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

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