Self-energy of the bogoliubov quasi particle

ANNALS

OF PHYSICS:

28,

Self-Energy

400-429

(1964)

of the

Bogoliubov

DONALD Department

of Physics,

Ohio

H. State

Quasi

Particle

KOBE University,

Columbus,

Ohio

The self-energy of the Bogoiiubov quasi particle, a linear combination of a particle and a hole which is used in superconductivity and nuclear superfluidity, is obtained by using the Green’s functions or propagators of field theory. The exact equation for the single quasi-particle propagator is an integral equation involving multiquasi-particle propagators. The equations for the multiquasiparticle propagators are approximated by assuming that one of the quasi particles entering the higher order propagators is uncorrelated with the others. These approximate propagators are then substituted into the equation for the single quasi-particle propagator. The single quasi-particle propagator is then composed of a Hartree-Fock type term, and terms representing exciton virtual emission and absorption processes. An exeiton here is a collective excitation composed of a bound pair of quasi particles. The Dyson equation is used to obtain the self-energy. A simplified expression is obtained if the pairing interaction approximation is made. The exciton propagators are obtained by solving a pair of coupled integral equations. I. INTRODUCTION

The phenomena of superconductivity was first explained by using a ground state wave function containing correlations between particles with equal ‘and opposite momentum and spin, and minimizing the ground state energy (1):The theory was reformulated by Bogoliubov (2) and independently by Valatin (S), making use of the idea of quasi particles first introduced in the theory of superfluidity by the former (4). The Bogoliubov quasi particles (QP)’ were defined in such a way that the correlated ground state was the quasi-particle vacuum. Therefore it was shown that a quasi particle was a linear combination of a particle and a hole with equal and opposite momentum and spin2 i.e., the quasi particle had a probability of being either a particle or a hole. The energy spectrum of the quasi particles had an energy gap between the ground state and the first excited state for a system with an even number of particles, which was needed to explain superconductivity. Since an energy gap was also observed in the 1 Since there are many types of quasi particles Bogoliubon was suggested by R. Brout (unpublished). quasi particle. 2 For a good review of the canonical transformation 400

(e.g.

the polaron, Henceforth, method

roton, QP will

see ref.

5.

etc.), the name be used here for

QUASI-PARTICLE

401

SELF-ENERGY

spectrum of even-even nuclei (6), the ideas were also applied with great success to nuclear physics (5--8). In the original treatment of superconductivity (1) and superfluidity in nuclei (5), the interactions between the Bogoliubov quasi particles were neglected. However, it was soon realized that the interactions between the QP caused collective excitations (9) which were needed to describe the Meissner effect in a gauge invariant manner (10, 11). In nuclear physics the collective excitations explained on a microscopic level the collective states considered on the macroscopic level as vibrational states of the nucleus (12, 13). The interactions between the QP results in collective states which are bound (or highly correlated) pairs of QP which Schrieffer has called excitons (l/t). There are presumably other collective states corresponding, for example, to bound quadruplets of QP. It was pointed out by Schrieffer (15) that the collective excitations would have an effect in modifying the energies of the single QP. The energy of a free QP is modified by exciton virtual emission and absorption processes. Keglectjng vacuum fluctuat,ion diagrams in which four QP can be simultaneously created or destroyed, Schrieffer calculated the effect of the virtual absorption and emission of an exciton on the single QP energy. However, the assumption that vacuum fluctuation diagrams can be neglected is questionable3 since their inclusion causes a qualitatively different behavior in the exciton propagators (I?‘). The lowest energy exciton with zero momentum always has zero energy if the vacuum fluctuation diagrams are taken into consideration. The two QF’ Green’s function has only poles on the positive real axis if the vacuum fluctuation diagrams are not considered. The inclusion of the vacuum fluctuation diagrams resu1t.s in the two QP Green’s function having poles on both the positiveand negative real axis, as is expected from the exact spectral representation (18). In this paper the approximate self-energy of a QP will be derived from the exact equation for the QP propagator. The approximation made will be the assumption that in multi-&P propagators, one of the QP passes through with only self-interactions, i.e. is uncorrelated with the others. The Dyson equation is then used to obtain the self-energy of the QP. The resulting expression includes many virtual exciton emission and absorption processes. The self-energy includes the effects of 1. a Hartree-Fock type term, 2. the virtual absorption and emission of an exciton, 3. the virtual emission and absorption of an exciton, 4. the virtual emission of two excitons and their annihilation, 3 In ref. 16 footnote fluctuation diagrams

4 Sawicki quotes A. Bohr and B. Mottelson as saying are negligible only for very special interactions.

that

the vacuum

KOBE

402

5. the creation of two virtual excitons and their absorption, and 6. the creation of an exciton and two QP and the annihilation of the exciton and two QP. In Section II, A the Hamiltonian for the system is given and the canonical transformation is made. Then in Section II, B the QP Green’s functions or propagators are defined, and in Section II, C the exact integral equation for the single QP propagator is given. In Section III approximations are made to obtain the approximate single QP propagator. By means of the Dyson equation in Section IV the self-energy of the QP is obtained and its effect on the free QP energy spectrum is discussed in the pairing interaction approximation. The exciton propagators are obtained in Section V by solving a set of coupled integral equations. Finally, the conclusion contains some remarks on the application of the equations. II.

THE

EXACT

SINGLE

QUASI-PARTICLE

PROPAGATOR

In this section the exact expression for the single QP propagator will be given. In order to obtain this expression, it is first necessary to define a QP and to write the Hamiltonian in terms of the QP creation and annihilation operators. Then the QP Green’s functions or propagators will be defined, and the method used to obtain the equations of motion generated by the Hamiltonian will be reviewed. A. THE

HAMILTONIAN

The Hamiltonian for a system of fermions interacting with two body forces is H = c

(er - p)u;‘ul - lAc

1

(1 21 V 134) altazta3u4

1234

(2.1)

For a system that is translationally invariant the numbers j = 1, 2, 3, and 4 are (j) = (kj , ui), the linear momentum and the spin (up/down = +/- ) of the particle j, respectively. For a system that is rotationally invariant the numbers (j) = (Jj , mj , Bi) which are the total angular momentum, its projection on the quantization axis, and any other quantum numbers of the particle j. The quantity p in Eq. (2.1) is the chemical potential, and ej is the single particle energy in the state j. The operators u> and aj are the creation and the annihilation operators respectively for a fermion and satisfy the usual fermion anticommutation relations. The potential is positive for attractive interactions and the matrix element of the potential (1 21 V 134) has been antisymmetrized. A canonical transformation to quasi particles can now be made which will have the effect of mixing the most important part of the interaction between particles of equal and opposite momentum into the single QP energy (5). The annihilation operator for a QP is Cfj

=

UjUj

+

V-j&j

(2.2)

QUASI-PARTICLE

403

SELF-ENERGY

where the coefficients are real numbers having the properties Uj Vj =

=

U-j

-V-j

=

Uljl

=

Vljl

(2.3)

uj2 + Vj” = 1 where 1j 1 = 1kj j or Jj . The properties of Eq. (2.3) insure that the creation and annihilation operators for QP, crf, and aj also satisfy fermion anticommutation relations. Equation (2.2) may be used to solve for oj, the particle annihilation operator, in terms of aj’ and aj , the QP creation and annihilation operators. When the expression for oj and its Hermitian conjugate is substituted into the Hamiltonian of Eq. (2.1) and put in normal order, the Hamiltonian becomes (19) H = i+lcz2.4

(2.4)

Hik

where

Hjk = I,2,Fj+k hjk(l,

2,

*-* ,j i- k)m+at+. . . afai+l . . . aj+k

(2.5)

and j, Ic = 0, 1, 2, 3, 4 are subject to the restriction that j + k = 0, 2, or 4. The coefficients hjk of the operators in Eq. (2.5) are given in Appendix A along with some of their properties. From Eq. (2.5) it is clear that the subscript j is the number of creation operators, and the subscript Ic is the number of annihilation operators in the term. The term Ho0 is the ground state energy, HII is the “kinetic” energy of the QP, and the other terms represent interactions between the QP. A fuller discussion is given in Section II of I* and in some other papers (5, 9). The parameters uj and vj have not yet been determined. If the ground state energy is minimized with respect to ui , the parameters can be evaluated. This process is equivalent to setting Hm = 0 which is what Bogoliubov (9) calls “eliminating the dangerous diagrams.” When this procedure is carried through the coefficient in the canonical transformation is Ut = M[l

+

(2.6)

(ej/Ej>]

where Ej = (6: +

AF)1’2

=

hII

(2.7)

is the kinetic energy of the QP. The coefficient Vi2can be determined 4 Reference 19 will be called as some of the results.

