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Notes on Greens Functions Theory for

Quantum Many-Body Systems

Carlo Barbieri

Theoretical Nuclear Physics Laboratory

Nishina Center for Accelerator-Based Science, RIKEN

April 2009

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Literature

A modern and compresensive monography is

W. H. Dickhoff and D. Van Neck, Many-Body Theory Exposed!, 2nded. (World Scientific, Singapore, 2007).

This book covers several applications of Greens functions done in recentyears, including nuclear and electroncic systems. Most the material of theselectures can be found here.

Two popular textbooks, that cover the almost complete formalism, are

A. L. Fetter and J. D. Walecka, Quantum Theory of Many-ParticlePhysics (McGraw-Hill, New York, 1971),

A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski, Methods ofQuantum Field Theory in Statistical Physics (Dover, New York, 1975).

Other useful books on many-body Greens functions theory, include

R. D. Mattuck, A Guide to Feynmnan Diagrams in the Many-BodyProblem, (McGraw-Hill, 1976) [reprinted by Dover, 1992],

J. P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MITPress, Cambridge MA, 1986),

J. W. Negele and H. Orland, Quantum Many-Particle Systems (Ben-jamin, Redwood City CA, 1988).

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Chapter 1

Preliminaries: Basic Formulae

of Second Quantization andPictures

Many-body Greens functions (MBGF) are a set of techniques that originatedin quantum field theory but have also found wide applications to the many-body problem. In this case, the focus are complex systems such as crystals,molecules, or atomic nuclei. However, many-body Greens functions still

share the same language with elementary particles theory, and have severalconcepts in common. To apply this formalism, one needs to use of the cre-ation/destruction operators of second quantization and the Heisenberg andinteraction pictures of quantum mechanics.

The purpose of this chapter is to gather the basic results of second quan-tization and pictures, so that they can be used for reference later on. Onethe way, we will introduce some of the notation to be used in our discussions.

1.1 A Note on Single-Particle Indices

The following conventions will be used in most of these notes.

When needed, the boldface r will be used to refer to the position ofparticles in coordinate space and k for momentum space. When internaldegrees of freedom are present, x(r, , , ...) will be used. However,most of the results to be discussed are valid for any general single particlebasis. Thus, we will use greek indices, ,,,. . ., to refer to all the states inthe baisis. In general, {i} is a complete set of orthonormalized one-bodywave functions and will be assumed to be discrete, unless it implies a loss of

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generality.1

In many-body theory one often starts from a product wave function de-cribing a set of non interacting particles that occupy given orbits (called thereference state). This could be a Slater determinant for fermions or a macro-scopic condensate orbit for bosons. It is customary to reserve the latin lettersh,i,j,k,l for the levels occupied in the reference state (which are called holeorbits) and m,n,p,q,. . . for the unoccupied orbits of the basis (the particleorbits).

Since removing a hole orbit from the reference state leads to a systemswith fewer particles, we extend the use latin hole indices to indicate statesof N 1,N 2,. . . particles. Here, N is the number of particles in thereferecnce state. Analogously, particle indices will be used to distinguishstates of N,N + 1,N + 2,. . . particles. This notation will include labellingexact many-body eigenstates of the Hamiltoninan. It should not be felt asunnatural: the eigenstates ofN1 and N+1 bodies can be seen as excitationsof the systems and directly are related to quasiholes and suasiparticle in theLandau sense.

1.2 Second Quantization

Most of the processes described by many-body Greens functions involve the

transfer of particles to/from the intital sysytem. Thus it is useful to extendthe Hilbert space to allow for states with different particle numbers. In factwe will use the Fock space which includes a complete basis set for each possi-ble number of particles, from zero (the vacuum) to infinity. The basis statesof the Fock space can be taken to be product of one-body wave functionsand must be automatically symmetrized or antisymmetrized (for bosons andfermions, respectively). Using Diracs bra and ket notation one can specifythe basis states just by saying how many particles n are contained in eachsingle particle orbit . For example, the state2

|n1 = 3, n2 = 0, n3 = 2, n4 = 2, n5 = 0, . . . , (1.1)contains a total of N=

i ni=7 particles, distributed over the orbits 1, 3

and 4. Obviously this must be a bosons state, or it would violate the Pauliexclusion principle with disastrous consequences. Note that, also due to

1What the {i} represent depends on the systems one wants to study (e.g., harmonicoscillator wave functions for nucleons in a nucleus or atoms in a trap, orthogonalizedgaussian orbits in a molecule, Bloch vectors in a crystal, and so on...).

