Vertex theory of many-fermion systems

ANNALS

OF

PHYSICS:

36, 67-95 (1965)

Vertex

Theory

of Many-Fermion I~RTHUR

Department

of Physics,

Systems

IJAYZER

Stevens Institute of l’echnology, Hoboken, Xew .Jersey

Castle

Point

Station,

A general theory of the weak-response of many-fermion systems is formulated in terms of the longitudinal vertex function, r. By including effects of exchange the vertex theory embeds the conventional dielectric theory, as expressed by the momentum-energy space factorization r = E-IF. Here i; is the proper vertex function and e-1 is the generalized dielectric constant. The integral equation for r is rederived via weak-response theory, then factored into separate equations for F and E- 1. Basic properties of I’, particularly gauge-invariance, are discussed. A nonperturbative approximation, called the v’ approximation, is introduced in which the interaction kernel of the integral equation for ? is replaced by a suitably chosen energy-independent effective interaction v’. The v’ approximation is shown to be a generalization of the time dependent Hartree-Fock approximation. By using the U’ approximation as an input to a more exact theory expressions are obtained for the quasi-particle lifetime and the coupling of the quasi particle to the collective mode. Some new physical consequences of the theory are (a) the momentum dependence of the quasi-particle scattering amplitude in a weak external potential, (b) the existence of an intrinsic longitudinal coupling constant renormalization associated with the static long wavelength limit of i;, and (c) the formal occurrence of a “spurious” pole of f coupled with a zero of eW1 for a sufficiently strong repulsive interactiou v’. The last bears a resemblance to a phenomenon recently noted in elementary-particle theory. A preliminary note on some of this work employing a simplified nongauge-invariant version of the v’ approximation was published earlier. Detailed results on the solution for r in the v’ approximation and in lowest order perturbation theory will be published in a second paper.

We propose here to consider some fundamental aspects of the coupling of a many-fermion system to a weak longit8udinal external potential. By an external potential is meant a given classical potential or the virtual potential due to an interacting nonidentical particle. In spite of t,he assumed weakness of the coupling to the potential, this problem is not a peripheral one. In fact, it goes to the heart of the matter since a weak external potSential can serve as a test probe bringing the

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model system into contact with observational reality. There are also, of course, many processes of intrinsic physical interest fitting this description. For example, when a sufficiently fast particle, identical or not, interacts with a manyfermion system, its coupling can be treated in Born approximation and is therefore effectively weak, even though the basic interparticle interaction may be strong. Previous approximate theories of the weak response of normal many-fermion systems are for the most part encompassed by the dielectric theory in which the response is characterized by the generalized momentum and energy-dependent dielectric constant E.I, 2 Aside from providing a complete account of polarization effects, the dielectric theory is general enough to include the existence of collective modes and their coupling to single-particle excitations. In fact, the collective mode is associated with an isolated pole of c-r as a function of the energy variable, and the square of the coupling constant of t’he collective mode to single-particle excitations is proportional to the residue of c-l at this pole. Enlarging the scope of the dielectric theory somewhat, we include under this heading also the equation of motion for the mixed density matrix or phasespace distribution function in the time-dependent Hartree approximation, or random-phase approximation, from which the dispersion equation for the collective modes may be derived.3v 4 The chief limitation of the dielectric theory is that it is essentially classical in concept. Exchange effects are treated as a secondary modification of what is regarded as a good starting approximation. But we cannot be sure that this is the case without investigating the exchange effects to a consistent approximation. The pure dielectric effects and the exchange effects are component parts of the complete vertex junction, r. The vertex function by definition determines the change 6L in the effective non-Hermitian and energy dependent Hamiltonian, L, for single-particle excitations under the influence of a weak external potential V.5 T,et us consider the matrix element of L between states of momen 1 In the language of Feynman diagrams e-i is determined by the totality of polarization insertions. 2 For a description of the dielectric theory and references to the extensive original literature on this subject, see ref. 1. 3 The phenomelogical theory of the collective mode oscillations was first given in ref, 3. 4 An appropriately formulated derivation of the theory of collective mode oscillations is given in refs. S and 4. These authors take as their starting point the time-dependent Hartree-Fock approximation for the density matrix. 6 The operator equation for the exact single-particle Green’s function, G, reads (ia/ - L)G = 1. For a discussion and interpretation of L in the case of a static external potential V, see, for example, ref. 6.

VERTEX

turn-energy tion6

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69

p- = p - q/2 and p+ = p + q/2. Then I is defined by the rela-

@P-l L IP+) = &UPwhere the Fourier transform V(q)

7 Pi.1 = Q,

a)v(Q)

(11

V(q) is defined by ’

= 1/(2~)~ /” V(5)eigr

cl’~ = (I?- IV (p+)

(2)

If we separate off the polarization part of the response, included in the generalized dielectric constant e, the vertex funct,ion may be writt,en in t.he factored form

rep, y) = c-‘(y)P(p,

y)

The dielectric t’heory of the response result’s when one replaces F(p, q) the proper vertex function, by unity. Exchange effects reside in i!(p, q). The results of bhe present investigation suggest that when the interparticle interaction is not weak enough to be t,reated in perturbation theory, it may not even be qualitatively correct to neglect exchange effects. The pure dielectric effects and the exchange effects are intimately linked. In a previous note,’ the form of r and 6-l in the long wavelength limit Q + 0 was written down explicitly on t,he basis of an approximate but nonperturbative t.heory. The formulation was in t.erms of an effective interaction IJ’. The expressions reveal an interesting property of r. For a sufficiently strong repulsive effective interaction, ? may exhibit a singularity, a pole, which is then cancelled by a zero of E’(q) at the corresponding value of q. This occurs in particular for st’rengths, C-I would the static limit qo/l q 1+ 0. For still larger interaction reverse sign leading to an interesting but apparently unphysical situation. For example, the effective static interaction between two external charges would reverse sign. Moreover, this would imply that the renormalization constant associated with the screening effect would become negative, a result which appears to be in contradiction with t#he spect,ral representation of the dielectric const.ant.g 6We consider only the longitudinal vertex function. In space-time coordinates, Eq. (1) is equivalent to 1‘ = -Lim(V + O)CJG-~/EIV(Z) which is the conventional definition of the ilnproper vertex function in quantum field theory. See also the concluding section of ref. 5. 7 We follow the convention that a = (a o , a) and ab = aobo - a.b. All vectors are fourvectors unless otherwise noted. 8 See refs. 6 and 7. 9Here we assume as in most of the following work that the basic two-body interaction v is repulsive. The screening renormalization constant may be defined as the static long wavelength limit (q -+ 0; QO/] q 1 --) 0) of ~-r(n). The conclusion that this limiting value is always positive (or zero in the case of long-range forces) follows from the dispersion-integral