I, since

the notation

of this

paper

will

by Eq. (2.3).

be used

here

as well

404

KOBE

The terms cj and Aj are defined in Eq. (A.6) and (A.7) of the Appendix. can be shown that the kinetic energy is independent of the spin. B. QUASI-PARTICLE

It

GREEN'S FUNCTIONS

In treating the interaction between the QP, it is convenient to use the Green’s functions or propagators of field theory, which describe the propagation with subsequent mutual and self interactions of the particles. In a previous paper (19), the general theory of QP Green’s functions was investigated, but it will be briefly reviewed here to establish the basic ideas and notation. The most general many-time causal Green’s function for the QP is defined in momentum and time space as a,,(

1, 2, . . * ) n;n + 1, . . . ) n + m) = Z’(T{LYl . . . c&&CC!+1. . . crt,+,))

(2.8)

where here (j) = (ki , aj , tj) for translationally invariant systems and (j) = (Jj , mj , ti) for rotationally invariant systems where tj is the time associated with particle j. The operator T in Eq. (2.8) time orders the product of creation and annihilation operators in the Heisenberg representation, with a positive (negative) sign for an even (odd) permutation of the order and puts the later times on the left. The expectation value with respect to the exact ground state of the interacting QP is used. The Fourier transform of Eq. (2.8) may be taken to give the Green’s function in momentum and frequency (energy) space G,, where the Fourier transform is defined as

where X-j is the frequency creation and annihilation

h-j

(energy) associated with particle j. The phases of the operators are chosen to have the opposite sign, so +Xj

if

O
-Xj

if

n
= !

(2.10)

The exact spectral representation of the Green’s function is obtained by substituting Eq. (2.8) into Eq. (2.9) and performing the integrals (18). The Green’s function Grimwill be called the “n + m line Green’s function” because of its graphical representation as a box with n lines going in and m lines coming out, as shown in Fig. 3 of I. The equation of motion for Grim is obtained essentially by differentiating c?hmwith respect to time maintaining the antisymmetry properties. The equations of motion are derived in Section III of I for the general case.

C.

THE

EXACT

SINGLE

QUASI-PARTICLE

SELF-ENERGY

QUASI-PARTICLE

PROPAGATOR

405

If the procedure outlined above is applied to the single QP Green’s function 8~ an integral equation can be obtained for the function Gn . This integral equation can also be obtained by letting n = m = 1 in Eq. (3.22) of I. It is convenient to define the single QP propagator as Gl(1) = c’ G11(11’)

(2.11)

where here 1’ = (kl’, aI’, X1’) for translationally invariant systems and x1’ is the frequency (energy) associated with the particle 1’. The prime on the sum represents a sum over all the primed momenta and an integration over the primed frequencies. The equation satisfied by G1is Gl(1) = G’(1)

(a, b)

+ c’ 4&(

1 3 2’ l’)@( l)Gzz( 1’ 2’ 3’ 4’)

Cc)

+ c’

67&( 1 3’ 2’ l’)G’( l)Glz( 1’ 2’ 3’ 4’)

(d)

+ c’

2&(

(e>

+ c’

8?rh:o(1 3’ 2’ l’)G’( l)G& 1’ 2’ 3’ 4’)

1 3’ 2’ l’)G’( l)Gsl( 1’ 2’ 3’ 4’)

(2.12)

(0

The free QP propagator is given by Go(l) = -[(27r)(X,

(2.13)

- El + iO)]-’

The vertex functions hiz , &I, & , IL:O in Eq. (2.12) and some of their properties are given in the Appendix. The graphical representation of Eq. (2.12) is given in Fig. 1, which can be thought of as an “expansion” in terms of the first interaction that the incoming

+--/--=-+

2

2

43 (b)

(0)

k)

(a) FIG.

1.

The exact graphical

(c)

equation

(1)

for the single quasi-particle

propagator

406

KOBE

QP can undergo. The interpretation of the terms is given in Section III of I. The multi-QP Green’s functions occurring in Eq. (2.12) and Fig. 1 are not known, but satisfy similar integral equations themselves. Actually, Eq. (2.12) is just the first equation of an infinite hierarchy of coupled integral equations. This set of equations cannot be solved exactly, but by making approximations it is possible to obtain the single QP propagator in terms of known quantities, as will be shown in the next sections. III.

APPROXIMATE

SINGLE

QUASI-PARTICLE

PROPAGATOR

Equation (2.12) cannot be solved for the single QP propagator G1 because f-’122,GB1, G13and Go4are unknown. These Green’s functions also satisfy coupled integral equations, from which it is possible to obtain approximations. The procedure followed in this section is to neglect certain graphs which cannot be easily treated, and which in most cases correspond to higher order processes. In this section the approximations will be discussed in detail, showing exactly what graphs are neglected. This analysis is worthwhile because a more refined calculation would include some of these neglected graphs. The basic approximation of this section is to assumethat one of the QP in a multi-&P propagator propagates with only self-interactions (20). This approximation is shown in Fig. 2, where the right side is understood to be an antisymmetric sum of terms. This approximation is better than the one used in Fig. 4 of I, in which one of the quasi particles propagates freely, becauseit includes more graphs. However, in I self-energy effects were neglected so the results there would be the same. A. “HARTREE-FOCK”

APPROXIMATION

The term in Eq. (2.12~) and represented graphically in Fig. 1 (c) involves the scattering of the incoming QP with one that has already emerged from the box, after having undergone all possible interactions. Since it is not possible to pair the two QP entering and the two leaving, it is not possible to use the two-time Green’s function Gszthat will be obtained in Section V to evaluate this term (21). Since this term is identical in form to the term for interacting particles, it is reasonable to make a Hartree-Fock type approximation (HFA) on this term (22). In this approximation the two QP entering the box are assumedto interact with themselves, but not with each other. This approximation is shown graphically in Fig. 2 for n = m = 2. Mathematically, this approximation is obtained by “factoring” the two QP propagator in an antisymmetric manner.5 The result is G&12 3 4) = -i[Gn(2

3Kh(~ 4) - Gn(1 3X&1(24)1

(3.1)

5 This approximation corresponds to neglecting correlations between two QP. A more rigorous justification can be made on the basis of the spectral representation of the manytime causal Green’s function given in ref. 18.

QUASI-PARTICLE

407

SELF-ENERGY

(b)

(a) FIG. 2. The

approximation

y2

used

for

the

multiquasi-particle

2/--r

&/r (b)

(0) FIG. 3. The

Hartree-Fock

propagators

Approximation

for

Fig.

1 (c) for

the

quasi

particles

This expression will be used to evaluate the term in Fig. l(c) and Eq. (2.12~). If Eq. (3.1) is substituted into Eq. (2.12~) and simplified, the following expression is obtained (2.12~) = Go(l) c’

- %rihzz(l 1’ 1’ l)G~(l’)Gr(l)

(3.2)

which is shown graphically in Fig. 3. Figure 3 (a) is the same as Fig. 1 (c) and in the Hartree-Fock approximation is equal to the incoming QP scattering with the QP that has already been scattered and has interacted with itself, as shown in Fig. 3 (b). The other scattered QP also interacts with itself and then emerges. Figure 3 (b) includes both the direct and the exchange term. At this point it is convenient to substitute Eq. (3.2) into Eq. (2.12) and neglect the other terms in the equation. The resulting equation can be solved for the single QP propagator in the HFA which is

E; + iO)]-'

(3.3)

E,' = El + x'4htz( 1 1’1’ 1)(&1+.

