close

Вход

Забыли?

вход по аккаунту

?

Brillouin-Wigner and Feenberg perturbation methods in many-body theory.

код для вставкиСкачать
Ann. Physik 5 (1996) 608-624
Annalen
der Physik
0 Johann Ambrosius Barth 1996
Brillouin-Wigner and Feenberg perturbation methods
in many-body theory
H. Keiter’ and S. Kililr
*,*
’ Institut fur Physik, Universitat Dortmund, D-44221 Dortmund, Germany
* Faculty of Natural Science and Arts, University of Split, 58000 Split, Croatia
Received 17 November 1995, revised version 3 June 1996, accepted 7 June 1996
Abstract. Brillouin-Wigner (BW) perturbation formulae can be rearranged into a form first proposed by Feenberg. Feenberg’s perturbation formulae also follow from a variational principle. They
are successfully tested at two typical problems, for which ordinary perturbation techniques completely fail. The first is to find conditions for the bosonization of the Tomonaga model perturbatively. The second is to clarify whether non-Fermi liquid behavior of the momentum distribution
function of the Luttinger model can be achieved perturbatively.
Keywords: Perturbation theory; Tomonaga model; Correlated electrons.
1 Introduction
From the historical point of view, as was demonstrated by Ljolje [l], the Feenberg
perturbation technique [2, 31 was the first physical theory beyond the BrillouinWigner perturbation theory, in which an infinite series (infinite number of terms) was
summed up and written in a general closed form. This was done just a year before
Feynman’s famous resumming in field theory. To quantum many-body theory it was
applied 25 years later. Today, almost 50 years after its discovery, it is expected that it
can be applied in different branches of theoretical physics, solid state, quantum
fluids, field theory . . .. From a formal point of view, Feenberg’s perturbation theory
until today is the most condensed one among the so called selfconsistent perturbatian
theories in the following sense: For a system with a finite number of states, all Feenberg formulae have a finite number of terms. This is easily seen from formulae like
(16) in Section 2 of the paper. In self consistent perturbation theories the quantities
one wants to calculate, e.g. energy, appear on the r.h.s of the perturbation formula as
well as on the 1.h.s. In this sense they are “holistic expressions” (p200) in which
each term contains unknown quantities. That is why they are not easy to use. But in
future we have to learn how to handle this kind of mathematical expressions.
In Section 2 we present a direct derivation of Feenberg’s wave function in which a
resummation of Brillouin-Wigner (BW) terms is easily recognised. In Section 3 we
* Supported by a grant from the cooperation on science and technology, Federal Republic of Germany - Republic of Croatia
1
~
,
,
i
I
H. Keiter and S. KiliC, Brillouin-Wigner and Feenberg perturbation methods
609
show how Feenberg's energy and wave function can be derived from the quantum
mechanical variational principle. In Section 4, using Feenberg's energy expression,
we derive the energy spectrum of the Tomonaga bosonized Hamiltonian [4] without
using Bogoliubov's transformation. This is partly done to show that Feenberg's perturbation theory can be applied to many-body problems. Though it in principle faces
the same difficulties as BW perturbation theory (the individual terms are not proportional to the volume of the system, which usually is considered as an essential weakness of the approach) it is shown, how these can be circumvented. The Bogoliubov
transformation is mainly avoided because of the following possibility: Considering
the Tomonaga Fermion Hamiltonian in the same way as the Boson one, and comparing the two treatments, we obtain three necessary conditions, which have to be fulfilled
if the bosonization procedure is allowed. They are compared with the traditional
ones.
We then turn to the second application. For the Luttinger model, which in contrast
to the Tomonaga Fermion model is exactly solvable, and which may show non-Fermi
liquid effects in the one particle momentum distribution function, we find an approximate Feenberg wave function for the ground state. From this we calculate the corresponding momentum distribution function. Some of the steps of the calculation are
similar to earlier ones. So in this part they will be sketched only. We then will give a
brief summary and outlook.
2 From Brillouin-Wigner to Feenberg perturbation theory
A derivation of BW-perturbation formulae starts from
where fi0 is the unperturbed Hamiltonian, and
projection operator
the perturbation. Introducing the
4f.l
which commutes with
&, one rewrites the foregoing equations into
I N ) = I n) + (EN - ri0,-'QnnP I N )
EN = e n + (n 1 A? 1 N ) .