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This pole of i; will actually be attained for sufficiently large coupling parameters if V’ is taken to be simply the elementary two body interaction v. If a more sophisticated choice of v1 is made, however, the conclusion is modified. It is reasonable for example to choose v’ as the static screened interaction e-‘v. In this case, the pole in ? does not appear due to the correlated behavior of g-l, which limits the growth of v’. One objection which could be raised against the previous treatment is the lack of gauge-invariance of the approximation. In the present treatment this defect is remedied by a self-consistent choice of the single particle propagators in the integral equations for r. In the long wavelength limit, the final results are modified only slightly (see Eqs. (98)-( lOO)), and the above conclusions remain the same. We can refer to the pole of i; therefore as a “spurious” pole on two grounds: first that it is apparently always cancelled by a corresponding zero of e-l, and second that it is probably not attainable within a fully self-consistent theory. It is interesting that an apparently similar phenomenon of the occurrence of a singularity of the proper vertex function and its cancellation has recently been noted in field theory models of elementary particles.” One can hope, then, that as an extra bonus the results presented here, when interpreted within the quasiparticle framework, may be relevant to the theory of elementary particles. The main purpose of the present work is to investigate the basic properties of the vertex function within the context of the theory of the weak response of many-fermion systems. The exact integral equation for the vertex function is rederived here on the basis of weak-response theory.“, I2 This equation is then factored into a separate integral equation for the proper vertex function and an equation giving the inverse dielectric constant in terms of the proper vertex function. The factored equations are treated in a nonperturbative approximation called the v’ approximation which we show can be viewed as a generalization of the time-dependent Hartree-Fock approximation. The approximation used in obtaining the results of the previous note is thereby improved and put on firmer grounds. representation of the retarded dielectric constant eIet and the positive nature of the spectral representation of Im 6. This shows that the static limit of g--‘(n) is positive (or zero) for all q for a parity-invariant interaction with v(q) > 0. lo See refs. 8 and 9. I1 The integral equation for the vertex function is closely related to the Bethe-Salpeter equation for the two-particle Green’s function. The generalization of the Bethe-Salpeter equation to many-fermion systems was first introduced in ref. 10. 12P. Nozieres (11) has discussed the integral equation for the vertex function and its role in providing a basis for the Landau theory of normal fermion systems.

VERTEX

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FIG. 1. A quasi particle is scattered by a weakly interacting function

of the quasi particle

is represented

SYSTEMS

external particle.

71

The vertex

by a circle.

In a second paper (I.%), hereafter called II, we derive detailed results for I? in lowest order perturbation theory and in the nonperturbative v’ approximation. In particular, the occurrence of the “spurious” pole of the proper vertex function is demonstrated for more general circumstances than previously found. The relation of the vertex function to real processescan be looked upon from two different points of view, with different experimental consequences, corresponding roughly to microscopic and macroscopic pictures. From the primary definition (1)) we can relate t’he vertex function to single quasi-particle scattering and bound state processes,as depicted in Fig. 1. In fact, t’he amplitude for the scattering process is proportional to ~‘(p, a). Alternatively, we can relate I? to properties of the bulk responseof the system and in particular to the collective mode. The frequency of the collective mode is still given by the isolated pole of c-’ as in the dielectric theory but (j? - 1) now modifies the determination of E1 as well as that of the coupling constant to the collective mode. Finally, the phase spacedensity matrix n( R, k) or its Fourier transform n(q, k) may be expressed in terms of P3, in the spirit of the cls.ssir! Landau description of the collective mode. In the working approximation we consider here, the v’ approximation, I’( p, a), t’urns out to be independent of E, the energy component of p. In this same approximation, the quasi particle is stable and, moreover, “wave-function” renormalization (occupation-number renormalization) effects are absent. This approximation also yields a stable collective mode associated with an isolated pole of c-* on the real 40axis. The stability of the collective mode is in accordance with the original Landau theory approximation asis the absenceof wave-function renormalization. The quasi-particle interpretation of r has several interesting aspects. Through f, the vertex function depends not only on the momentum-energy transfer q as in the dielectric theory but also on the incident momentum. The exxneri13See also the discussion

given in Section III.

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ment’al verification of this momentum dependence in quasi-particle scattering processes represents a very direct check on the calculation of r. In the quasi-particle description I’(p, a) is a sort of form factor of the quasi part’icle indicating the strength and shape of the coupling of the quasi particle to the medium. If we separate out the screening effect contained in c-l, we may say that r is the intrinsic form-factor of the quasi particle. This intrinsic size of a quasi particle may be ascribed to ground-state fluctuations in t,he effective interparticle potential, much as the intrinsic size of the electron in quantum-electrodynamics may be ascribed to the zero-point fluctuations in t(he vacuum electromagnetic field. The value of ( r - 1) in the static 4 -+ 0 limit can be regarded as a momentum dependent coupling constant, renormalization in the sense that it affects the forward scattering amplitude of the quasi particle in a weak static external field.‘4 Part of this renormalization is attributable to the familiar screening effect, as in quantum electrodynamics. It turns out, however, that (i; - 1) is also nonvanishing in this limit and therefore an intrinsic coupling constant renormalization due to longitudinal vertex effects is really present in many-fermion systemsI It does not cancel the “wave-function” renormalization as is the casein quantum electrodynamics, where this cancellation is a necessary consequence of Ward’s identity and gauge invariance. In many-particle theory, the intrinsic coupling constant renormalization is consistent with the appropriate Ward’s identity.16 Because of the delicat’e nature of the mathematical formulation, involving an isolated singularity of the vertex function at q = 0, the gauge-invariance condit,ion is investigated in the present paper in some detail. I. BASIC

PROPERTIES

OF

THE

VERTEX

FUNCTION

The vertex function I? is defined by Eq. (1) which relates p to the change in t,he effective single-particle Hamiltonian, L, under the influence of a weak external potential 8. L can be written as the sum (4) I4 It is a peculiarity of many-fermionsystems that the q + 0 limit of r and related functions is not well defined unless one specifies in addition the ratio go/[ q 1. For the definition of the coupling constant renormalization we choose the static limit pa/l q [ -+ 0. I5 See refs. 6 and 7. The distinction between longitudinal coupling and transverse coupling, as of a charged particle to a magnetic field, is important. There is no transverse coupling constant renormalization. 18 The four Ward identities for many-fermion systems are derived in refs. 11 and 13. The existence of the intrinsic longitudinal coupling constant renormalization and its compatability with the appropriate Ward’s identity were noted in refs. 6 and 7.

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where S is t’he self-energy operator of the quasi particle.” For convenience in calculating I‘ we shall amune that the response of the aysten1 to V is not a retarded one, as one should have for a classical “driving” potential, but is transnlitted through a Feynman t’ype vertex function appropriate to the “virtual potential” of an iuteract#ing particle. The task of recovcring, if desired, the retarded vertex function from its Feyrman counterpart, alt,hough in general rather vexing, is straightforward, as we shall shorn, in the approximation used in the later calculations. In terms of k’eynnlan diagrams, it is almost but not quite correct to say t,hat r is given by the sun1 of all possible diagrams consistent, wit’h the triangular blob of Fig. 1. This is t#he standard prescription of quantum electrodynamics (QED). The point,s of entry and emergence of the fermion line may be coincident. If so, these diagrams, when multiplied by V(y), contribute to the screened potential,

T=cq)= t-v(y). This descril)tion is not quite accurate for many-fermion syst,enm because it overlooks the possible change of t)he chemical pot’ential ,A due to 1’. Thus, we have Liin 6L (p- , p+) = Lim aup-, p+) V(q) + y v-0 v-0 dl’(d

(b)64(q)