(3.4)

G’(1) = -[(27r)(X1

-

where

The integral in Eq. (3.2) can be done6if the exact spectral representation of the B The HF term that appears in Eq. the canonical transformation. The term in In the limit as A + 0 this term vanishes. occurred in the order 3’ 2’ 1’ at time t1 , a Eq. (3.2). (This point, was suggested by A. representation is used in the integral, the

(A.6) is Eq. (3.4) Because factor of Kromminga.) advanced

a particle HF term which is due to is a QP HF term, not the particle term. the operators in Eq. (2.12~) originally exp (i~ri) is implicit in the integral in When the exact single QP spectral part contributes nothing and the re-

408

HOBE

single QP propagator is used (18). The single QP propagator dressed in the HFA will be used later in this section.7 In the next subsection the next term in the single QP propagator equation will be approximated. B.

SINGLE

ANNIHILATION

AND

TRIPLE

CREATION

PROPAGATOR

The single annihilation and triple creation propagator G1, corresponding to one incoming QP and three outgoing QP occurs in Fig. 1 (d) for the single QP propagator. In this section an approximation will be used based on the integral equation for G13which involves six line propagators. The exact graphical equation for G13is similar to Fig. 1 except that each box has two more lines emerging from it, and Fig. 1 (b) is multiplied by Go2. The mathematical equation for G13can be obtained from Eq. (3.22) of I and is G13(1234) = f:

c’ 2nrh;,+,(

1 1’ 2’ 3’)G”( l)Gd--n,n+2(3’2’ 1’ 2 3 4) + iGO(l)[Wo@

4) - 814G02(2 3)l

The approximation of Fig. 2 will now be made in Eq. (3.5) and the result is shown in Fig. 4 where the GMterm has been neglected becauseit has a minimum of three vertices, and corresponds to a higher order process. Figure 4 (e) can be combined with Fig. 4 (a) and the equation can be divided through by the coefficient. The result is the renormalization of the entering free QP propagator to give the propagator in the HFA given in Eq. (3.3). The terms in Fig. 4 (c) and (f) are rather intractable and will thus be neglected. Figure 4 (g) is zero in first order and will be neglected in higher orders. Therefore, just the terms in Fig. 4 (b) and (d) will be used for G13with the entering free QP replaced with the HFA dressedQP. The mathematical expression for the approximate single annihilation and triple creation propagator G13can be obtained by neglecting all but the terms involving Gn, and GS3in Eq. (3.5), replacing the free propagator with the HFA dressed propagator, and factoring the Green’s functions G, and GZ3. When this The factor (al,al,) is the expected number of tarded part contributes a factor -i(af,c~~,). QP with momentum ki’ in the exact ground state. It is zero if the BCS ground state is used, so it would be expected to be small. Since the probability of finding a QP with high momentum in the ground state is small, the function (af,czY1,) should cause the sum to converge. To evaluate (af,~l,) it is necessary to sum over all states the residues of the retarded part of the single QP propagator. 7 The Hartree-Fock Approximation (HFA) is just a special case of the h-approximation described in ref. 23. Any excitation with a finite lifetime is called quasi particle there.

QUASI-PARTICLE

409

SELF-ENERGY

(b)

(d)

(e)

+

+

(f) FIG. 4. The approximate graphical propagator G,, after the approximation

(a)

equation for the single annihilation of Fig. 2 has been made.

and triple

creation

is done the result is Gn(12

3 4)

= ax’

(a> 18aiG’( l)Gl(i&( + az’

l%riG’(

1 1’ 2’ iz)God( 1’ 2’ is id)

(b)

l)Gl(i&(

(c)

1 iz 1’ 2’)G22(1’ 2’ ia i,)

(3.6)

where Q. = (l/n!)~p(-l)P 1s ’ the antisymmetrization operator in which P is the permutations of the appropriate numbers and ( - 1)’ is the sign of the permutation. The term Eq. (3.6b) corresponds to Fig. 4 (b) and Eq. (3.6~) corresponds to Fig. 4 (d), with the incoming QP dressed in the HFA. The approximation to G13 can be substituted into Eq. (2.12d) and Fig. 1 (d) to obtain the contribution to the single QP propagator. The result of substituting the approximate G13 into Fig. 1 (d) is shown in Fig. 5. The Green’s function G22 describes the propagation of two QP that can interact in all possible ways, and the Green’s function Go4describes the creation of four QP which can also interact in all possible ways. It will be shown in Section V that these Green’s functions satisfy two coupled integral equations. They both have poles at free pair states and bound pair states. A bound pair of QP is a collective excitation known as an exciton. For this reason the Green’s functions G22and GM will be called exciton propagators. In Fig. 5 (b) the QP absorbs a virtual exciton, propagates with only a scattering interaction with itself, and then absorbs the other virtual exciton which

410

KOBE

FIG.

5.

(aI (b) (cl

The approximate

contribution

to the single quasi-particle

propagator

from

Fig. 1 (d).

was created with the first one. Then the QP interacts with itself in al1 possible ways before emerging. In Fig. 5 (c) the QP absorbs an exciton, propagates with self-interactions, and then emits an exciton. It then interacts with itself before emerging. The mathematical form of Fig. 5 is obtained by substituting Eq. (3.6) into Eq. (2.12d) which gives the contribution to the single QP propagator from this term. The result is (2.12d) = c’

(a) -36r%h&l,

-2’ + 4’, -4’, 1 - 2’) ‘h&2’ + 3’, -3’, 1 - 2’, 1)

G’( l)Gl( 1) /’ G’( 1 + 2’) (b) ‘G&2’ + c’

(3.7)

+ 3’, -3’, 5’ + 4’, -4’)

- 36,r2&1( 1, 2’ + 4‘, -4’, 1 + 2’) .h13(1 + 2’, 1,2’ + 3’, -3’) G’(l)Gl(l)

/‘G’(l

+ 2’) *G&Z’ + 3’, -3’, 5’ + 4’, -4’)

The integral in Eq. (3.7) is just over the primed frequencies. Equation (3.7b) corresponds to Fig. 5 (b) and Eq. (3.7~) corresponds to Fig. 5 (c). The integrals in Eq. (3.7) can be done exactly with the help of the exact spectral representation of the Green’s functions, which will be given in a later subsection. C. TRIPLE

ANNIHILATION

,LND SINGLE

CREATION

PROPAGATOR

The procedure followed in obtaining an approximate contribution from Fig. 1 (e) is similar to the procedure of the last subsection. The exact graphical equation

QUASI-PARTICLE

411

SELF-ENERGY

(b)

(d)

FIG. 6. The exact graphical propagator G3r .

(a)

equation

for the triple annihilation

q+;y:--&qy (a) (b)

FIG.

7. The approximation

(f)

for the triple annihilation

and the single creation

(cl

and the single creation propagator

for the triple annihilation and single creation propagator GS1is obtained from the adjoint of Fig. 2 of I and is shown in Fig. 6. The interpretation of the graphs is similar to that of the last subsection, except now that Fig. 6 is an “expansion” in terms of the last interaction the QP undergoes. The mathematical form of Fig. 6 is obtained from Eq. (3.25) of I. The approximation of Fig. 2 will again be made in Fig. 6 for GS1, The term in Fig. 6 (d) will result in the renormalization of the emerging free QP in the HFA. The terms in Figs. 6 (e) and (f) will be neglected in this approximation. The approximate graphical expression for Gsl is shown in Fig. 7.’ The contribution to the single QP propagator from Fig. 1 (e) can be obtained by substituting Fig. 7 into it, which is shown in Fig. 8. Figure 8 (c) describesthe emission of a virtual exciton by the QP and its absorption, The emissionof two virtual excitons and their annihilation is shown in Fig. 8 (b) . * The emerging QP is assumed to be completely dressed, and the entering QP is dressed only in the HFA. This interchange of the propagators is convenient later, and is valid since essentially the same class of graphs is being summed.