The wave function renormalization constant for the perturbed system:
~
I
2-' = ( N I N )
=1
+ ( N I n?Qn(EN - fio)-2Qnn3
I N) 2 1
(7)
610
Ann. Physik 5 (1996)
is related to the energy via
z = 8EN
-.
8&n
Eqs. (5), (6) and (8) form the basic equations of BW-perturbation theory. The energy
formula of Rayleigh-Schriidinger perturbation theory follows from Eqs. (5) and (6)
by iterating Eq. (5) and inserting the result into Eq. (6). Iterating then energy and
wave function in powers of 1 yields the RS-energy formula and the RS wave function. Vice versa, BW-perturbation theory may be viewed as a resumed RS-perturbation theory.
Further resummation of BW-theory is possible. This was done first by Feenberg,
and can be found in detail in Ref. [2]. Here a method proposed in [ l ] is used.
Iterating Eq. (5), one gets
00
I N ) = I n ) + C [ ( E N - fiO)-lQnnv]k1 n) .
(9)
k= 1
Inserting now the projectors from Eq. (4) explicitly, the first few terms of the expansion read:
Here, RI is the eigenvalue of the resolvent in the state I 1):
(EN - fro)-' I I) = RI I l ) ,
and Vln is the matrix-element of the perturbation (1= 1). Sums on intermediate
states (all different from I n)) have been left out for clarity.
Combining those terms, in which the last matrix element everywhere is Vln (i.e.
rn = I in the 3rd term, r = 1 in the 4th etc.), we arrive at
Here, the prime in the last two lines indicates that all indices inside the bracket are
pairwise different from each other. The indices at the square brackets denote the first
state, all states are different from I n), as before. Next, the square brackets are
summed as follows: For [ . . - I I we define an irreducible part with respect to the state
I 1):
41 = RI(VII VImRm V m / VImRm VmrRr VrI * . *) 9
(12)
+
+
+
which contains only intermediate states different from I l ) . Then
[ * . ' ] / = ( 1 + 4 / + 4 : + . . * ) = ( 1- q l ) -
I
.
61 1
H. Keiter and S. Kilii, Brillouin-Wigner and Feenberg perturbation methods
The Feenberg energy E r has the same form as the BW energies, obtained from inserting the iterated series @. (10) into Eq. (6); but one starts from the initial state
I I) and the intermediate states have to be different from 1 n) and I I ) :
ElF = El
+ V//+ V/mRmVm/ + V/mRmVmrRrV r / + VIrnRrnVmrRr VrsRsVsr + .
*
.
(15)
Combining those terms, in which the last matrix element everywhere is Vmr etc. as
before, one finally arrives at
The prime at the sum again indicates, that all indices have to be pairwise different
from each other.
Eqs. (14) and (16) together with Eqs. (6) and (8) form the basis of kenberg’s perturbation theory. Since they result from rewriting the corresponding BW @. (3,
one
should expect that all properties of the BW theory can be translated into the Feenberg
form. We note in passing that all Feenberg formulae consist of a finite number of
terms, if the system has a finite number of states, like an NMR multiplet.
3 Variational principle in Feenberg’s perturbation approach
Starting from the. quantum-mechanical variational principle for the ground state energy Eo
We use a ‘generalized’ Feenberg trial state, obtained from Eq. (14) by replacing there
the energy denominators in the following way, e.g.
With
P l j I I) = f i o I ,)
+ PP I 1) = + (I I P I
(El
YL(2)))
I ,)
612
Ann. Physik 5 (1996)
and 1 lu,(z)) follows from Eq. (14) with the replacements just mentioned. This is the
Feenberg energy of Eq. (16) written with the aid of a projection operator. One could
also introduce operators which take care of the fact that all intermediate states in Eq.
(16) have to be different from each other and from I 1) and I n), but this is not really
necessary for the formulation of the variational principle.
) ) the r.h.s. of Eq. (17)’ we decompose it in the following way
Inserting I v / ~ ( z on
Let us suppose that we may write the numerator of the first fraction as
Looking for the extremum with respect to z , we find
(22)
=O.