(5)

where Z(p) is the eigenvalue of’ S in the absenceof the poteutial.‘* To compute r we must, divide by I’(q). Because of the singularity of the second term at’ q = 0 it is clearer to work temporarily wit]h a discrete norlnalization, where 64(q) is replaced by 6,,0 . Since J?should be independent of V, the ratio 6,/l’(q) in the limit of weak V should be independent of the form of V. Let, us choose a constant “poten?ial” V = a, which is simply a shift in the origin of the single-particle energies. In this case 8p = 1’ = V(0) = a l’rom (5) and (1) we have then in t)he discrete nonnalizatjion 17 In t.erms of Feynman diagrams, 2 is the totality of irreducible self-energy insertions between two points of the exact fermion propagator. The class of diagrams for which the two points of the insertion coincide is called the polarization part of the self-energy. The remaining class of diagrams is the exchange or mass-operator part of the self-energy. The polarization and exchange parts of Z will be denoted here by P and M respectively. For further discussion of the operators Z and 1; see ref. 5. 18 To appreciate the way in which Z(p) depends upon P, the reader may find it helpful to study the explicit calculation of Z(p), p, and aZ(p)/ap in lowest order perturbation theory. This is given by Eqs. (90)-(95) of Section III provided that one replaces U’ by ~1, the basic two-body interaction. The first term of (90) is the polarization part of Z, the second is the exchange part.

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Therefore

in either the continuous

or discrete normalization

if q # 0

r(p,q) = Limmp-, p+) Y+O

dV(d

This term can be computed according to the triangle diagram prescription The singular term of (7) can be evaluated by studying the dependence of X(P) on p. The relation (7) holds independently of the size of the space-time “box” provided that V(q) = (p- 1 V 1p+) for normalized states 1p). Let us note that the delta function term in (7) does not contribute to most processes of physical interest such as the scattering of quasi particles by a weak external agent, the collective mode oscillations or even the static screening effect. We are indeed interested in the long wavelength limit q + 0 but we are not interested in the isolated point q = 0. The isolated q = 0 term in I’, however, necessarily affects the calculation of the total number of particles N in the system. It is needed to maintain the invariance of N under the influence of V. Moreover, it is related to the gauge invariance of the theory.lg Suppose that we have a static potential, V, and we perform the change of gauge v-+v’=v+a

(9)

From the definition of G as a ground state expectation value of field operators +, and the Heisenberg equation of motion for $, we must have G(V’; t - t’) = G( V; t - t’)e’+“”

(10)

or, in an energy representation,

G(V’; E’) = G(V; E)

(11)

where 19 Conservation of the number of particles and the invariance of the Hamiltonian to longitudinal gauge transformations are equivalent since in nonrelativistic field theory the number operator is the generator for these transformations. The actual rigorous verification of the constancy of N in the presence of V is a delicate matter because the behavior of r in the p + 0 limit is so singular. However, if one accepts the admittedly nonrigorous derivation given here one sees from Eq. (20) that the delta function term in (7) guarantees that ~-1 is unity at the isolated point q = 0 in a discrete normalization. This in turn ensures the vanishing of the induced change in N, at least in the case of a static potential V.

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E”=E+a Now,

in an energy representation

(12)

we have the equation

[E - V -

z;,(V, E)]G(V;E)

= 1

where we have noted that Z depends implicitly on cc. From gauge-dependent definition of P it follows that Z,*(V’,

E’)

= &(V,

(13) (11) and (13) and the

E)

(14)

where p’, V’, E’ = p + a, V + a, V + a, E + a This means that for weak static potentials uv,

El

(15)

V, Z is of the form

= Z;(E - P, v - PCL)

(16)

In t#he discrete normalizat,ion, we can look at a diagonal matrix element of Eq. (14) in a moment’um representation in the limit V + 0. Because 8~ = 6E = SV(q = 0) = a, this gives a linear relation between the partial derivatives of 2: with respect to V, P and E. Returning now to a continuous normal ization, we can formulate this relation as Lim v-0

, p+) + WP) Lim mpq+o dV(d dP =,I‘@

; mP) dE

-0

(17)

where w = qo;

g=lIsL

po = E

(18)

Here we have assumed that, unlike I’, the first term of (17) is continuous in the limit 4 -+ 0, w/q --j 0. We have derived in this way one of the Ward identities for many-fermion systems.20 From (8) and (17) we can write Lim (q -+ 0, w/y--j

0)r(p,

q) = 1 - -mp) d/J

- -WP) dE

On the other hand, at the isolated point q = 0 in a discrete normalization, vertex function from (7) and (17) satisfies the relation up, eo See refs. ii and 13. The and diagrammatic structure the present derivation.

0)

=

1 -

(19) the

a.ap) aE

derivations given in these references of the theory rather than directly

are based on the algebraic on gauge invariance as in

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and therefore in a discrete normalization, I’(p, q) is discontinuous in the st’atic long wavelength limit by the amount aZ(p)/+. Let us note that the polarization part of 2 is independent of E, and therefore only the exchange part of Z, the mass operator M, contributes t,o the last term of (19). As noted above the corrections to r in the static long wavelength limit give rise to a renormalization of the effective coupling constant for the scattering of a quasi particle aside from the customary screening correction contained in the dielectric constant. Here we define the momentum dependent coupling constant renormalization g(p) as the ratio of the Born approximation scattering amplitude in the static long wavelength limit to its value in the absence of an interparticle interaction : g(p)

= Z”(P)

Linl (q + 0, w/g + 0) F(P, q)

(21)

where

Z”(p)= [l - !??!$I-’ Z”(p)

is

(22)

the occupation number renormalization constant.21 From (19), WP)

g(p) = 1 - Z’(p) T

(23)

Let us separate the effect of screening by writing g(p) =

g0(p)

Lirn (4 -+ 0, w/q --+ O)~-‘(a)

(24)

go(p), the intrinsic (to the quasi particle) coupling constant renormalization is given by g0(p)

= Linl

(n + 0, w/p +

O)z2(p)

RP,

n)

(25)

One verifies in the static long wavelength limit the separate Ward identities -1 t = 1 - dP/dp and I? = 1 - e(aM/ap + aM/dE) (26) where M is the mass operator or exchange part of Z and P is the polarization part. A lowest order calculation shows that aM(p) is nonvanishing while dM(p)/dE = 0. Therefore, (go(p) - 1) is nonvanishing in lowest order. Indeed, the lowest order diagram of Fig. 5(b) yields the well-known resulP 21 The considered noted in 22 For Feynman normalization source of

occupation number renormalization constant for many-fermion systems was in refs. 14and 10. Its occurrence in the quasi-particle scattering amplitude ref. 16. the interaction Y, we conform to standard practice in adopting the conventional normalization of the Fourier transform f(n) =Jf(x) exp (inx) d4x. Note that of V(p) defined in (1) is different. The conventional normalization is the factors of (27r)4 in the general formulas of the following section.

first was

the the

VERTEX

M(P)

THEORY

= -

OF

1/(8r3)

where kF is the Fermi momentum.

