and dynamical correlation factors.
The following rules hold:
The statistical correlation factors constitute closed, separate loops
-
and satisfy the statistical loop rules discussed before.
The external particles I and 2 are associated with the function
- f2(r12).
With each internal particle i is associated at least one function
-
h(rik).
Each integration on particle coordinates implies a factor .p
-
2.2. FHNC equations. Once we have specified all the terms which con-
tribute to the two-body distribution function, we must device a technique
to sum all these contributions. In the case of an unpolarized system this
sum can be expressed in terms of four functions which satisfy a set of
integral equations )4( known as FHNC equations. These functions will be
denoted here as Ndd(r12), Nde(r12), Nee(r12) and Ncc(r12), where the
subcripts ,d e and o stand for dynamical, exchange and cyclic, respec-
tively. Each function N is the sum of an infinite number of terms with
structures symilar to those involved in eq. (2.25). The terms corres-
ponding to Ndd(r12) include the particles I and 2 only with dynamical
correlation factors, those corresponding to Nde(r12) include the par-
ticle I only with dynamical correlation factors and the particle 2 with
two statistical factors. Those corresponding to Nee(r12) (Ncc(r12)) in-
clude both the external particles with two (one) statistical correlation
factors and an arbitrary number of dynamical ones.
It is to be noted that for unpolarized systems, there is only one sta-
tistical function l(kFrij)J while for polarized systems there are, in
general, v different statistical functions and, therefore, v different
chains of statistical factors. As a consequence, to sum up the terms
contributing to the distribution function of a polarized system, it is
convenient to introduce a larger number of functions, which are denoted
N(I'J)( ) N(I'J)(r12 ) N(I'Jl(r12 ) and N(I'J)(r12 ) However. when
as -dd r12 ' de ' ee cc '
the external particle 1 (or 2 is connected only by dynamical correla-
tion £actors, no dependence on the state I (or J) occurs, so that we
have: