Completeness of pair wave functions

Volume 102A, number 9

PHYSICS LETTERS

11 June 1984

COMPLETENESS OF PAIR WAVE FUNCTIONS Heinz BARENTZEN i and M.D. GIRARDEAU

Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, OR 97403, USA Received 9 April 1984

A rather general proof is given of orthonormality and completeness of pair wave functions. Because of the close relationship between RPA and the pair-theory method, the pattern of the proof also applies to RPA wave functions.

Recently a pair-theory approach [1] has been worked out for the one-band Hubbard model. The pair hamiltonian obtained is an exact reformulation o f the original Hubbard hamiltonian in terms of pair operators, which satisfy elementary boson commutation relations. The pairs, which are the elementary particles in this theory [2], are characterized by pair energies Ep(q) and pair wave functions ¢bkp(q ). Here q is the total quasimomentum of the pair, and the quantum number p labels the various pair states. As was shown in ref. [ 1 ], the Hubbard pair spectrum consists o f one bound state (spin wave mode) and a set o f continuum states. For the validity o f the pair-theory method it is crucial that the pair functions constitute a complete orthonormal set [2]. In ref. [1] this has been taken for granted, i.e., it has been assumed that the following relations are satisfied:

fpp,(q) = ~ Cb*kp(q)rbkp,(q) = 6pp, k (orthonormality),

(1 a)

gkk'(q) = ~ dPkp(q)cb*k'p(q) = 6kk' P (completeness) ,

state and continuum). Since completeness of the

rbkp(q ) is so important for the pair-theory method, a rigorous proof of eqs. (1) is highly desirable. In the present work an attempt is made to provide such a proof. Since, as will be seen below, our proof does not depend in any essential way on the details of the Hubbard model, it seems to be applicable to a much broader class of pair functions than merely those associated with that particular model. This work is similar, in aim and spirit, to Brout's [3] paper on pair excitations in the high density electron gas, where he proved that the scattering states and the plasma mode together constitute a complete set. Brout's proof, however, seems to be less general than ours. Formally the pair functions are defined as the solutions of a certian eigenvalue equation (eq. (6.3) in ref. [1]). For q = 0 the solutions are simply d~kp(O) = N-f/2-exp(ik,Rp), where Rp denotes a lattice vector, while the corresponding eigenvalues are given by b~p(0) = U(1 - 6po ). Since it is quite obvious that the q~kp(O) constitute a complete orthonormal set, we may concentrate on q 4 : 0 in the following. F o r q ~ O, the pair functions related to the Hubbard model are given by [ 1 ]

dPkp(q) = (U/N)[Cok(q) --Ep(q)] -INp(q), (1 b)

where the sum over p runs over all pair states (bound

where Cok(q) denote the particle-hole ( p - h ) energies and where

Np(q) = ~ (Pkp(q) " 1 Permanent address: Max-Planck-lnstitut for Strahlenchemie, D-4330 Miilheim an der Ruhr 1, Fed. Rep. Germany. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2a)

k

(2b)

The pair energies for q ~ 0 are determined by the zeros 393

Volume 102A, number 9

PHYSICS LETTERS

of the characteristic function [ 1 ] ~',q)

= 1 +U~

[ E _ cog(q)l_ 1

(3)

where the sum extends over the first Brillouin zone. As mentioned above, the explicit forms of the functions cog(q), Ep(q), and Np(q) are irrelevant for our proof so they will not be given here (the reader is referred to ref. [1] for the details). There is a strong similarity between eqs. (2) and (3) on the one hand and analogous equations resulting from RPA treatments [4] o f many-electron systems on the other hand. This is no accident, as has already been shown by Wentzel [5]. in fact, a comparison shows that the RPA treatment [6] o f the Hubbard hamiltonian yields results quite similar to those obtained in ref. [ 1]. Therefore, because of the close relationship between the two methods, the pattern of our proof also applies to RPA wave functions. In order to prove that the pair functions satisfy eqs. (1) it will be necessary to assume that the following inequalities hold:

cog(q) --/=cok'(q) Ep(q)4:Ep,(q)

for for

k =/=k ' ,

(4a)

p--/=p',

(4b)

(all q 4= 0). We now see more clearly why the case q = 0 has been treated separately. The reason is that for q = 0 the p - h energies reduce to a constant and, hence, eq.(4a) could never be satisfied. Next it will be shown that eq. (4a) implies (4b). The latter clearly holds, if p refers to the bound state and p' to one of the continuum states and vice versa. If both p and p' refer to continuum states, eq. (4b) also holds because of (4a) and the easily proven fact that D(E, q) possesses exactly one zero Ep(q) between two successive poles cog(q). Hence, eq. (4b) holds provided (4a) can be shown to be valid. Obviously the latter is generally valid for single-particle energies o f the form eg ~ k 2, as is the case in the electron gas. It also holds for tight-binding models provided the lattice is non-alternant as, e.g., the fcc lattice. In alternant lattices [7], however, eq. (4a) is not true in general. Due to the "perfect nesting" property o f these lattices it happens here that cog(q) = cok'(q) for k 4= k'. This difficulty can, however, be easily overcome by constructing [7] a smaller Brillouin zone. In order to avoid inessential complications of the proofs, alternant lattices (in connection with tight394

11 June 1984

binding models) will henceforth be excluded from our considerations. With some minor modifications, however, the proofs given below can be extended to alternant lattices as well. After these preliminaries the proofs are now straightforward. To show orthonormality, consider first the case p 4: p'. From eqs.(l a) and (2a) we obtain

fpp, =(U/N)2NpNp,*

~k (cog - e p ) l ( o o k - E p , ) -1

= (U/N)2N*pNp,(Lp - l'.'p,)- 1 X ~

[(cog -

Ep)'-1 __ (COg _ Ep,)- 11 .