2In this case the corresponding product wave function would be the symmetrizedS[1(r1)1(r2)1(r3)3(r4)3(r5)4(r6)4(r7)].

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Pauli, we can only say how many particles are in each orbit but not which

particles.The completeness relation in Fock space is

I =nmaxn1=0

nmaxn2=0

nmaxn3=0

nmaxn=0

(1.2)

|n1, n2, n3, . . . n, . . . n1, n2, n3, . . . n, . . . |

and includes the vacuum state |0 |n = 0, . nmax is 1 for fermions and for bososns. States with different number of particles are orthogonal bydefinition.

The second quantization formalism introduces the creation operator cwhich adds a particle in state to a vector of the Fock space. Its self-adjoint c removes a particle from the same state and is called destructionorannihilationoperator. Their effect on Fock states is the same as that for thecration and annihilation of harmonic oscillator quanta in the linear oscillatorproblem

c|n1, n2, . . . n, . . . =

n + 1 |n1, n2, . . . n + 1, . . . , (1.3)c|n1, n2, . . . n, . . . =

n |n1, n2, . . . n 1, . . . , (1.4)

where we momentarily neglect a conventional signs that appears for fermions[see Eqs. (1.11) and (1.12) below]. Note that destroying and empty states(c|n = 0 = 0) gives the c-numer zero, and not the vacuum state |0.From these relations it follows that cc|n = n|n yields the number ofparticles the state . The operator for the total number of particles is then,

N =

cc . (1.5)

The eigenvalues of N are non negative integers and its eigenstates are wavefunctions with a definite number of particle. By applying Eqs. (1.3) and (1.4)to these states, one can see that the following commutation relations mustapply,

[N, c] = c ,N, c

= c . (1.6)

Equations from (1.2) to (1.6) are valid for both boson and fermions.Eq. (1.4) makes it impossible to create Fock states by removing a parti-cle from an alredy empty orbit. However, there is still no restriction on thenuber of particles that can be added in the case of Fermions. The correctPauli statistics is imposed by choosing different commutation and anticom-

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mutation relations3,c, c

= , , [c, c] =

c, c

= 0 , for bososns (1.7)

c, c

= , , {c, c} =

c, c

= 0 , for fermions (1.8)

With both these relations, Eq. (1.6) is still valid. At the same time the an-ticommutator cc

= cc ( cc=0) imposes the antisymmetrization of

fermionic wave functions and restrict the occupation of each orbit to n=0,1only.

To create the basis (1.2), one simply acts several times on |0 with creationoperators. The normalized many-body states are given by

|n1, n2, , n, = 1n1! n2! . . . n! . . .

(c1)n1(c2)

n2 (c)n |0 ,(1.9)

which simplifies in the fermion case because it can only be n!=1,

|n1, n2, , n, = (c1)n1(c2)n2 (c)n |0 . (1.10)Note that only for the case of bosons Eq. (1.9) is independent on the orderin which the creations operators are applied. For Fermoins a phase signis introduced by changing the order and one must put extra care to avoidconfusions. Ususally one chooses a specific order in the

{}

and then stickto it. Once this is done the correct fermionic version of Eqs. (1.3) and (1.4)

c|n1, n2, . . . n, . . . = 0,n()s

n + 1|n1, n2, . . . n + 1, . . . , (1.11)c|n1, n2, . . . n, . . . = 1,n()s

n |n1, n2, . . . n 1, . . . , (1.12)

withs = n1 + n2 + n3 + + n1 . (1.13)

The creation operator for a particle in position r of coordinate space isindicated by (r). If {u(r)} are the single particle wave functions of ageneral orthonormal basis, the creation (and annihilation) operators in the

two representation are related via a unitary transormation,

(r) =

cu(r) , (1.14)

c =

dr (r)u(r) . (1.15)

3Here, [A,B] ABBA and {A,B} ABBA are the commutator and anticommu-tors, respectively. Later on we will also use the more compact notation [A,B]

ABBA

to indicate both at the same time. Otherwise, [ , ] without sign will always be a com-mutator.