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/- d3kv(k

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- p)8(k,

-

lkl)

(27)

Then, we obtain from (27) (28)

where v,,(k, p) is the angular average of v(k - p). Then, from (25) and (26) go(p) = 1 + 1/(2~*Wwo(k~,

p)

(29)

This result agrees with that derived as a limiting case from the direct calculation of i; in lowest8 order perturbation theory, given in II. It is interesting to note that had we used the expression (20) for r at the isolated point’ q = 0, in a discrete normalization, we would have obtained t,he result, from (21), that g(p) - 1 = 0. Therefore, the existence of t’he coupling constant renormalization, including the screening effect, is dependent on the discontinuity of the vertex function at q = 0. In the following work, we shall not be furt,her concerned with the isolated singularity at Q = 0. It will be understood that the formulas writt,en down are not intended to apply at this isolat,ed point, though they will apply to the limit,ing value of F as q -+ 0. II.

THE

INTEGRAL

EQUATIONS

FOR

THE

VERTEX

FUNCTION

We shall derive the basic integral equation for r by studying the behavior of the single-particle Green’s function, G, in the presence of a weak time-dependent, external potential, V. G, of course, satisfies the operator equat’ions G’G The inverse Green’s function

= I’&-~ = 1

(30)

has the form

Let us agree that a bar line over G or B or a related operator denotes that quantity in t,he absence of V. Then 2, G, - and 8-l are diagonal in momentumenergy space. Under the influence of V, these quantities undergo small changes and are no longer diagonal. From (30), the first order variat,ions sat,isfy the equations (-62 - VjC? + (8)-‘6G = 0 (32a) c;l( -62 Equat.ion

- V) + 6G(C?-’

(32a) can be rewritten

= 0

(32b)

in the form

6G = G(V + szz)c;*

(33)

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FIG. 2. General Feynman diagram for 6~ illustrating the relation between 6x, sG, and the irreducible particle-hole interaction function I. Since G is the exact Green’s function, containing all self-energy insertions, the irreducible interaction function must be used.

which is also obvious from a diagrammatic analysis. We can express 6Z through SG and the irreducible particle-hole interaction function, 1.23From Fig. 2 we can write, in a momentum-energy-spin projection representation,

.(k- ul 16G ( k+ al')

(34)

u + u’ = Ul + Ul’ Here the barred notation @(k&) for p(k) in the Fermi sea indicates a hole in the corresponding initial (final) state. More precisely, for any value of the momentum, the bar indicates that the corresponding labelled line is leaving (entering) the Feyman diagram. In accordance with the convention introduced in (l), k, and p* denote k f q/2 and p f q/2 respectively. The simplest Feynman diagrams for I are shown in Fig. 3, through the first two orders of perturbation theory. Since 6G and 62 are diagonal in the spin indices, with the diagonal value independent of the projection of the spin and since we sum over the spin variable u1 , only the singlet (spin-zero) part of I affect’s the right hand side of (34). That is, the symmetric combination / UC) + [ -u - 5) is an eigenstate

of the

particle-hole

spin

operator

with

eigenvalue

zero

and

therefore F (urq I lUll71)= 10&.,/

(35)

where I0 denotes the spin-zero part of I. In lowest order, corresponding to diagrams

(a) and (b)

we havez4

28 The irreducible particle-hole interaction function is the kernel of the Bethe-Salpeter equation for the two-body particle-hole Green’s function. See refs. IO and 11. 24 To avoid confusion, the reader is reminded that v is the elementary two-body interaction in distinction to V which denotes the external potential.

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(cl

(d)

\

/

(e)

\

FIG. 3. Feynman diagrams for the irreducible particle-hole interaction function I through the two lowest orders of perturbation theory. (P-P+1 I0 V-i+)

=+ Wd

- 4~ - k)

We can now omit reference to the remaining write, in particular for the relation (34),

(p- 1a I p+>= -i i Let us further

abbreviate

g4

trivial

(36) spin-dependence.

J& p+ 1I0 1 k- i+)(k-

We

/ 6G 1 k+)

(37)

this relat’ion to read

6Z =--is

d4k106G

If we substitute (38) 6G whose kernel is 1’: [(+i AlternaCvely, tion for 6X,

into (33)

we can suhstitut’e

equation

an operator

/- d4kIo6G) - V] c + (C?)-?G

622 = --i The integral

we obtain

(38) integral

equation

= 0

(33) into (38) and obtain an integral

s

d4k10@2Z + V)C?

for the vertex

function

results

for

(39) equa-

(40) finally from the recog-

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nition that r is defined by the relation,

in abbreviated

notation,

for weak

62 + v = rv and therefore

(41)

from (40) r=1--i

Written comc#

s

l”t%c;l d%

out in full, in a momentum-energy

(43)

representation

this equation

rep, a) = 1 - i / $$$ 20cp- p+ ; k- L++)G(k-)r(k, &&CC+) FACTORIZATION

V

OF

THE

INTEGRAL

EQUATION

FOR

be-

(43)

r

The structure of the diagrams for the irreducible interaction function 1’ has a kind of connectivity which provides a useful factorization of t,he eyuat,ion for the vertex function (8). We can divide the diagrams for 1’ into two categories, those which can or cannot be broken into two disconnected and time-separated pieces by cutting a single dotted line. “Time-separat’ed” means that one of the two separated parts contains the incoming particle-hole pair, p- , p, . Let us call a diagram that can be disconnected in this manner a a-reducible diagram. This type of reducibility is possible even though 1’ must be irreducible as far as fermion lines are concerned and therefore its diagrams cannot be broken into two disconnected and time-separated parts by cutting only two internal fermion lines. The simplest r-reducible diagram is shown in Fig. 3(a). In fact this is the only r-reducible diagram in 1’: For, suppose that a more complex r-reducible diagram exists. The dotted line to be cut would join on at one or both of its ends to a pair of internal fermion lines.26 If this pair were to be cut instead of the dotted line, the diagram would be again disconnected and time-separated and t#herefore this diagram would be reducible with respect t’o fermion lines, contrary to assumption. We shall separate off the r-reducible part of 1’ and write, using (36)) -JO According

to the above discussion,

= I0 - 2v(q) Jo is r-irreducible

(44) and consists

of all dia-

25 This agrees with ref. 11, formula ((iJO), except that the spin dependence has been eliminated. The present derivation of the integral equation for the vertex function has the advantage, for our purposes, that it proceeds directly from the defining equation for the vertex function, Eq. (1). 26 More accurately, at least one member of the pair would be an internal line. If only one member of the pair is internal, one is necessarily dealing with a self-energy insertion in an external line. This type of diagram is also excluded from the irreducible kernel 10.