(5)

k

where use has been made of eq. (4b) and where, for simplicity, the parametric dependence on the total wave vector q has been omitted. Because of D(Ep) = 0 for all p, where D(E) is given by eq. (3), we have the additional relation

NU•

(cog

Ep)- I = 1

(allp).

(6)

Eqs. (5) and (6) now immediately show that the pair functions are orthogonal: fpp, = 0 for p :~p'. For p = p' eqs. (la) and (2a) yield

fpp = (U/N) 2 [Np I2 ~k (cog - Ep)- 2

= --(U/N)[Np [20D(E3/OI',"[E=Ep .

(7)

We now observe thatNp remains undetermined by eqs. (2a) and (2b). We use this freedom to satisfyfpp = 1. This choice [ 1 ] fixes Np up to a phase factor and we obtain:

INp h-2 = -( U[N)DD(E)/aE bE=Ep •

(8)

This completes the proof of orthonormality, In order to prove completeness we need the inverse o f the characteristic function D(E). Form eq. (3) we infer that D(E) is meromorphic with only simple poles and zeros at E = cog and E = Ep, respectively. Moreover, D(E) -+ 1 for E -+ ~, whence it follows that D(E) can also be written as

D(E) = l-I (E - Ev)/ ~ (l:: - cog ) . p

(9)

Volume 102A, number 9

PHYSICS LETTERS

Form the properties o f D ( E ) ii follows that D - I ( E ) must also be meromorphic with simple poles and zeros at E = Ep and E = cok, respectively, and the same limiting behaviour as D(E). The most general function satisfying these requirements has the following form: D-I(E) = I +

~ c p ( E - E p ) -1 ,

(10)

P where Cp is the residue o f D - l ( E ) at E = Ep. By using standard techniques, the Cp can be evaluated from the inverse o f e q . (9). The result is

,

H (cok -eok,)/ [I (¢ok -Ep). k'

(17)

p

(k',k)

Consider now the residue of D(E) a t E = cok. From eq. (3) we obtain directly: Res(6ok) =

U/N.

(18)

Alternatively, Res(~ok) can also be calculated from eq. (9). A simple calculation yields the following:

(19) P

(11)

/ (k'~g)

On comparing eqs. (17) and (19) we find

or, by using eq. (8),

Cp = -(U/N)INp

OD- 1(E)/DE IE=,~ k

Res(ook) = U/N

CP = ~k (EP - C°k) / ~p' (Ep - Ep,) (p'~p) = 1 / [ a D ( E ) / a E ] E=Ep

11 June 1984

i2 .

(12)

OD-I(E)/OEIE=,,ok =

l/Res(cok)=N/U.

(20)

Hence, D - I(E) is given by D - I ( E ) = I - NU ~pI N p I 2 ( E - E p ) - I

(13)

To prove eq. (lb), consider first the case k ~k'. From eqs. (lb) and (2a) we obtain

gkk' = (U/N)2(C°k' - C°k)- 1 X ~ INp 12l(~,

- E p ) - I - (,~,. - E p ) - I ] ,

p

(14) where use has been made o f e q . (4a). Since D - l ( c o k ) = 0 for all k, eq. (13) yields the additional relation

US Np

INp 12(~k

-

Ep) -I

= 1

(allk).

(15)

References

The last two equations show that gkk' = 0 for k 4: k'. For k = k', eqs. (lb) and (2a) yield

g** = (V/N)2 ~ , INp

p

12(~, - ~.))-2

= ( U / N ) O D - 1(E)/OE IE=,~ k '

From eqs. (16) and (20) we now easily deduce the closure relation gkk = 1 for all k. This completes the proof of eq. (lb). In summary, a rather general proof has been given of orthonormality and completeness of pair wave functions. Originally, the need for such a proof arose in connection with the pair-theory method [ 1,2] applied to the Hubbard model. In the form presented here, however, the pattern of the proof applies to a much broader class of pair functions than merely those related to the Hubbard model. In particular, it applies to RPA wave functions because o f the close relationship [5] between RPA and the pair-theory method.

(16)

where the last equality follows from eq. (13). By using again the inverse o f eq. (9), the r.h.s, of (16) can be rewritten as follows:

[1] [2] [3] [4]

H. Barentzen, Phys. Rev. B28 (1983) 4143. M. Gimrdeau, J. Math. Phys. 4 (1963) 1096. R. Brout, Phys. Rev. 108 (1957) 515. D. Pines, The Many-body problem (Benjamin, New York, 1961). [5] G. Wentzel, Phys. Rev. 108 (1957) 1593. [6] A. Blandin, in: Theory of condensed matter (IAEA, Vienna, 1968) p. 691. [7] J. Des Cloizeaux, J. Phys. Radium (Paris) 20 (1959) 606; E.G. Larson and W.R. Thorson, J. Chem. Phys. 45 (1966) 1539.

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