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It follows that to create a particle in a state one simply superimposes

eigenstates of position with weights given by the corresponding wave func-tion,

| = c|0 =

dr u(r)|r . (1.16)where

|r = (r)|0 . (1.17)is a particle localized in r. Analogously, one can extract the wave funcioncorresponding to a one-body Fock state by

u(r) = r| . (1.18)

These relations are extentded to states of any particle number, where

|r1, r2, . . . rN = 1N!

(r1)(r2) (rN)|0 . (1.19)

and the first quantization wave function of an N-body Fock state is

r1, r2, . . . rN|n1, n2, . . . = (r1, r2, . . . rN; {n}) . (1.20)

1.2.1 Examples

If

|r

and

|p

are eigenstates of position and momentum one has

r|r = (r r) , (1.21)r|p = 1

(2h)3/2eirp/h . (1.22)

Given the Fock state |=cc|0, the corresponding antisymmetrizedwave function in first quantization is

(r1, r2) = r1r2| (1.23)=

120|(r2)(r1)cc|0 (1.24)

= 12

u(r1)u(r2)0|cccc|0 . (1.25)

Using the (anti)commutator relations (1.7) and (1.8) one finds thatccc

c|0=(,, ,,) |0, with the upper (lower) sign refer-

ring to bosons (fermions). This leads to the usual symmetrized andantisymmetrized product wave functions of Slater type,

(r1, r2) =1

2{u(r1)u(r2) u(r1)u(r2)} . (1.26)

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1.3 Operators in Fock Space

1.3.1 Operators

Let O = O(r) be a one-body operator that acts independently on each par-ticle of the systems. The expression for its matrix elements in coordinatesystems depends on the number of particles N and is

r1, r2, . . . rN|O|r1, r2, . . . rN = Nj=1

(ri ri) N

i=1

O(ri) . (1.27)

The corresponding results for a generic Fock states can be related to the

latter by inserting a completeness relation based on the position states (1.19),n1, n2, . . . |O|n1, n2, . . .

=Ni=1

dr1

dr2

drNn1, n2, . . . |r1, r2, . . . rN

O(ri)r1, r2, . . . rN|n1, n2, . . . (1.28)

=1

N!

Ni=1

dr1

dr2

drNn1, n2, . . . |(r1)(r2) (rN)|0

O(ri)0|(r)N (r)2(r)1|n1, n2, . . .

=

1

(N 1)! dr1 dr2 drNn1, n2, . . . |(r1)(r2) (rN)O(r1)(r)N (r)2(r)1|n1, n2, . . .

In the last line we have removed the sum over the interacting particle sinceit gives N times the same contribution. The projection on the vacuum state|00| can be subsituted with the identity becose the operators (r) ((r))have already annihilated all the particle contained in the ket (bra) vectors.

We still need to perform the integration on the coordinates r2 to rN.This is easlily done remembering that

dr(r)(r) is the particle number

operator in coordinate space (1.5). One starts integrating over rN to get a

factor of 1, then the integral overrN1 gives 2, and so on up to cancellingthe factor (N 1)! at the denomonator. Finally,

n1, n2, . . . |O|n1, n2, . . . =

dr n1, n2, . . . |(r)O(r)(r)|n1, n2, . . . .(1.29)

The one-body operator in second quantization representation is therefore

O =

dr (r)O(r)(r)

=

o cc , (1.30)

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where o are the metrix elements in the generic basis

{

}, as can be verified

inseting Eq. (1.14) in the latter equation

o =

dr u(r)O(r)u(r) . (1.31)

For a two body operator (symmetric in the exchange of ri and rj),

V =Ni

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with

w, =

dr1

dr2

dr3

u(r1)u(r2)u(r3)W(r1, r2, r3)u(r1)u(r2)u(r3) . (1.38)(1.39)

1.3.2 Expectation values

Lets asume that that we have a state |N of a system of N particles. Thisdoes not need to be a basis vector and can be any Fock state, for example, an

exact solution of the Scrodinger equation. The expectation value of a one-body operator O can be calculated with a simple sum involving the one-bodyreduced density matrix, which is defined as

= N|cc|N . (1.40)

By comparing to Eq. (1.30), it is easily seen that

N|O|N =

o = T r (O) . (1.41)

The diagonal matrix elements of the density matrix give the ex-pecteation value of the operator cc, which is interpreted as quantifying ofthe occupation of the single particle orbit in the state |N. From Eq. (1.5)one finds the obvious results that summing over all occupations must givethe total number of particles,

T r () =

= N . (1.42)

These results are particularly interesting since the theory of many-bodyGreens functions does not attempt any calculation of the full many-body

wave function. Rather the focus is on determining directly quantities raltedto the density matrices, which are calculated in terms of basic excitationmodes of the system. Thus, even if one does not compute the completeground state wave function, Eqs. (1.41) and (1.42) tell us that it is stillpossible to extract the expectation values of interesting observables.