This can be fulfilled by setting the 2nd and 3rd line separately equal to zero, yielding two conditions for z:
Next we will show that Feenberg’s perturbation theory is obtained, if we assume that
also a third condition on z holds:
We may identify K ( z ) by decomposing
(vNI=
(n 1 + ( y N- n I in Eq. (21)
613
H. Keiter and S. Kilid, Brillouin-Wigner and Feenberg perturbation methods
Inserting the definition of K ( z ) from the last two lines of Eq. (26) into Eq. (25), the
matrix element (tyNI P I w N ) is cancelled, P can be left out in front of ho, and
one obtains
v
This equation can be satisfied by the normalization (n I
value of z
WN(Z))
= 0 at the following
(which is equivalent to Eq. (24) with condition Eq. (25) inserted there) and by
Z=
(IYN(4
I fio + f I W N ( 4 )
(V/N(Z)
IW
N W
(29)
Obviously, z in the last two equations is the energy in the state I v / ~ ( z ) )So,
. Eqs.
(25) and (24) are compatible with each other.
We still have to show, whether Eq. (23) is compatible with the other two conditions on z. Using Eq. (25) once more, Eq. (23) reads
With Eq. (28) this Eq. is turned into
Inverting the derivatives and identifying z as E N , one obtains
This corresponds to the normalization condition for the wave function (Eqs. (7) and
(8)), and thus completes the derivation of Feenberg's perturbation theory from the
quantum-mechanical variational principle.
614
Ann. Physik 5 (1996)
4 Some new applications of Feenberg's perturbation method
In many-body theory the Feenberg perturbation method was applied to the following
problems: The Bogoliubov model of the interacting Bose gas (ground state and the
elementary excitations) [I], ground state of the high density electron gas [6] and the
interaction between excitations in Bose systems [7]. In this section we want to apply
it to two well known problems:
I. The energy spectrum of the Tomonaga Hamiltonian and
11. The momentum distribution in the ground state of the Luttinger Hamiltonian.
I. We will derive the energy spectrum of the Tomonaga Hamiltonian for the Bose
and the Fermi case in order to get new insight into the bosonization procedure.
1. Boson case
For the boson case we use the Tomonaga Hamiltonian for the one-dimensional electron gas recast into the standard Bose Hamiltonian
fi=ko+Q
(33)
where
(34)
(35)
relations hold in the standard
vk is the Fourier transform of the potential energy, VF is the Fermi velocity of the
particles. The relations (33) to (35) show that the one-dimensional electron gas is described by boson excitations of the electron gas.
In the standard treatment [8], the Hamiltonian is diagonalized by a Bogoliubov
transformation, leading to the energy spectrum as in Eqs. (47) and (48). In Feenberg's
approach the derivation is clumsier. Using the method of successive approximations,
a lot of terms are cancelled. The remaining ones can be turned into a structure typical
for this approach: continuous fractions. By this procedure we also resolve the aforementioned volume dependence problem. The intermediate states to be used are listed
in the appendix (part 1).
They are eigenstates of the Hamiltonian f i 0 in the occupation number representation. Let ket I l ) =I 14) represent the eigenstate with one excitation with momentum
q'(h = 1) and energy mq(h = l), i.e.
H. Keiter and S. Kilii, Brillouin-Wigner and Feenberg perturbation methods
615
If we denote 1 l q ) = 1 and similarly the other allowed states, the Feenberg energy formula reads:
Before solving this equation by a successive approximation method, a few remarks
seem to be in place here. The allowed states can be found with the aid of diagrams.