412

KOBE

The mathematical expression which results from this procedure is (2.12e) = c’

(a) - 36?r2ih1& 1, 1 + 2’, -2’ + 4’, -4’) ‘hH(1 + 2’, 1,2’ + 3’, -3’) G’(l)Gl(l)

j’ G’(1 + 2’) b) .G4,,(2’ + 3’, -3’, 5’ + 4’, -4’)

+ c’

(3.8)

- 36?r2&( 1, 1 - 2’, 2’ + 3’, -3’) &(2’ G’(l)Gl(l)

j’G’(l

+ 4’, -4’, 1 - 2’, 1)

- 2’)

cc>

.G22(2’ + 3’, -3’, 5’ + 4’, -4) Equation (3.8b) corresponds to Fig. 8 (b) and Eq. (3.8~) corresponds to Fig. 8 (c). The integrals in Eq. (3.8) will also be done later. D. THE

QUADRUPLE

CREATION

PROPAGATOR

In this subsection an approximation will be obtained for Fig. 1 (f) which contains the quadruple creation propagator. Since three QP annihilate with the incoming QP, it is not possible to pair the emitted QP together. Therefore the two-time Green’s function GM which will be obtained in Section V is not applicable, for the same reason the two-time G2zcould not be used in Fig. 1 (c). The exact graphical equation for GM can be obtained from Fig. 6 by replacing the three entering lines with three emerging lines, and omitting the last term. This equation in mathematical form is obtained from Eq. (3.25) of I. In order to obtain an approximate expression for GM that may be used in Fig. 1 ( f ), the approximation of Fig. 2 will again be made. In this approximation the term involving GM is zero, and the term involving Ga will be neglected. Part

(0) FIG. 8. The contribution

(b)

to the single quasi-particle

(c)

propagator

from Fig. 1 (e)

QUASI-PARTICLE

413

SELF-ESERGY

of the term involving the C&bwill renormalize the emerging QP propagator in the HFA and the other part will be neglected. The G33can be factored and an approximate expression for GO4can be obtained. The approximate contribution to Fig. 1 (f) of the single QP propagator can be obtained by substituting approximat,e GO4in graphical form into it, which gives Fig. 9. This figure shows the incoming QP annihilating with an exciton and another QP. These latter were created along with another QP that interacts with itself before emerging. The mathematical contribution to Eq. (2.12) which results from this procedure is (2.12f) = c’

288?r2iGo( 1 )Gr ( 1)h,o( 1, - 1 - 2’, 2’ + 4’, -4’) .hC4(-1

- 2’, 1,2’ + 3’, -3’)

(3.9)

I G’( - 1 - 2’)G& 2’ + 3’, -3’, Yj’ + 4’, -4’)

I

which is the mathematical form of Fig. 9. The internal QP propagator is in the HFA. The integral in Eq. (3.13) will be done later after the spectral representations have been discussed. E. SPECTRAL

REPRESENTATIONS

The integrals in Eqs. (3.7), (3.8), and (3.9) can be done if the spectral representations of the causal Green’s functions are used. The Green’s functions in the above equations are all many-time causal Green’s functions in frequency space, since each particle has a frequency associatedwith it. In a previous paper (18) the relation between the many-time causal Green’s function and the twotime causal Green’s function was shown in frequency space. The spectral representation of the two-time Green’s function can be obtained by integrating the spectral representation of the many-time Green’s function over all frequencies -

2 0

II-

4

(a)

FIG. 9. The contribution

(b)

to the single quasi-particle

propagator

from Fig. 1 (0

414

KOBE

except one in the following $dX-1;

1 + 2, -23

manner

+ 4, -4) m dXz dXa dh G,,( 1 + 2, -2, 3 + 4, -4) =s -w

(3.10)

where n + m = 4 and (n, m) = (0, 2,4). The function snrn depends only on the frequency X-1 but on all the momenta (j) = (kf , cj) or (Jj , mj). The two-time causal Green’s function $j’nm can be written as a sum of an advanced part $jzm and a retarded part ~2, in the following manner (3.11)

s nm = $gn + &n

The advanced part ~2~ is analytic in the upper X-1 plane and has simple poles in the lower half plane. The retarded part S$, is analytic in the lower X-1 plane and has simple poles in the upper half L1 plane.g It is due to these analyticity properties that the integrals in the above equations can be performed. The spectral representations involve the exact excitation energies of the system. The exact eigenstates 1s) of the total Hamiltonian H of Eq. (2.4) are assumed complete, and 10) is the exact ground state. The excitation energy ws = E, - E. is the energy E, of the exact state 1s) above the energy E. of the exact ground state. In order to write the spectral representations in a simpler form, it is convenient to define the state amplitudes $412)

=

(0

1 QIm

x,(1 2)

=

(0

I altazt

(3.12)

I s>

(3.13)

Is).

l3ecause of the anticommutation relations, the amplitudes in Eqs. (3.12) and (3.13) are antisymmetric in their arguments. The spectral representation of the two-time Green’s function can be written as I;?&

; 1 + 2, --2,3

+ 4, -4) _

*

2 2 c Psnm z- 8

(1 + 2, -2,3 x-1

3= ws f

+ 4, -4)

(3*14)

io

where the upper (lower) index and sign corresponds to the advanced (retarded) part. The function p&,,, is defined for the various four-line Green’s functions as 9 The advanced part of the function in time space contains a O(t) and the retarded part of the function contains a @(- t). The step function ff is zero for negative arguments and unity for positive arguments. Therefore, it is clear why these functions are called the advanced and retarded parts of the causal function.

QUASI-PARTICLE

415

SELF-ENERGY

d-d1

2 3 4)

=

h(l

2)&*(3

4)

(4

&(l

2 3 4)

=

x8(1

2)$,*(3

4)

(b)

dm(1

2 3 4)

= &(l

2)x,*(3

4)

(cl

(3.15)

so the residues of the Green’s functions in Eq. (3.14) can be seen to be proportional to products of state amplitudes. The function ~5%~~is defined for the various four-line Green’s functions as ~$1

2 3 4)

=

o&d1

2 3 4)

= &*(l

&(

1 2 3 4)

xs*(l

2)x*(3

4)

(4

2)x,(3

4)

(b)

= xs*( 1 2)+,(3

4)

Cc)

(3.16)

so the residues of the retarded Green’s functions are also proportional to products of state amplitudes. The spectral representations and their analyticity properties are needed to evaluate the expression for the self-energy of the QP, which is given in the next section. IV.

SELF-ENERGY

OF

THE

QUASI

PARTICLE

In this section the approximate self-energy of the QP will be obtained by using the Dyson equation and the results of the last section.” The effect of the virtual emission and absorption of excitons on the energy of the QP will be discussed in the pairing interaction approximation. A. THE

DYSON

The self-energy

EQUATION

of a particle Gl(1)

is defined by means of the Dyson

= @(l)

equation”

+ G0(1)2sz(1)Gl(l)

(4.1)

where Z ( 1) is the self-energy of the particle (26). The Dyson equation is shown graphically in Fig. 10. If Eq. (2.13) for the free QP propagator is used in Eq. (4.1) and the equation is solved for G1 the equation Gl(1)

= -(2?r)-‘[Xl

- E, + z(l)

+ iO]-’

is obtained which shows why Z( 1) is called the self-energy spectral representation of the Green’s function shows that 10 Some recent calculations of the self-energy 24. I1 The factor 2~ occurs here because of the transform.

for

normal

fermion

(4.2)

of the QP, The the systems

function are

has given

ref.

difference

in the

definition

of the

Fourier

in

416

KOBE

poles at the exact excitation Gl(1) will have singularities

energies (18). From Eq. (4.2) it can be seen that when

--XI + El = X(1, A,)

(4.3)

where (1) is the momentum of the QP. Thus it is only necessary to look for solutions to Eq. (4.3) to obtain the single QP energies, which will be perturbed from their noninteracting energy El . B. SELF-ENERGY

OF THE QUASI

PARTICLE

The approximate self-energy of the QP can be obtained graphically by substituting Figs. 3, 5, 8 and 9 into Fig. 1 and equating it to the Dyson equation in Fig. 10. Then the free and dressed propagators cancel and leave an explicit expression for the self-energy which is given in Fig. 11. The mathematical form of Fig. 11 is obtained by substituting Eqs. (3.2), (3.7), (3.8), and (3.9) into Eq. (2.12) and equating it to the right side of the Dyson equation in Eq. (4.1). The integrals can be done by using the spectral

q--t-

(a)

+-@-,-

(b)

FIG.