It is also useful to insert the complete set of eigenstates {|N1k } of the(N-1)-particle system into (1.40)

= cc =k

N1k |c|N2 . (1.43)

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This result is interesting because the overlap function

N1k

|c

|N

gives the

probablility amplitude that the system collapses into a state |N1k after aparticle has been removed from the state . This quantity can be probedin a measurment that involves the suddend ejection of a particle.4 One alsosees that what is observed in particle emission experiments to one final stateshould not be directly interpreted as occupation numbers, since Eq. (1.43)requires a sum over all possible final states.

In an analogous way, we introduce the two-body reduced density matrix

, = N|cccc|N . (1.44)With this definition the expectation value of a two-body Hamiltonian be-

comes5

N|H|N =

t +1

4

,v,

= T r (T) +1

4T r (V) . (1.45)

1.4 Pictures in Quantum Mechanics

The time evolution of a quantum mechanical system is determined by theScrodinger equation. There exist different approaches to keep track of time

dependence, which are commonly referred to as pictures. The three mostrelevant are the Schrodinger, the Heisenberg and the interaction (also calledDirac) pictures. The last two are important for our discussions because theyare used in the definition of Greens functions and to develop their expansionin Feynman diagrams.

In the Scrodinger picture the wave function carries all the time depen-dence, as described by the corresponding equation,

ihd

dt|S(t) = H |S(t) . (1.46)

If one knows the state of the system |t0 = |S

(t = t0) at time t0 theevolution at later times is fixed by Eq. (1.46). This can be formally solvedto give the result

|S(t) = U|t0 , (1.47)4This is only a first order approximation to the measured cross section and care must

be taken to undertsand additional effects, such as final state interactions. Nevertheless,knock out experiments are one of the best tools available to understand the many-bodydynamics of a system.

5Note that the factor 14

appears since we are using fully symmetrized (or antisym-metrized) matrix elements ofV, Eq. (1.35).

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where U is the time evolution operator

U U(t, t0) = eiH(tt0)/h . (1.48)

The idea of the Heisenberg picture is simply the opposite: keeping thewave function constant at the time t0 while the operators evolve. Since U is aunitary operator, one simply inverts the time propagation of the Schrodingerstate and instead applies it to the operator,

|H = U |S(t) = |t0 , (1.49)OH(t) = U OS U . (1.50)

With these definitions the expectation values of Heisenberg operators remainunchanged and commutation rules evolve according to Eq. (1.50), as long asthey are evaluated at equal times [AH(t), BH(t)] = U

[AS, BS] U. Theevolution of Heisenberg operators is given by

ihd

dtOH(t) =

OH(t), H

. (1.51)

which is valid only for time-independent Schrodinger operators (OS = OS(t)).Note that the Heisenberg Hamiltonian does not depend of the time since it

commutes with itself.In the interaction picture is a hybrid between the former two. In thiscase one splits the hamiltonian in two parts, H = H0 + H1. One drivesthe evolution of the operators and the other the evolution of wave functions.Let H0 be the part that applies to the operators. Thus one defines thecorresponding time evolution operator

U0 U0(t, t0) = eiH0(tt0)/h , (1.52)

which is applied to the operators and used to partially invert the evolutionof the Schrodinger state (the correct expression for U is given further below),

|I(t) = U0 |S(t) = U|t0 , (1.53)OI(t) = U0 O

S U0 . (1.54)

The corresponding equations for time evolution are

ihd

dt|I(t) = HI1 (t) |I(t) , (1.55)

ihd

dtOI(t) =

OI(t), H0

, (1.56)

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where H0 reamins time independent but HI1 (t) evolves according to

HI1 (t) = U0 H1 U0 . (1.57)