VERTEX

THEORY OF MANY-FERMION

SYSTEMS

81

grams of 1” except Fig. 3(a). In lowest order the matrix element of Jo reduces to v( p - /I). Sote t)he change in sign in (44).” The criterion of T-reducibilit,y when applied to r leads to the previous fact.orization of the vertex function into t,he r-irreducible proper vert’ex fun&on, i!‘, and the inverse dielectric constant t-l. Now from the struct)ure of t’he Feynman diagran:s cC1can be expressed in terms of i; alone*’ cl”kc;l(k-)f?(k+)f’(k,

-1 1

(1)

(4.5)

When this is substituted int’o Eq. (43) we obt’ain an integral equation for i? that is the same as that for r except that ( -Jo) replaces the kernel I’:

g+ ;k-L+)i%, y) i;(p, y)=1+L sc~“k~(k-)~(k+).~“(~ (46)

(2rj4

The pair of equations, (4;i) and (46), represent a fact’orization of the original integral equation (43). This factorization permits us to estimate the changes in the t.heory brought about by t’he inclusion of effects due to the proper vertex function. SYMMETRY

ANL) ANALYTIC

PROPERTIES

OF THE VERTEX

~JNCTION

One of the basic differences bet’ween the field t’heory of normal fcrmion systems and quantum electrodynamics (QED) is t’hat the former t,heory is not, invariant to charge conjugation since particles and holes occupy nonoverlapping regions of momentum space. For this reason several processeswhich give vanishing contribution to QED do cont’ribute in the many-fermion case. One consequence, already noted, is that the intrinsic longitudinal coupling constant renormalization is nonvanishing in the many-fermion case. Another is that, the triangle closed loop diagram (Fig. 4) which vanishes (Furry’s theorem) in QED does not do so in many-fermion theory.2g Even though charge conjugation is eliminated as a symmetry property, other symmetries remain: parity, time-inversion, and t,he “substitution rule,” which permits bending of the external lines of a Feynnlan diagram. The exist27In ref. 6, the signof

the corresponding equation,Eq. (4), is in error. 28 See for example the discussion in ref. 1, p. 53. The quantity in brackets in (45) is 1 + VT where r is the totality of irreducible polarization insertions or bubble-diagrams. The explicit appearance of ? in (45) indicates that one of the two corners of the bubble diagram must be completely equipped with proper vertex corrections. The normalization employed here agrees with that of ref. 16, see especially equations (21) and (31) of that reference, where s is called K and the function ?r is called &. 29 The diagram in question determines the lowest order contribution to the nonlinear part of the dielectric response. See refs. 7 and 16.

82

-------aA LAYZER

X

lb)

(a)

4. Diagrams containing triangular closed loops. virtual potentials. In quantum electrodynamics (a) and jugation invariance (Furry’s theorem), but they do not lem. These diagrams contribute to the nonlinear part of FIG.

'

The crosses represent external or (b) cancel because of charge concancel in the many-fermion probthe dielectric constant.

ence of the reversed diagram in Fig. 4 is associated with time-inversion invariance. Of main interest here is the fact that the interaction function 1’ exhibits the symmetry property

10(p-) @+; Id,)

= IO(p+, $L ; k+L)

(47)

which follows from the reversibility of Feynman diagrams and is a reflection of parity plus time-inversion (PT) invariance. The integral equations (45) and (46) now show that e-l, ?, and P must also be even functions of Q, that is, f(P, Q) = RP, -cl> C’(q) = e-y-q>

(48a) (48b)

and therefore, r(P, 4) = m

-n>

(48~)

As in the caseof I’, these properties also follow directly from the reversibility of the associated Feynman diagrams. In the caseof a parity-invariant interaction v, c-’ has the stronger symmetry property ?(q, w) = E-l( -q, w) (@a) and therefore E1(q, -w)

= r’(q,

+w)

(49b)

The symmetry properties (49a) and (49b) of e are well-known and follow directly from the spectral representation of the retarded dielectric constant eret.30The spectral representation also yields analytic properties in the complex w-plane. Let us note that E,t , since it is real in space-time satisfies the condition *a See, for example, ref. f .

VERTEX

THEORY

OF

4a)

MANY-FERMION

SYSTEMS

83

= a-4)

For a parity invariant interaction, then, eret(q, w) = &(q,

-w)

Therefore, the imaginary part of 6ret is an odd function of w rather than an even function as in the case of the present E. The connection between E and E,,,, is hhe following. Through the spectral representation3” we can construct a function e’(w) analytic in the entire complex w-plane except for a branch cut along the real axis, where the imaginary part of E’ changes sign. c’ coincides with cretin the upper half-plane and along the upper edge of the real axis. The Feynman 6 of this paper agrees for w > 0 (w < 0) with E’ on the upper (lower) edge of the real x-axis. E’then obeys the symmetry property in the entire complex w-plane (for parity invariant v): +I, WI = 4% -WI (5la) Analytic properties for r in the exact case are more difficult’ to establish because of the extra variables of the four-momentum p. Fortunately, the analytic properties of i; are straightforward analogues of t’hose of e-l in the approximation which we shall use in II, the v’ approximation, to solve the factored integral equations. This approximation is defined and discussedin the following section. The most noteworthy feature of the approximation is that r(lc, a) is independent of the energy component ko of F;. The analytic properties of e-l in the v’ approximation are the sameas presented above for t,he exact case except for the occurrence of isolated poles marking the frequencies of stable collective modes. We show in II that in the v1 approximation one can define for fixed k and q in the complex plane analytic extension l!’ of r, with the symmetry property, analagous t’o (51a) i;‘(k, q, w) = I?(k, q, -w) (5lb) Aside from the possible occurrence of isolated real poles, F’ is analyt,ic in the complex w-plane except for branch cuts on the real w-axis. Across these branch cuts, the imaginary part of I?’ changes sign and the real part is continuous. Isolated poles of f’ may occur for certain values of t,he strength of interaction and of the energy variable w. But, if they do occur, they are cancelled by corresponding zeros of t-1.31V 32 It follows from t’heseanalytic properties of r that a retarded vertex fun&ion rrrt may be deduced from r in just the same way, discussed above. that Erct is deduced from t. s1Comparethe discussiongiven in the introduction about this point. 32Analytic properties of the vertex function in the U’ approximation are summarized in the Analyticity Theoremstated in Section III.

84

LAYZER

G

FIG. 5. Diagrams for the self-energy operator in the Hartree-Fock approximation, (a) selfand (b), and in the exact theory, (c) and (d). In (a) and (b), 0 is to be determined consistently. In the v’ approximation, one replaces v by v’ in diagram (b). In (c) and (d), G and p are the exact Green’s function and vertex function.

III.

NONPERTURBATIVE

APPROXIMATIONS

R/lost of the complexity of the integral equations (43) and (46) lies in the irreducible interaction function Jo. The simplest approximation would seem to be the replacement of Jo by its lowest order evaluation v(p - k). The singleparticle Green’s function C? could be chosen as the free propagator Go. An improved approximation would result from a self-consistent choice for 8. We shall now show that this approximation, suitably defined, may be identified with the conventional time-dependent Hartree-Fock (THF) approximation. The THF approximation has been shown to provide a basis for the Landau theory of normal Fermi liquids and to be equivalent to other widely employed methods. In the next subsection, we present a generalization of this approximation, the U’ approximation, which allows greater freedom in the choice of Jo. The genuinely nonperturbative nature of these approximations is shown by the fact that they correctly predict a collective mode of the phonon or plasmon type with an energy spectrum that depends in a nonanalytic way on the interaction coupling const,ant. Let us recall that the conventional time-independent Hartree-Fock equations arise from approximating Z by the lowest order diagrams 5(a) and 5(b) .33 33 For a more ample, ref. 6.

detailed

discussion,

in the

case of a static

external

potential,

see,

for ex-

VERTEX

THEORY

OF

MANY-FERMION

85

SYSTEMS

That is, (p-

j &IF

/ p+)

=

--i

/

[(2v(q)