-@= (0)

10. The

(cl

Dyson

equation

in graphical

form

L2lb)

+-EL +&k +&3[d)

(a)

2

2

+

kI!ifzF

(f)

(0)

FIG.

11. The

self-energy

of the

quasi

particle

in graphical

form

QUASI-PARTICLE

representations Z( 1) = C’

of Section III,

417

SELF-ENERGY

E. The result for the self-energy

is

4hn*( 1 1’ 1’ 1)(&l’)

+ c’

18&(

(a, b)

1, 2’ + 4’, -4’,

1 + 2’) . h13(1 + 2’, 1,2’ + 3’, -3’)

$$i( - Al + E:+T ; 2’ + 3’, -3’, + c’

18&(1,

-2’

+ 4’, -4’,

2’ + 4’, -4’)

1 - 2’)

. h31(2’ + 3’, -3’,

S&(--X, + c’

+ E:-y ; 2' + 3', -3',

lS&(

-2'

1 - 2’, 1)

+ 4', - 4')

+ c’

(d)

1, 1 - 2’, 2’ + 3’, -3’) . h31(2’ + 4’, -4’,

s;&l

(c>

- E:-p ; 2' + 3', -3',

187&(1,

1 + 2’, -2’

1 - 2’, 1)

2' + 4', -4')

(4

(4.4)

+ 4’, -4’) . h13(1 + 2’, 1,2’ + 3’, -3’)

s&( - XI + E:+Y ; 2’ + 3’, - 3’, -2’ + C’

-1447&rJ(

1, -1

(0

- 2’, 2’ + 4’, -4’) * h04( -1

@‘d -xl

+ 4’, -4’)

- 2: 1, 2’ + 3’, -3’)

- E:+Y ; 2’ + 3’, - 3’, 2’ + 4’, -4’)

k>

The terms in Eq. (4.4) are represented graphically by the corresponding terms terms in Fig. 11. Eq. (4.4b) is the “Hartree-Fock” part of the self-energy, which is independent of the frequency. The remaining terms represent the effect of virtual excition emission and absorption processes on the single QP energy. These terms depend on the frequency, so Eq. (4.3) must in general be solved graphically, or by a process of successive approximation.12 Schrieffer (15) neglected vacuum fluctuation diagrams and only obtained a term similar to Eq. (4.4~). Because he neglected the poles of the exciton propagators, he obtained the causal function instead of the retarded function. Equation (4.4) for the self-energy of the QP is still very complicated, but it can be evaluated using the expressions for the exciton propagators sz2 and so4 which will be derived in Section V. However, in order to simplify the expression, I2 If the energy shift due to the self-energy is small, a good approximation is to replace X in Z(1, X) in Eq. (4.3) with E1 . Then the perturbed energy of the QP is h = El - Z(1, E,) which is an explicit expression for the energy.

418

KOBE

the pairing interaction next subsection. C.

THE

PAIRING

approximation

INTERACTION

can be made, which

will be done in the

APPROXIMATIOX

In the pairing interaction approximation, which has been very useful in the theory of superconductivity (1) and for nuclei (5), it is assumed that only particles with equal and opposite momenta and spin interact with each other for translationally invariant systems, or the same angular momenta but opposite projections for rotationally invariant systems (i.e. only particles with momenta (j) and (-j) interact with each other). This interaction is mathematically convenient, but even though it is important in forming the correlated ground state, there is no a priori reason to think that it would be more important than any other interaction between QP (26). However, for simplicity the pairing interaction approximation will be used to investigate its consequences. The expression for the self-energy in Eq. (4.4) is much simplified in the pairing interaction,13 because the expression vanishes unless the momenta (2’) and (4’) are both zero. If the spectral representations of the retarded and advanced Green’s functions in Eq. (3.14) are used, Eq. (4.4) in the pairing interaction approximation may be written in a simpler form14, ” Z( 1) = C’

4h,,( 1 l’l’l)(a:~al~) -

36 c 1A,(l) 8

- 288 c

/'[XI - EI - w, + iO]-'

1Bs( 1) I’[& + El + ws -

(4.5)

iO]-’

where

A,( 1) = c’

hsr( 1, 2’, -2’,

1)x8(2’,

-2’)

(4.6) +

C’h13(1,

2’, -2’,

m42’,

-2’)

and B,(l) The the Eq. the

= C’h0*(

-1,

1, 2’, -2’)&(2’,

-2’)

(4.7)

functions in Eqs. (4.6) and (4.7) can be evaluated much more easily than coupled functions which occur in Eq. (4.4). Equation (4.5) may be used in (4.3) to find the effect of the virtual emission and absorption processes on single QP energy.

i3 See Section V, B for the definition of the pairing interaction approximation. i4 The self-energy in Eq. (4.5) has the same form, i.e. a meromorphic function, that exact self-energy in a normal system should have according to ref. 27. I5 The self-energy in Eq. (4.5) vanishes in the limit of infinite volume, so BCS theory solve the problem exactly in the pairing interaction approximation in this limit.

the will

QUASI-PARTICLE

419

SELF-ENERGY

In Section V, D it will be shown how the magnitude and phases of the state amplitudes C#J~ and x8 appearing in Eqs. (4.6) through (4.8) may be determined. The phase of C&and xp can be evaluated only to within the same additive constant. However, since they occur in Eq. (4.5) multiplied by a complex conjugate function, the constant phase cancels out. Therefore, it is possible to calculate the self-energy from Eq. (4.5) in the pairing interaction approximation. D. EFFECT

OF EXCITONS

ON THE SINGLE

QUASI-PARTICLE

ENERGY

Since we still do not know the state amplitudes and excitation energies that occur in the retarded and advanced exciton propagators in Eq. (4.4) or in Eq. (4.5), it is not yet possible to discuss quantitatively the effect of the virtual emission and absorption of excitons on the single particle spectrum. The next section will be concerned with the evaluation of the exciton propagators. However, it is possible at this point to discuss the qualitative effect of the virtual exciton emission and absorption processes.Equation (4.5) for the self-energy in the pairing interaction approximation can be plotted qualitatively as shown in Fig. 12. The function --X1 + Ei can also be plotted in Fig. 12, and the intersections of the straight line with 2( 1) are the solutions to Eq. (4.3). For very weak interactions the self-energy function will coincide with the real axis, except for spikes at the asymptotes. In this case the energy shift due to the virtual emission and absorption processeswill be essentially zero. As the interaction is increased, the self-energy will appear as shown by the solid lines in Fig. 12. The asymptotes are at exciton energiesshifted by & . The only solu-

FIG. 12. The graphical solution of Eq. (4.3) to obtain the single-quasi-particle energy. Solid circle indicates perturbed quasi-particle energy and El denotes the unperturbed energy. For strong coupling shown by the dotted line the energy shift is large.