Normally is it assumed that H0 is simple enough to allow for an exactsolution of the many-body problem. The remaining part HI1 (t) (possibly asmall perturbation) may then be used to evolve |I(t). This last correctionleads to the exact solution of the problem. However, HI1 (t) is now dependentof time and Eq. (1.55) one cannot be solved for U by simply exponentiatingas done for U and U0. The formal solution for the time evolution operatorof a state in interaction pictue can be found in standard textboooks. Here,we just show the result,

U(t t0)= 1 +

n=1

1

n!

ih

n tt0

dt1

tt0

dt2 tt0

dtnTHI1 (t1)H

I1 (t2) HI1 (tn)

= expi

h

tt0

dtTHI1 (t

)

, (1.58)

where the last line is used as a symbolic notation for the one above andT[ ] is the time ordering operator, defined in such a way that the latesttime apperas at the far left.

The expansion (1.58) is of particular importance since it is central in

applying perturbation theory to Greens functions and to derive the cor-responding rules for Feynman diagrams. The interaction picture has alsoanother powerful application in quantum field theory: by applying a smallfictitious external perturbation to the system, it is possible to derive usefulrelations among Greens functions. This apporach leads to self-consistentequations for the propagators and shows how to construct approximations ofpropagators that satisfy conservation laws. We will discuss these points inbetter details later on.

1.5 Exercises

Derive equations (1.33) for the two-body operator in second quantiza-tion.

Derive the Hartee-Fock equations in second quantization Derive the matrix elements of the Hamiltonian H between the 1p and

2p1h configurations.

Derive the matrix elements of H between the 1h and 2h1p states.

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Chapter 2

Basics of Many-Body Greens

Functions

2.1 Propagation of One Particle

Let us consider a particle in free space described by a single particle Hamil-tonian h1. Its eigenstates and eigenenergies are

h1|n = n|n . (2.1)

In general, if we put the particle in one of its |n orbits, it will remainin the same state forever. Instead, we immagine to prepare the system in ageneric state |tr (tr stands for trial) and then follow its time evolution.If the trial state is created at time t=0, the wavefunction at a later time t isgiven by [see Eq. (1.47)]

|(t) = eih1t/h|tr=

n

|neint/hn|tr . (2.2)

The second line shows that if one knows the eigentstates

|n

, it is rel-

atively simple to compute the time evution: one expands |tr in this basisand let every component propagate independently. Eventually, at time t, wewant to know the probability amplitude that a measurement would find theparticle at position r,

r|(t) = r|eih1t/h|tr=

drr|eih1t/h|rr|tr

=

drn

r|neint/hn|rr|tr (2.3)

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drG(r, r; t)tr(r

) ,

which defines the propagator G. Obviously, once G(r, r; t) is known it canbe used to calculate the evolution of any initial state. However,there ismore information included in the propagator. This is apparent from theexpansion in the third line of Eq (2.3): first, the braket n|r = n|(r)|0gives us the probability that putting a particle at position r and mesuring itsenergy rgiht away, would make the system to collapse into the eigenstate |n.Second, the time evolution is a superposition of waves propagating withdifferent energies and could be inverted to find the eigenspectrum. Immaginean experiment in which the particle is put at position r and picked up at r

after some time t. If one can do this for different positions and elapsed timesand with good resolutionthen a Fourier transform would simply give backthe full eigenvalue spectrum. Such an experiment is a lot of work to carryout! But would give us complete information on our particle.

We now want to apply the above ideas to see what we can learn by addingand removing a particle in an environment when many others are present.This can cause the particle to behave in an unxepected way, induce collectiveexcitations of the full systems, and so on. Moreover, the role played by thephysical vacuum in the above example, is now taken by an many-body state(usually its ground state). Thus, it is also possible to probe the system by

removing particles.

2.2 One-Body Greens Function

In the following we consider the Heisenberg description of the field operators,

s(r, t) = eiHt/h s(r) e

iHt/h , (2.4)

where the subscript s serves to indicate possible internal degrees of free-dom (spin, isospin, etc...). We omit the superscrips H (Hiesenberg) and S

(Scrodinger) from the operators since the two pictures can be distinguishedfrom the presence of the time variable, which appears only in the first case.Similarly,

s(r, t) = eiHt/h s(r) e

iHt/h , (2.5)

For the case of a general single-particle basis {u(r)} one uses the followingcreation and annihilation operators

c(t) = eiHt/h c e

iHt/h , (2.6)

c(t) = eiHt/h c e

iHt/h , (2.7)

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which are related to s(r, t) and s(r, t) through Eqs. (1.14) and (1.15).