All of them up to fourth order are presented in the appendix (part I). Terms with odd
number of factors V in the numerator are zero. Thus third order, fifth order etc. terms
are zero as a consequence of the interaction operator form. The matrix elements are
calculated in the appendix. The first two successive approximations in (37) read
The third one includes the second order term with the zeroth successive approximation in the denominator:
(n includes all possibilities)
Ej2' = wq
+ V,l+
wq - 3wq
Similarly the third and fourth order successive approximations are obtained:
(the 3rd order term is zero),
616
Ann. Physik 5 (1996)
The states nl,n2 .... are given in the appendix. In the same way the Feenberg energies
in the denominators are derived
Inserting Eqs. (41) and (42) into Q. (40), and performing the expansion up to second order one finds:
4 4 )
2-4V;
= Ej') +
1
4P2
-20, - 4vq - 4
-2w4
$5
c
p(#*q)
4
v;
4P2
-20, - 4vp - -2w,
(43)
All other terms up to fourth order are cancelled. One can continue with the same procedure in the sixth order and so on. We conclude that the final result should read
Sums of continuous fractions we perform as usual. Introducing
y=- f
"-Y
(45)
where f = 4 v i and a = -20, - 4vq and solving the quadratic equation we find the
physical solution
+ oz + r
y = -aq - 2v,
40,vq,
617
H. Keiter and S. KiliL, Brillouin-Wigner and Feenberg perturbation methods
or in the final form
where
The expression (47) with the relation (48) is a well known result for the Tomonaga
one-dimensional model of the electron gas. The way it was derived seems to be typical for coming to grips with Feenberg's perturbation approach.
2. Fermion case
In the Fernion case the Hamiltonian (33) is expressed as follows:
(49)
k,s
(50)
where the Fermion operators anticommute, i s .
t
{akl
,SI 7
ak2,s2} = &I
,k$sI
,SI
1
t
t
{akl JI 7 a k z ~ 2 = i U k l ,sI 3 'k2,s2 } =
*
Instead of the excited states we now calculate the ground state energy. In this case
the ground state of the unperturbed Hamiltonian is the filled up 'Fermi line'
[ - k ~+, k ~ and
] the ground state is given by
10) =I
l-kF..
.I k F ;
)
.
Excited states and corresponding matrix elements needed for the calculation are given in the appendix (part 2). Using again Feenberg's formula (37) with I 1)
0) and
the matrix elements for the Fermi case, one finds in the 4th order successive approximation
ZI
where
618
Ann. Physik 5 (1 996)
Now we compare the relation (51) with (43) or (44), which both contain ground state
energy and excitations. Let us mention that the 3rd and 4th order terms are not given
in Eq. (51) because of the fact that they do not appear in the relation (43). Furthermore, there is an exact cancellation of some terms from the 4th order, which appear
in the Bose case also, and the terms from the denominator of the 2nd order, which
are expanded to become 4th order terms. The relation (51) transforms into relation
(44) (in which the terms related to the excitations are excluded) after the approximations:
= oq
=
vq
v.
v,
(53)
where
=
The first relation in (53) is well known in the bosonization procedure of the Tomonaga Fermion model, see e.g. [5]. The other two are more or less in spirit of the corresponding ones in the original Tomonaga paper [9], though somewhat different.
Since ordinary perturbation expansions fail for this model, self- consistent ones had
to be used to derive conditions for the bosonization. In our opinion the derivation of
the relations (53) has a more general meaning: It shows the way, how other Fermion
systems could be mapped onto a Bose system using Feenberg perturbation techniques.
11. Next we want to calculate the ground state wave function and the momentum distribution function for the Luttinger model. This differs from the Tomonaga model in
the essential feature that it has an additional Dirac sea, which makes it exactly solvable. The Luttinger Hamiltonian reads [ 5 ] :
H. Keiter and S. Kilii, Brillouin-Wigner and Feenberg perturbation methods
v-Pa:klslaIki+psia~k2+p~2a2k~s2}
619
(55)
where both kinds of Fermion operators satisfy the anticommutation relations
and where i,j = 1,2.
In order to obtain the momentum distribution in the ground state
of any sort i of particles, we have to know more about the wave function yo. The
derivation of wave functions is usually more complicated than energy calculations.