420

KOBE

tion which goes smoothly into the unperturbed energy as the interaction is decreased is shown by the solid circle. The other solutions correspond to the energy of an interacting &I’ and exciton. If the self-energy is complex, there would be a real and imaginary part to Eq. (4.3) and Fig. 12 would correspond to the real part. The imaginary part would be given by a similar figure. As the interaction is increased still further, the self-energy could become as shown by the dotted line in Fig. 12. Thus for strong coupling the decrease in the single QP energy would be larger than for weak coupling, as is expected. In the next section it will be shown how the advanced and the retarded exciton propagators may be obtained, so that the quantitative effects of virtual exciton emission and absorption processes may be calculated. V. EXCITON

PROPAGATORS

In this section the exciton propagators $22and So4needed to evaluate the selfenergy will be determined. The poles of these two-time Green’s functions give, in addition to the energies of two noninteracting QP, the energy of bound pairs of QP, which have been called excitons (14, 28). Therefore the Green’s functions sz2 and so4 will be called exciton propagators. Unfortunately the retarded and advanced functions needed for the self-energy in Eq. (4.4) cannot be calculated directly, since the equations of motion for these functions as defined here takes us out of the function space of retarded and advanced functions.‘6 However, the equations of motion for the causal functions remain within the function space of causal functions, since only causal functions occur in Eq. (3.22) of I. If the causal exciton propagators are known the excitation energies and the state amplitudes needed to construct the retarded and advanced functions in Eq. (3.14) can be obtained. Unfortunately, the excitation energies and state amplitudes cannot be obtained analytically and resort must be made to numerical methods. A. COUPLED

INTEGRAL

EQUATIONS

FOR THE EXCITON

PROPAGATORS

In I a set of two coupled integral equations for the Green’s functions Gzz and GO4was obtained by considering one of the QP in the exact equations to pass through without any interaction. The self-energy effects were neglected. The derivation is given in Section V of I.17A somewhat more general set of equations is obtained by replacing the product of the two free QP operators in Eqs. 16Other advanced and retarded functions which are averages of (anti) commutators can be defined which have equations that remain in the space of retarded and advanced functions (29). 17Another set of equations for studying the two particle bound state for nonideal fermi systems has been given by 1’. V. Tolmachev (SO). He includes the effect of the two particle bound state on the single particle Green’s function.

QUASI-PARTICLE

=

22-r (0)

421

SELF-ENERGY

72

+

0

4

+ i

(b)

FIG.

13. The

two

integral equations

coupled

for the

exciton propagators

(5.9) and (5.10) of I with the product of two dressed propagators. The graphical form of these equations is given in Fig. 13. If a change in variables is also made in Eqs. (5.9) and (5.10) of I an integration and a summation over a delta function can be performed. The new set of coupled integral equations for the exciton propagators is” G&l

+ 2, -2,

3 + 4, -4)

= ~‘4?rihn(l

+ 2, -2,

-2’,

1 + 2’)G1( 1 + 2)G1( -2) .G22(1 + 2’, -2’,

+ x’24?rih,o(l

+ 2, -2,

-2’,

-1

*G&l + &I&4

- L,s+Gi(l

3 + 4, -4)

+ 2’)G1(1

+ 2)G1( -2)

+ 2’, -2’,

3 + 4, - 4)

(5.1)

+ 2)G(--2)

and Ga( -1

+ 2, -2,

= c’

3 + 4, -4)

47r&z( - 1 + 2’, -2’,

-2,

- 1 + 2)G1( -1 -Gw( -1

+ c’24ti&d(

-1

+ 2, -2,

-2’,

+ 2)G1( -2)

+ 2’, -2’,

1 + 2’)G1( -1 .G22(1 + 2’, -2’,

3 + 4, -4)

(5.2)

+ 2)G1( -2) 3 + 4, -4)

18 This set of equations was also considered in momentum and time space by V. M. Galitskii (Sf). He converted his equations directly to the method of approximate second quantization of ref. 9. Reference 32 gives a detailed conversion of these equations to the method of approximate second quantization.

422

KOBE

Equations ments. B.

and (5.2) are antisymmetric

(5.1)

SOLUTION

OF

TIIE

in the first two and last two argu-

EQUTIONR

In order to solve Eqs. (5.1) and (.5.2 i some assumptions must be made about the form of the interaction. If the vertex functions are sums of separable functions, then the integral equations can be solved exactly. Since the pairing interaction approximation is of such importance in the theory of superconductivity and for nuclei, a generalization of this approximation will be used. The matrix elements of the potential will be assumed to be separable (1 + 2, -21 V /-2’,

1 + 2’) = g(l + 2, -2)g(l

where g = 0 if 1 1 + 2 1, ( -2 1 > K, some cutoff g are real and have the following properties d12)

= g(-1,

g(12)

= -g(2

+ 2’, -2’)

momentum.

(5.3)

The functions

-2)

(5.4)

1)

Belyaev (5) and Kisslinger and Sorenson (7) have assumed that for the short range force all matrix elements of the potential are zero, except the one in Eq. (5.3) for kl = 0, which are assumed constant. The type of matrix element given in Eq. (5.3) was used in I along with a Coulomb interaction which will not be considered here. The vertex functions in Eqs. (5.1) and (5.2) can be obtained from Eqs. (A.3) and (A.4) and areI 4h22( 1 + 2, -2, = -U(l

-2’,

1 + 2’)

+ 2, -2)U(l

+ 2’, -2’)

-

V(1 + 2, -2)V(l

+ 2’, -2’)

(Ti.5)

and 24hh,,(l + 2, -2,

-2’7

-1

+ 2’) = -U(l -

+ 2, -2)V(-1

V(1 + 2, -2)U(-1

+ 2’, - 2’) + 2’, -2’)

(5.6)

where U( 1 + 2, -2)

= g(l + 2, -2)Ul+zU-z

(5.7)

19 In ref. 16 Schrieffer considers the whole vertex function h2 for the T-matrix to be separable, As we have seen, it is not the ‘P-matrix that enters into the self-energy, but the Green’s function. Thus, free propagation of two QP is also included. He also assumes the vertex function hro to be zero, which is neglecting vacuum fluctuation diagrams. It is more reasonable to assume that the matrix element of the potential is separable, so the vertex functions are sums of two separable terms of the same order. In this way the vertex function may be expressed directly in terms of the coupling constant and the coefficients in the canonical transformation. The coupling constants of other workers, e.g., ref. 7, may then be used.

QUASI-PARTICLE

423

SELF-ENERGY

and v(l,+

2, -2)

= g(1 + 2, -2h+22)-2

(5.8)

The other terms in the vertex functions are not separable and are zero in the pairing interaction approximation, except for some forward scattering terms. These types of terms will be neglected here, although the forward scattering terms could be treated exactly and the others treated by perturbation theory. The other vertex function ho4 can be obtained from Eq. (5.6) by means of Eq. (A.1). If Eqs. (5.5) and (5.6) are substituted into Eqs. (5.1) and (5.2), a set of two equations is obtained which can be converted into a set of four linear algebraic equations by the method used in Section VI of I. These four linear algebraic equations can be solved and the results substituted into the original integral equations to give Gzz(l + 2, -2,

3 + 4, -4) = 27rG1( 1 + 2)G1( -2)G1(3 {U(l

+ 2, -2)02(l)

+ 4)G1( -4)6&(

1) (5.9)

+ V(1 + 2, -2)01(l)] + i61&~ - 8--2,3+41Gl(l + 2)G1( -2)

and God -1

+ 2, -2,

3 + 4, -4)

= 27rG1( - 1 + 2)G1( -2)G1(3 fU(-1 The D-functions

+ 2, -2)D,(l)

are determinants D(1)

+ V(-1

I

F,,

1)

(5.10)

+ 2, -2)D,(l)j

given by

1 + Fit, + F,

=

+ 4)G1( -4)8&(

-I- Fit,

Dl(l)

1 1 + Kiu + F, = / F- + F+ WV z(1L

LA(l)

=

F:v + Ku

(5.11)

1 + F;fv + FL U(1 + 4, -4)

(5.12)

V(1 + 4, -4)

and

The F-functions Fit, = C’tiU(fl

UC.1 + 4, -4)

Fit, + Ku

V(1 + 4, -4)

1 + FL + F,,

(5.13)

are defined as + 2’, -2’)V(fl

+ 2’, -2)Gl(fl

+ 2’)GI(-2’)

(5.14)

424

KOBE

and the other functions can be obtained by replacing U with T’, or 17 with U. Equation (5.14) has a sum over the momentum and an integration over the frequency. If the dressed QP propagator in Eq. (4.2) is used in Eq. (5.14) the integral over the frequency cannot be easily done because the residue theorem cannot be used since the self-energy is a function of the frequency. If, however, the propagators are evaluated in the HFA the integral can be done because the Hartree-Fock propagator in Eq. (3.3) has a simple pole. The integral in Eq. (5.14) can then be done using the residue theorem, and is

m 7l-i s &‘G’( -03 In the case where (6.5) of I.