&

-

dp

-

WI

(k- 1 GHF

1 k+)

(52)

After making this replacement in G-’ we must solve the equation G&GEF = 1 self-consistently for GHF .sa We shall define the time-dependent Hartree-Fock (THF) equations by using the samet#wodiagrams, Figs. 5(a) and 5(b), but interpreting them in the presence of t,he external pot’ent,ial, V. We have then z HF

=

GTHF

&SF

+

GHF +

=

(53)

$2

6’G

(54)

The first variations 6’2 and 6’G satisfy the analogues of Eqs. (38) and (33) $2 = -i

(z’)HF 6’G d4k

(55)

S’G = GHF(V + 6’Z)GEF

(56)

s

where (p-f~+[

(~*)HF

/k-i+)

E

2v(q)

-

V(p

-

k)

(57)

Furtjhermore, we define the vertex function in this approximation by the relation 6’2 = I’THFV

(58)

When we compare these equations with the corresponding ones of the previous section and trace through the derivation of the integral equation for I’, we seet.hat t.he THF approximation leads to a transcription of the results of the previous section in which 2, G and r are replaced by their THF counterparts and one also makes the replacements e + c&F 1’

-+

&’

=

%‘(q)

-

(59) V(p

-

k)

(60)

This applies also to t2hefact’ored integral equations (45) and (46) wit’h t,he introduction of the THF i; and EC*.Here we must make the replacement JO+ v(p - k)

(61)

To tie in the vertex theory in the THF approximation t,o the classic Landau theory of the collective mode it is sufficient to give the connection between r and the oscillatjions of the mixed density mat,rix n( R, k, t) . The Fourier trans-

86

LAYZER

form of the latter with respect to R is given by the expression3* n(k,

q; t> = (d+(hdt>)

(62)

where &=kfq/2 and a ground state expectation value of second-quantized operators senberg representation is indicated. As a check on this specification of n(q, k; t) one can verify that

s

d3kn(q, k;

s

d3pn(q, k;

(63) it in the Hei-

t) t) t) = n(n, t)

(64)

= n(k,

(65)

where n(p, t) is the Fourier transform of the ordinary density and n(i& t) is the occupation number of the state with momentum k. We can express n(q, k; t) in terms of the fluctuation of the single particle Green’s function : n(q, k, t) = (-i)GG(k-t; We introduce

the energy-transform

with

from

k; w)emiwt

(67)

(66) shows that n(q, k; w) = (--i)

Now,

(66)

n(q, k; w) by the relation

n(q, k; t) = & 1 dwn(q, Comparison

k+t+)

/ dko 6G(k+ , k-)eikoO+

(33), for a given “driving” 6G(k-

, k+)

potential,

= G(k-)r,et(k,

V, we have

qK”(k+)V(q)

where the retarded vertex function is used.35 In the THF approximation, as in the more general v’ approximation following work, I’(k, q) is not a function of ICO. Therefore, n(s, k; w) = f(k

(68)

q; w)rret(k,

(69) of the

n>v(a>

14 Our purpose in introducing the mixed density matrix is to make contact formulation of the Landau theory of the collective mode given in refs. ~4. 56 We assume in the present discussion that one has a given classical potential, necessitates a retarded response.

(76) with

the which

VERTEX

THEORY

OF

MANY-FERMION

SYSTEMS

87

where

f(k, q, w) = s dkoG(k-)G(k+)eikoO+ The integration

is straight’forward (‘W-tf(k,

q) =

(71)

and yields the result e(k+ - k,)8(k, - k-) 0 - E(k+) + E(k) + 2ie

(72)

Here, we have discarded the time-reversed term, which has a singularit’y in the lower half of the complex zu-plane.35 E(k) in this expression is the Hartree-Fock energy of the state k. Once r is determined from the integral equations (43), n(q, k; w) may be determined from Eq. (70)) for a given “driving” potential V( 4). An isolated pole ‘of the vertex function at w = w* corresponds to a self-sustaining oscillation of n(q, k; t) at the frequency w*, which is the collective mode. The only difference between Eq. (70) and the corresponding equation in the time-dependent Hartree approximation (no exchange) is that in the latter, we have the replacement of r by e-l. The result (70) indeed might have been guessed beforehand. The derivation that has been given confirms this guess. It confirms also our identification of the approximation under consideration with the conventional time-dependent Hartree-Fock approximation.36 THE

v’ APPROXIMATION

Instead of replacing the kernel Jo in Eq. (46) by the lowest order interaction 0, let us replace it by an arbit.rary, local Hermitian and energy-independent or instantaneous potential v’: Jo -+ v’(p - k)

(73)

At the same time, let us replace the propagator (? in (46) by the Green’s function 8, for a suitable Hermitian and energy-independent effective Hamiltonian h:

( ) ii-h

c’,=l

This approximation obviously includes the THF approximation but it is somewhat, more general. We shall refer to it as the v‘ approximation. The 8’ approximation has the simplifying mathematical feature that I’~,(P, ,J) is independent of E, the energy component of p, as is clear from the form of the 36A more direct, identification may be made through the integral equation for n(p, k) which can be derived from Eqs. (39) in the THF approximation and (66) and which agrees with that given in refs. S and 4.

88

LAYZER

integral equation (43). Then, since, in addition, Gh has only simple poles at the eigenvalues of h, one can immediately integrate over the energy component of the integration variable k. The actual solution of the int’egral equations in this approximation will be discussed in II. The question of the choice of o’ remains open. Once 8’ is chosen it will be seen that h is determined. In a sense the proper choice of U’ creates a second self-consistency problem. A reasonable choice is to pick U’ so as to include all polarization corrections to v. With this choice, v’ would be the static screened interaction v’ = vt’(q, 0). After solving the integral equations for r and cC1 in the IJ’ approximation we must then adjust the “input” v’ so that it agrees, approximately, with the value v” = UC-‘(q, 0) obtained using the output value of c-l. To choose h and C!& consistently we should, by analogy to the THF case, determine Q by a self-consistent procedure entirely similar to t*he HartreeFock procedure except that v1 is used instead of v in the mass operator diagram of Fig. 5(b).3’ This method of choosing G,, is not so arbitrary as it may sound. In fact, it, is the only simple way to assure that t’he determination of r is consistent with the gauge invariance condition ( 19). The same remark holds true even in t,he simpler THF approximation. That is, once we make the approximation J -+ v in bhe interaction kernel of t,he integrel equation (46)) we are led to t,he Hartree-Fock choice for the input 0 if we wish to maintain the gauge-invariance of the result.3s The symmetry and analyticity properties of r in t,he v’ approximation are discussed in detail in II. We show there that I‘,, has the analytic properties summarized by the following Analyt’icity Theorem, some of whose results have been mentioned above. ANALYTICITY THEOREM. For a parity invariant interaction v’, the functions F = I?,, , Fvf and E;? for jixed p and Q can be extended to functions F’ analytic in the complex w plane except for the real axis and having the symmetry property F’(P,

q, -w)

= F’(P, q, +w)

(75)

The functions F’ have branch cuts on the real axis, across which Im F’ is discontinuous and changes sign. The branch cuts extend over the region of possible particle-hole excitation energies (for jixed q) 0 < 1w 1 < I.‘$x [E(b+)

- E(k-)] (76)

&=kfq/2 37 In the Eq. (74) has lowest-order tion of Gh , 3* See the

uniform case considered here, the self-consistent determination of the f$ of a somewhat trivial aspect since the self-energy operator must coincide with the perturbation result with v + u’. In the nonuniform case, the determinaof course, would no longer be trivial. See, for example, ref. 6. following section on gauge invariance.