For the Luttinger Hamiltonian it is possible to derive a form of the wave function
yo, which may be used successfully in studies of the momentum distribution function. The general form of the Feenberg ground state wave function reads (14):
where qm = Eo - EC. Because of the special form Of the Luttinger Hamiltonian we
are able to recognize all states Ivi) and all intermediate states which appear in relation (55). Of course we start from the state:
We use the notation where in the first line of the ket is the first sort of particles (index 1) and the second line is the second sort of particles (index 2). For instance
is a state in which one particle of the sort 1 from the state tl is excited in the state
r ) + a l , and at the same time one particle Of the sort 2 is excited in the state
rl - a l . The total momentum is conserved of course. The intermediate states
1p2),)713), ... equal 1 YI).The matrix elements can be calculated similarly to the case
I. Introducing them into the relation (55), we find
620
Ann. Physik 5 (1996)
where in the third sum i = 1,2,3. The factorials come because of the non-repetition
of the summation states. In a short written form the wave function reads
where
and
Now it is easy to calculate the momentum distribution. From eq. (57) one finds
In the same way for the numerator of the momentum distribution in Eq. (54) one
finds
The sum of the 6's can be transformed into a product - as in the last term. Introducing
H. Keiter and S. Kilid, Brillouin-Wigner and Feenberg perturbation methods
62 1
one finds
Thus the momentum distribution may be written in the form
The relation (67) is quite general as long as the approximation (61) is valid. It can be
studied for an explicit potential in the same way as was done by Mattis and Lieb
[lo] and with similar results. Here we just investigate non-Fermi liquid behavior. Indeed, its existence is a general property of the solution. Obviously &, = $ is reached
for both parts, if
a
where we supposed that for k -+ k,= and Q + 0 yia = y . This relation may be studied in more detail for toy-models. Let us mention that in the relations from (57) to
(62) all a's adopt values greater than zero. We conclude that non-Fermi liquid behavior can be derived from the approximated wave function (61).
5 Summary and outlook
Feenberg's perturbation approach was derived from the Brillouin-Wigner one, and from
the quantum-mechanical variational principle. In an application to the Tomonaga model
it was shown that the method of successive approximations came to grips with the
extensivity problem of self-consistent perturbation expansions for large systems.
Also, bosonization conditions were found perturbatively. Furthermore, an approximate
momentum distribution function for the Luttinger model was derived. This shows, that
generalized self- consistent perturbation approaches can be useful in many-body theory.
There are many more models to which the present approach can be applied. The present
approach can also be applied to resolvent perturbation theory, where our preliminary
results seem to be promising.
6 Appendix
In this appendix we write up 1st all Boson states and 2nd all Fermion states which
are needed for the calculation up to the fourth order and calculate corresponding matrix elements.
622
Ann. Physik 5 (1996)
I. Boson case
In the second order of Feenberg’s perturbation formula (38) there are two states (and
two possibilities of summing):
Besides these in fourth order we have:
(and there are three possibilities of summing). In 6th order there are 22 possibilities
of summing. Matrix elements are:
Diagonal:
v[[= (1, I v I 14) =
v k +2vq
k
Nondiagonal:
H. Keiter and S. KiliC, Brillouin-Wigner and Feenberg perturbation methods
Vn,,, = d5 *
623
Jz.2 V p
Vndni= d5.a. 2 V q
2. Fermion case
In this case the following states are used:
Let us note that in the 'Fermi sea' only holes are marked. The matrix elements of the
operators (SO) in the states above are (if we assume spin-independent interactions):
Diagonal:
Nondiagonal:
624
Ann. Physik 5 (1996)
References
S. Kilic, K. Ljolje, Fizika 4 (1972) 195
E. Feenberg, Phys. Rev. 74 (1948) 206
H. Feshbach, Phys. Rev. 74 (1948) 1548
P.M. Morse, H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York 1953
G.D.Mahan, Many-Particle Physics, Plenum Press, New York, London 1990
S. KiliC, Fizika A1 (1992) 135
S. Kilic, C.E. Campbell, to appear in Z. Physik B
L.D. Landau, E.M. Lifshitz, Lehrbuch der Theoretischen Physik. Bd. 9 $25 4. Auflage, Akademie
Verlag GmbH, Berlin 1992
S. Tomonaga, Progr. Theoret. Phys. (Kyoto) 5 (1950) 544
D.C. Mattis, E.H. Lieb, J. Math. Phys. 6 (1965) 304
Документ
Категория
Без категории
Просмотров
4
Размер файла
621 Кб
Теги
many, feenberg, wigner, perturbation, method, brillouin, body, theory
1/--страниц
Пожаловаться на содержимое документа