1 + 2’)G’(

-2’)

the bare propagators

= S[X~ - E:+zp - E:, + io]” are used, Eq. (5.15)

reduces

(5.15) to Eq.

C. EXCITATION ENERGIES From the spectral representations of the Green’s functions in Eq. (3.14) it can be seen that the exact Green’s function has poles at the excitation energies of the system. Since approximations have been made, Eqs. (5.9) and (5.10) do not have the simple form of the exact spectral representations. However, the singularities of the approximate Green’s functions should correspond to the excitation energies. When Eqs. (5.9) and (5.10) are integrated with respect to XZ, XB , and X4, it can be seen from Eq. (5.15) that the resulting function will have singularities when the equation

XI - E:+r - E;, = 0

(5.16)

is satisfied. This singularity corresponds to the energy of two HFA QP propagating along at the same time. The Green’s function G22 and Go4 in Eqs. (5.9) and (5.10) also have singularities when D(1)

= 0

(5.17)

which corresponds to the exciton energies. It is for this reason that the Green’s functions GZZ and GM are called exciton propagators. Equation (5.17) was investigated in Section VI of I for certain special cases. It was shown there that D(0, 0) = 0 so that a zero momentum exciton state exists with zero excitation energy. According to Baranger, this state is a spurious state and should be neglected (8). Equation (5.17) can be solved graphically for the exciton energies. A general plot of D as a function of X in the pairing interaction approximation is given in Fig. 10 of I from which the exciton excitation energies can be determined. If the excitation energiesare known it is only necessary to calculate the

QUASl-PARTICLE

state amplitudes, and then the retarded and advanced for the self-energy can be determined. D.

STATE

425

SELF-ENERGY

Green’s functions

needed

AMPLITUDES

As can be seen from the exact spectral representations in Eq. (3.14) the residues of the functions szz , ~04 , and 640 are products of state amplitudes. The residues can be determined from the functions C&z and Go4 obtained in Eqs. (5.9) and (5.10) by first integrating them as in Eq. (3.10) to obtain the twotime exciton propagators ~~2 and so4 . The residues of $&,, for the excitation energy w, are from Eq. (3.14) PL(l

+ 2, -2,3

+ 4, -4)

= f ; hliy (x =F W8f - *

iO)$&i;

1 + 2, -29 3 + 4, -4)

(5.18)

Therefore, the products of state amplitudes given in Eqs. (3.15ab) and (3.16ab ) can be obtained from Eqs. (5.9) and (5.10). The product of state amplitudes for ~20 and ~2~0can be obtained by taking the complex conjugate of Eqs. (3.15b) and (3.16b), respectively, and changing variables. Equation (5.18) is satisfactory to use to evaluate the residues for the free states but not for the exciton states, since the result would be indeterminate. For the exciton states it is necessary to use 1’Hopital’s rule, so Eq. (5.18) becomes pL(f

+ 2, -2,3

+ 4, -4)

(5.19) f kl;t D’(l j-lN,,(h; 1 + 2, -2,3 + 4, -4) -t I where N,, = D$j’,,,, and D’( 1) is the derivative of D( 1) with respect to x1 . These residues may be used in Eq. (3.14) to construct the retarded and the advanced Green’s functions from the approximate expression for the causal Green’s functions. The retarded and advanced Green’s functions may be used in Eq. (4.4) for the self-energy. However, in the pairing interaction approximation it is more convenient to use Eq. (4.5) and the associated Eqs. (4.6) and (4.7) to evaluate the selfenergy. In this form the amplitudes are uncoupled from each other, so the magnitude and phases of the amplitudes & and x8 must be known. The squares of the magnitude of the state amplitudes are thus obtained from Eq. (5.18), Eq. (3.15a), and Eq. (3.16a) and are =-f

I &Cl + 2, -41” =- i:iF(X - *

- w.p + iOjl;zdX;

1 + 2, -2,

1 + 2, -2)

(5.20)

426

KOBE

and I X8(1 + 2, -4”

= -;

p

09

(x + W8- iO)S22(X; 1 + 2, -2, 1 + 2, -2)

(5.21)

For Eqs. (4.6) and (4.7) the momentum (1) must be set equal to zero in the above equations. To complete the determination of the functions CJ*and x8 it is necessary to determine their phases. It is not possible to determine the exact value of the phase, but only to within the same additive constant. However, from Eq. (4.5) the functions always occur as absolute values squared, so the additive constant will cancel out in the expression for the self-energy. If the argument of the function in Eq. (3.15a) is taken for the special case argb(l

+ 2, -2)4,*(0,0>

= arg 4,(1 + 2, -2)

- arg&(O, 0)

(5.22)

the argument of A to within the additive constant arg &(O, 0) is obtained. If the arguments of the functions in Eqs. (3.15b) and (3.16a) are subtracted for the special case arg xd0,0)08*(0,

0) - arg xs*(l + 2, ---2)x40, 0) = arg

x8(1

+

2, -2)

(5.23) - arg &CO, 0)

the argument of xs is obtained to within the same additive constant. Eq. (5.19) may be used to obtain these arguments. In this section it has been shown how the causal exciton propagators can be obtained by solving a pair of coupled integral equations. In the last two subsections it was shown how the retarded and advanced exciton propagators which are need-d in the QP self-energy expression in Eq. (4.4) could be constructed by finding the excitation energies and state amplitudes which could then be used in the exact spectral representations of Eq. (3.14). Thus, this section gives a procedure for completely determining the QP self-energy in this approximation. VI.

CONCLUSION

In this paper the self-energy of the Bogoliubov quasi particle was obtained by making approximations on the exact integral equation for the QP propagator. The QP self-energy has a Hartree-Fock term and terms that correspond to virtual exciton emission and absorption processes.The effect of the QP selfenergy would appear in the modification of the energy gap in a superconductor (1, 9, 14). However, the phonon field must be included in a refined calculation, since it is only eliminated to second order (3s).

QUASI-PARTICLE

427

SELF-ENERGY

The effect of the QP self-energy would be most easily observed for energy levels in nuclei. Some preliminary calculations using the single particle energies, coupling constants, energy gaps, and chemical potentials of Kisslinger and Sorenson (7) indicate that in lowest order the virtual exchange of a pair of QP has a small contribution.” Thus, it would also be expected that the effect of virtual exciton processes would be small. unless the energies of the excitons were small. Presumably, the self-energy effects should result in better agreement with experiment. A more realistic potential could be used, and the angular momentum of the pair of QP coupled to different values. The inclusion of the long range quadrupole interaction should also give better agreement with experiment. The work here provides the framework in which these questions can be answered. The possibility of applying these ideas to liquid helium is also open. APPENDIX.

THE

COEFFICIENTS

IN

THE

HAMILTONIAN

For the sake of completeness the properties of the functions hjk in Eq. (2.5) willbegivenhere(19).Thefunctionhik(1,2,3, . . . ,jl,j;j+ 1, . . . ,j+k) is antisymmetric in the first j arguments, and antisymmetric in the last k arguments. Since the Hamiltonian is Hermitian, the functions hjk have the property hj*k(l, 2, . . . , j + k) = hkj(j + k, j + k -

1, . . . , 2, 1)

(A.11

The function h’ik is just the function hik multiplied by 6(X, + X2 + . . . + xi xj+l - . . - Xj+k) where X, corresponds to the frequency associated with particle n = 1, 2, . . . , j + k. The functions hjk can be written explicitly in terms of the coefficients in the canonical transformation and the matrix elements of the potential. They are ho0 = c

1

[(el - P) + %q +

hzz (1 2 3 4) = -Ml

4/4

c

12

(1 -

(1 21 V 11 2)7~2%1 1 1 v

21 V 13 ~)(UCUZU~U~ + WW~) + (1 -31 v (4 -2)(u1wu4 - (2 -31 v 14 -

h&l

2 3 4) = ?&

(A.21

12 - 2)ululuzvz

[-

(1 21 V I-3

+ (1 31 v l-2

+ zhu2u3214)

(A.3)

l)(wzu3~4+ wMwJ4)1 -4)(u1u2w4

+ twwu~)

-4)(u1uzu3v~ + u1u2u3u4)

+ (3 21 v I - 1 -44)(u1w3u4 20 This remark is in agreement with that of B. L. Birbriar, who were neglected in his work because they are small (cf. ref. IS).