VERTEX

E(p) is the real < k, < (k+(.

where

THEORY

energy

OF

MANY-FERMION

spectrum

89

SYSTEMS

corresponding

to C?(p)

and / k- 1

In addition to the above analyticity properties, E;/, may have an isolated real pole at w = 18 lying outside of the region (76). This pole corresponds to a stable collective mode. f’,, may also exhibit an isolated pole outside the region (76) for an appropriate strength v f interaction. IJ’ it OCCUI’S, howeve?, this pole is cancelled b?~ a corresponding zero of 6-l.

The v’ approximation, in spite of it)s apparent generality, is still limited by the energy-independent and Hermitian character of t’he approximation for Jo. In the input t,o this approximation there is still no occupation number, or “wave function,” renormalization and quasi particles are completely stable. In t’he following we show ihat’ this approximation may be “transcended” t)o asrive at a det’ernlinat’ion of G t’hat does include quasi-particle lifetime aud renonnalizalion effects. The discussion makes essential use of the properties of r &ted in t#heAnalyticity Theorem. QUASI-PARTICLE

LIFETIME

Let us assumet’hat a solution F,J for r in the v’ approximat~ion has been obtained. We can now use this approximate vertex function to boost, the determination of G. That is, we can substitute I?~(for the exact vertex functiou in the massoperator diagram 5(d) and t’hen solve for G self-consistently. The resultant G now is not of the t’ype C&. The effective Hamiltonian for G is no longer Herrnitian or energy-independent. One of the new pieces of information we pick up in this way is the lifet,ime of single-particle excitations. Let us assumethat a quasi particle of moulentum p is nearly st,able and has the energy2?(p) which we can assumeto be real to a first approximat’ion. The decay rate of the quasi particle, by perturbation theory, has the value -1 T = Ini Z(p,E) = Im M(p,Z) (77) where M is the mass-operat,or. We have assumedt’hat for this value of the mentuni Im M(p,E)/E

<< 1

mo-

(78)

The aualyt,ic expression for M is GYP - aMq>rd(p

- p/Z),

rJ)leiso+

(79)

If we limit ourselves to a range of p such that the quasi particle is uearly st,able and a sufficiently small range of momentum transfer 4 the Green’s fun&on G(p - q) will also describe nearly stable single-particle excitations. Let us assume that particle-number renormalization effect’s are also small. Therefore,

90

LAYZER

consistent with the approximation (78) we can assume, in calculating Im G, that the G(p - q) of Eq. (79) can be replaced by a quantity of the type 8, :

G(P - n> ---) (8~ - d = [PO - qo - E(P - a> + i4p where I? is the real energy spectrum of G and e(k) = sgn The ~0 integration in (79) has a pole due to G(p - CJ) in On the basis of the analytic properties of I? described Theorem given above, we can assert that the imaginary part yields only this pole term. We have therefore, Ini M(P,lZ)

= (2~)~~ / d3qv(q) Im I?&

- all-’

(80)

( kp - 1k I) . this approximation. in the Analyticity of the qo integration

- q/2, qlO[l p - q 1 - kF1 (81)

where now 40 = Z(P)

- RP

- a>

(82)

This result is a generalization of a corresponding result of the dielectric theory in which one makes the familiar replacement cm1-+ I’ (17). Equation (81) describes the decay transitions of a nearly stable quasi particle of momentum p above the Fermi surface to a nearly stable quasi particle of momentum p - q also above the Fermi surface. We recall that in the Y’ approximation e-l will have an isolated pole corresponding to the emission of a quantum of the collective mode, of energy w = c(q). In this region of energy, then p has the form

A(P, n,a)

1

where

A (P, q, 4 cw

= Res. r(p - q/2, q, ti) = f(p - q/2, q, ti) Res. E-l(q, ti) Res. c?(q, ti) = [&

From -1 ?-e

(84)

dq, w)]IIG

(77) and (81) the decay rate for this mode of decay has the form =

cw3

J 2

ddA(P,

Q, @Ml P - Q I

- bl

4ti

- B’(P) + &P

We may define a squared coupling constant usual manner by means of the relation3’

for this “boson”

a The coupling constant description is patterned after theory. One regards the collective mode as an independent

elementary particle

- (1)) decay in the

particle S-matrix which interacts with

VERTEX

-1 Tc

=

(27c2

Comparison

/- &

THEORY

9cTP,9MlP

OF

MANY-FERMION

- q I - hM~

SYSTEMS

- &-I)

+ E’(p - q))

91 (87)

of (56) and (87) shows that we can make the identification (24&P,

q) = A(P, q, a)v(q)

(88)

Thus the momentum-dependent factor i;( p - q/2, q, ti) modifies the determination of gc2. Let us turn to the proof of the assertion that the pole term (81) is the sole contribution to the expression for Im G in the approximation (80). This result follows from the symmetry and analytic properties of I’ given in the Analyticity Theorem of the previous subsection. We can use the symmetry property (75) to restrict the qo integration to the region (0, CQ ). At the same time, we can replace G(qo) by the symmetrical combination G, = f?(qo) + G( -qo). If we now rotate the contour of the qo integration to t’he positive imaginary axis we pick up the pole contribution (81). The integral along the imaginary axis itself is necessarily real due to the symmetry of pwland G, in the complex qo-plane. The line integral along the imaginary axis may Oherefore be discarded leaving only the pole term.40 If lifetime effect’s are appreciable the quasi-particle picture is no longer appropriate and one must resort to the self-consistent determination of G described in the first paragraph of this subsection. We can write G(p) in the form G(p) =

[po

- p2/2m - v(0)kr3/(37r2) - M(p)]+

(79’)

The pair of equations (79’) and (79)) with I’,,? a given input to (79)) may be converted to a single integral equation for either G(p) or M(p), thus defining the self-consistency problem. GAUGE

INVARIANCE

We have assert’edthat the U’ approximation is consistent with gauge invariance. This means, in particular, that p,r should satisfy from (19) the equation

Lim(p -+ 0; w/q + o)r,? = i - aPp> ~ fb where B is the self-energy operator corresponding to the Gh of Eq. (74), the latter being determined self-consist’ently in the manner described above. Here single-particle excitations via the coupling constant g. , in accordance with the usual Feynman rules for a boson. This type of description was employed by Dubois (17) in his treatment of the plasmon mode of an electron gas. 40 The derivation given here is a generalization of one given in ref. 17 for the dielectric theory of the electron gas.