(-4.4)

+ 2.‘1uzu32)4)1 says that

these

diagrams

428

KOBE

h31(1 2 3 4) = 46

I (121 V I-3 4)(UIW&

+ W2%zk)

+ (3 11 v 1-2 4)( ulv2u3u4

+

hu2v3u4)

+ (2 31 v I-1

+

‘%u2v3v4)]

4>( h%u3u4

(A.5)

The other coefficients can be obtained by using Eq. (A.l). The quantities e1and A, which appear in Eqs. (2.6) and (2.7) are defined as 61 = (el - P) - F (1 21V 12 l>v,”

(A.61

and Al = xc

2

(1 ---II

v I-2 2)wJz

(A.7)

The quantity Ai which is usually called the energy gap because of Eq. (2.7) is usually assumed to be independent of the momentum. ACKNOWLEDGMENTS

I would like to thank Professor R. L. Mills, Professor C. P. Yang, and Mr. R. Prabhu for many stimulating and informative discussions relating to this work, and to the rest of my colleagues in the Department of Physics at the Ohio State University for their interest and encouragement. The final details of the manuscript were completed at the University of Lausanne, and I would like to express my gratitude to Professor G. Wanders for his hospitality during my visit there. RECEIVED:

November

18, 1963 REFERENCES

1. J. BARDEEN, L. ,9. N. N. BOGOLIUBOV,

N.

J. R. SCHRIEFFER, Phys. Rev. 108, 1175 (1957). i Teoret. Fiz. 34, 58 (1958), Transl. Soviet Phys.JETP 7, 41 (1958); Nuovo Cinlento [lo] 7, 794 (1958). J. G. VALATIN, Nuovo Cimento [lo] 7, 843 (1958).

3. 4. N. N. BOGOLIUBOV, 6. S. T. BELYAEV, Kgl.

COOPER,

Zh.

AND

Eksperim.

J. Phys.

6. A. BOHR, B. MOTTELSON, 7. L. S. KISSLINGER AND

USSR 11, 23 (1947). Videnskab. Selskab., Mat. fys. Medd. 31, no. 11 (1959). AND D. PINES, Phys. Rev. 110,936 (1958). R. A. SORENSON, Kgl. Danske Vi’idenskab., Mat. fys. Medd. 32,

Danske

no. 9 (1960). 8. See also V. G. SOLOVIEV, Kgl. Danske Videnskab. Selskab., Mat. fys. Skr. 1, no. 11 (1961); C. J. GALLAGHER, JR., AND V. G. SOLOVIEV, Kgl. Danske Videnskab. Selskab., Mat.fys. Skr. 2, no. 2 (I~~~);M.BARANGER, Phys. Rev. 120,957 (1969);V. G.SOLOVIEV, Zh. Eksperim. i Teoret. Fa’z. 43, 246 (1958), Transl. Soviet Phys.-JETP 16, 176 (1963). 9. N. N. BOGOLIUBOV, V. V. TOLMACHEV, AND D. V. SHIRICOV, “A New Method in the Theory of Superconductivity.” Academy of Sciences Press, Moscow, 1958. Translation: Consultants Bureau, New York, 1959; P. W. ANDERSON, Phys. Rev. 112, 1900 (1958). 10. 11. 12.

Rev. 117, 648 (1960). Phys. Rev. 116,795 (1959). S. T. BELYAEV, Zh. Eksperim. i Teoret. Fiz. 39, 1387 (1966), Transl. Soviet Phys.-JETP 12, 968 (1961); S. T. BELYAEV AND V. G. ZELEVINSKY, Nucl. Phys. 39, 582 (1962). Y.

NAMBU,

G. RICKAYZEN,

Phys.

QUASI-PARTICLE

SELF-ENERGY

429

AND M. G. UBIN, Zh. Eksperim. i Teoret. Fiz. 41, 898 (1961), Transl. Soviet Physics-JETP 14, 641 (1962). 14. J.R. SCHRIEFFER, Physica (Suppl.)26,5124 (1960); A. BARDASISAND J.R. SCHRIEFFER, Phys. Rev. 121, 1050 (1961). 16. J. R. SCHRIEFFER, Nucl. Phys. 36, 363 (1962). 16. J. SAWICKI, Alzn. Phys. (N. Y.) 13, 237 (1961). f7. D. H. KOBE, Ph.D. Thesis, University of Minnesota, 1961 (unpublished). 18. See e.g. D. H. KOBE, Ann. Phys. (N. Y.) 19,448 (1962) where other references are given 13.

D. F. ZARETSKII

to spectral representations. D. H. KOBE AND W. B. CHESTON, Ann. Phys. (N. Y.) 20,279 (1962) (referred to as I). P. C. MARTIN AND J. SCHWINGER, Phys. Rev. 116, 1342 (1959); V. P. GACHOK, Zh. Eksperim. i Teoret. Fiz. 40,879 (1961), Transl. Soviet Phys.-JETP 13,616 (1961). 21. R. PRABHU, private communication (1962). 22. L. P. KADANOFF AND G. BAYM, “Quantum Statistical Mechanics,” Chap. 3 and Appendix, Benjamin, New York, 1962. 23. A. J. LAYZER, Phys. Rev. 129, 897 (1963). 84. A. J. LAYZER, Phys. Rev. 129, 908 (1963); S. V. MALEEV, Zh. Eksperim. i Teoret. Fiz. 43, 1044 (1963), Transl. Soviet Phys.-JETP 16, 738 (1963); M. YA. AMUS'YA, Zh. Eksperim. i Teoret. Fiz. 43, 287 (1962), Transl. Soviet Phys.-JETP 16, 205 (1963); V. M. GALITSKII, Zh. Eksperim. i Teoret. Fiz. 34, 151 (1958), Trand. Soviet Phys.JETP 7, 104 (1958). 2.5. See e.g. D. PINES, “The Many-Body Problem,” p. 52. Benjamin, New York, 1961. 26, R. L. MILLS, private communication (1963). 27. S. V. MALEEV, Zh. Eksperim. i Teoret. Fiz. 41, 1675 (1961), Transl. Soviet Phys.-JETP 14, 1191 (1962); V. L. BONCH-BRUEVICH, Dokl. Akad. Nauk SSSR 147, 1049 (1962), Transl. to be published. 28. See also V. G. VAHS, V. M. GALITSKII, AND A. I. LARKIN, Zh. Eksperim. i Teoret. Fiz. 42,1319 (1962), Transl. Soviet Phys.-JETP 16,914 (1962); K. MAKI AND T. TSUNETO, Progr. Theoret. Phys. Kyoto 28, 163 (1962). 29. N. N. BOGOLIUBOV AND S. V. TYABLIKOV, Dokl. Akad. Nauk SSSR 126,53 (1959), Transl. Soviet Phys.-Doklady 4, 589 (1959). 30. V. V. TOLMACHEV, Dokl. Akad. Nauk SSSR 144, 1015 (1962), Transl. Soviet Phys.Doklady 7, 511 (1962). 31. V. M. GALITSKII, Physica (Suppl.) 26, S174 (1960); Zh. Eksperim. i Teoret. Fiz. 34, 1011 (1958), Transl. Soviet Phys.-JETP ‘7, 698 (1958). 32. D. H. KOBE, Ann. Phys. (N. Y.) 26, 121 (1963). 33. J. R. SCHRIEFFER AND Y. WADA, Bull. Am. Phys. Sot. II 8, 207 (1963). 34. B. L. BIRBRIAR, Zh. Eksperim. i Teoret. Fiz. 42, 479 (1962), Transl. Soviet Phys.-JETP 16, 336 (1962). 19. 20.