92

LAYZER

we have noted that in our approximation Z is independent of E, the energy component of p. A heuristic graphical interpretation of the fulfillment of (89) is as follows. The integral equation in the v’ approximation is equivalent to summing up a set of ladder triangle diagrams with ‘?ungs” represented by the factor [2v(O) v’(p - k)]. This set of diagrams will be gauge-invariant if we can regard it as derived from a corresponding set of self-energy diagrams by the process of inserting a cross in a fermion line in all possible ways. One might guess that this will be bhe case if the self-energy diagrams are also of the ladder type. However, unless t’he cross is symmetrically placed, there will arise self-energy insertions in one of the two legs of the vertex triangle diagram, producing vertex diagrams other than the simple ladder type originally considered. The only way this situation can be avoided is to “saturate” the fermion lines of the self-energy diagrams with insertions of the class considered so that they will be unchanged by the addition of another self-energy insertion. The self-consistent choice of C?hin the V’ approximation achieves this kind of saturation. It is easy to directly verify Eq. (89) by calculating both sides of the equation. A simplifying feature is that for uniform systems the self-consistent Green’s function 6, may be calculated exactly. In fact, the self-energy operator 2 is the same as in lowest order perturbation theory with the replacement v -+ v’: z,,(~)

= z(~; kF) = v(o) ;$

- &

1’” dk ~I’(P,

k)

where vO’(p, k) is the s-wave component or angular average of v’(p - k). To evaluat,e the RHS of (89) we need in addition the following relations

(91)

ap dab ; kF) =

-=

akF

vF + Lim(k

--)

+ kF) aE($

kF)

kF)

aJ%k;

akF

kF) akF

(93)

where vF = Lim(k From these relations,

we obtain after an elementary

a% -= all

calculation

(27r”>-‘~2v(o> - Bo'(P, 111 p + (2n”)-‘[2v(O) - vo’(1, l)]

(95)

VERTEX

THEORY

where the dimensionless

OF

parameter

MANY-FERMION

93

SYSTEMS

0 is given by the relation

In (95) we have used the units fi = ??A= lcs = 1

(97)

The calculation of I’,,, in the static long wavelength limit is also very simple. The k. integration may be performed immediately as in (71) yielding a factor of the type (72) with a denominator w - E(k+) + E(k) which in the static limit approaches 0lc.a. The step functions in (72) limit the k integration in the long wavelength limit to the region of the Fermi surface. The vertex function may then be pulled out of the integral on t’he right hand side of the integral equation. The latter then reduces to an algebraic equat8ion. The final result is in the static long wavelength limit. r -1

e

-

( 27r2)-‘p-‘oo’( p, 1) l = 1 - (2??)-‘P-‘uo’(1, 1) -(2r2)-‘p-Y2u(0)) l = 1 + (27?)-W[2(v(O)

- (27r2)-‘p-‘[2v(O) r - l = 1 + (27rz)-lp-l[2v(o)

(98)

(99)

- ?Jol(l, l)] - z&l, l)] - fI<(l, l)]

(loo)

Formulas (98)-( 100) agree with (95) and (89)) as well as the separate equations (26) for ? and c-l. These formulas are the same as those previously obtained except for the factors of p-‘.41 From (90) to (96)) /3 has the explicit value p = 1 - 2 (27r2)-’ J,” czk k2dYp,

k)

(101)

where vi’l’(p,

k) = Lim(p

-+ kF) $

bo’(P, k)l

SUMMARY

We show that the dielectric theory of the weak response of many-fermion systems may be embedded in the more general vertex theory, which properly 41 See ref. 6. In this f% in obtaining explicit

paper, results.

the

unperturbed

Green’s

function

Go was

used

instead

of

94

LAYZER

takes account of exchange. We consider in varying degree general questions related to the longitudinal vertex function, I’: symmetries, analyticity of I? in the complex energy plane, the gauge-invariance condition and the renormalization of the quasi-particle coupling constant. We show the existence of an intrinsic longitudinal coupling constant renormalization and its consistency with gaugeinvariance. On the basis of the primary definition (1) of I’ in terms of the weak response of the system, the integral equation for I? is rederived, then “factored” into the separate equations (45) and (46) for P and c-‘. In the time-dependent HartreeFock (THF) approximation, the interaction kernel J of the integral equation for P is replaced by the lowest order interaction, v. In the more general v’ approximation J is replaced by a suitably chosen energy-independent effective interaction, v’. If the single-particle Green’s function G in the input to the THF and v’ approximation is properly chosen, the gauge-invariance condition (89) will automatically be satisfied. The determination of I’ in the v1 approximation can be used as the input for an improved determination of the single-particle Green’s function, G. This improved value of G will incorporate occupation-number renormalization and quasi-particle lifetime effects. The inverse lifetime, 7-l) can be expressed in terms of Im G. In a consistent lowest order approximation in lifetime effects, Im G is given by t’he pole contribution (81), which generalizes a corresponding result of the dielectric theory. The influence of the proper vertex function on the determination of the coupling constant of the quasi particle to the collective mode is given in Eq. (88). When lifetime effects are large enough to render the quasiparticle picture inappropriate, the coupled equations (79) and (79’) permit the self-consist’ent determination of G(p). The analytic properties of I’ in the v’ approximation as a function of w = QO, for fixed p and q are similar to those of the inverse dielectric constant e-l. However, r may have a pole which is then cancelled by a corresponding zero of E-I. These results are stated but not proved in the Analyticity Theorem of the last section. Although many of the results obtained for the vertex theory in the v’ approximation are natural generalizations of corresponding results in the dielectric theory, it should be kept in mind that the derivations employ simplifying properties of r in the v’ approximation that do not necessarily apply to the exact vertex function. The derivation of the stated analytic properties of I? in the v’ approximation will be given in II. A lowest order calculation of ?, presented there, facilitates this derivation. In II we also give an explicit solution of the integral equations for r in the v’ approximation in several cases of interest. RECEIVED:

June 4, 1965

VERTEX

THEORY

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MANY-FERMION

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95

REFERENCES D. PINES, “The Many-body Problem.” Benjamin, New York, 1961. L. LANDAU, Soviet Phys. JETP 6. 101 (1957). J. GOLWTONE AND K. GOTTFRIED, Nuovo Cimento [Xj 13, 849 (1959). 4. H. EHRENREICH AND M. COHEN, Phys. Rev. 116, 849 (1959). 5. A. LAYZER, Phys. Rev. 129, 897 (1963). 6. A. LAYZER, Phys. Letters 13, 121 (1964). 7. A. LAYZER, Bull. Am. Phys. Sot. 9, 604 (1964). 8. B. GESHKENBEIN AND B. IOFFE, Phys. Rev. Letters 11, 55 (1963). 9. C. GOEBEL AND B. SAKITA, Phys. Rev. Letters 11, 293 (1963). 10. V. GALITSKI AND A. MIGDAL, Soviet Phys. JETP 7, 96 (1958). 11. P. NOZII%RES, “PropriCtks Gkn&rales des Gaz de Fermions,” Chapters Dunod, Paris, 1963. 12. A. LAYZER, unpublished. 13. P. NOZIBRES AND J. LUTTINGER, Phys. Rev. 137, 1423 (1962). 14. A. MIGDAL, Soviet Phys. JETP 6, 333 (1957). 16. L. LANDAU, Soviet Phys. JETP 7, 104 (1958). 16. A. LAYZER, Phys. Rev. 129, 908 (1963). i7. D. V. DUBOIS, Ann. Phys. (N.Y.) 7, 174 (1959); ibid. 8, 24 (1959). 1. 2. 3.

IV and VI.