In most applications the Hamiltonian is split in a unperturbed part H0and a residual interaction

H = H0 + V . (2.8)

The N-body eigenstates of the full Hamiltonian are indicated with |Nn ,while |Nn are the unperturbed ones

H |Nn = ENn |Nn , (2.9)H0 |Nn = E(0),Nn |Nn , (2.10)

The definitions given in the following are general and do not depend on

the type of interaction being used. Thus, most properties of Greens functionsresult from genaral principles of quantum mechanics and are valid for anysystem.

2.2.1 Definitions

The two-points Greens function describies the propagation of one particleor one hole on top of the ground state |N0 . This is defined by

gss(r, t; r, t) = i

hN0 |T[s(r, t)s(r, t)]|N0 , (2.11)

where T[ ] is the time ordering operator that imposes a change of sing foreach exchange of two fermion operators

T[s(r, t)s(r

, t)] =

s(r, t)

s(r

, t) , t > t ,

s(r, t)s(r, t) , t > t ,(2.12)

where the upper (lower) sign is for bosons (fermions). A similar definitioncan be given for the non interacting state |N0 , in this case the Heisenbergoperators (2.4) to (2.7) must evolve only according to H0 and the notationg(0) is used.

If the Hamiltonian does not depend on time, the propagator (2.11) de-pends only on the difference t t

gss(r, r; t t) = i

h(t t)N0 |s(r)ei(HE

N

0)(tt)/hs(r

)|N0

ih

(t t)N0 |s(r)ei(HEN

0)(tt)/hs(r)|N0 . (2.13)

In this case it is useful to Froutier transform with respect to time and define

gss(r, r; ) =

d eigss(r, r

; ) . (2.14)

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By using the relation

() = lim0+ 12i

+

dei

i , (2.15)

one obtains

gss(r, r; ) = gpss(r, r

; ) + ghss(r, r; )

= N0 |s(r)1

h (H EN0 ) + is(r

)|N0 (2.16)

N0 |s(r)1

h + (H

EN0 )

i

s(r)|N0 ,

In Eq. (2.16), gp propagates a particle from r to r, while gh propagates ahole from r to r. Note that the interpretation is that a particle is addedat r, and later on some (indistiguishable) particle is removed from r (andsimilarly for holes). In the mean time, it is the fully correlated (N 1)-body system that propagates. We will discuss in the next chapter that inmany casesand especially in the vicinity of the Fermi surfacethis motionmantains many characteristics that are typical of a particle moving in freespace, even if the motion itself could actually be a collective excitation ofmany constituents. But since it looks like a single particle state we may still

refer to it asquasiparticle

.The same definitions can be made for anyorthonormal basis {}, leadingto the realtions

g(t, t) = i

hN0 |T[c(t)c(t)]|N0 , (2.17)

wheregss(r, t; r

, t) =

u(r, s)g(t, t)u(r

, s) , (2.18)

and

g() = N0 |c1

h (H EN0 ) + i c|N0 (2.19)

N0 |c1

h + (H EN0 ) ic|N0 .

Equations (2.17) and (2.19) are completely equivalent to the previous ones.These may look a bit more abstract than the corresponding Eqs. (2.11)and (2.16) but are more general since they show that the formalism can bedeveloped and applied in any orthonormal basis, without restricting oneselfto coordindate space.

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2.2.2 Lehmann Representation

As discussed in Sec. 2.1 for the one particle case, the information containedin the propagators becomes more clear if one Fourier transforms the timevariable and inserts a completness for the intermediate states. This is sobecause it makes the spectrum and the transition amplitudes to apper ex-plicitely. Using the completeness relations for the (N 1)-body systems inEq. (2.19), one has

g() =n

N0 |c|N+1n N+1n |c|N0 h (EN+1n EN0 ) + i

(2.20)

k

N0 |c

|N1k

N1k |c|

N0

h (EN0 EN1k ) i.

which is known as the Lehmann repressentation of a many-body Greensfunction1. Here, the first and second terms on the left hand side describe thepropagation of a (quasi)particle and a (quasi)hole excitations.

The poles in Eq. (2.20) are the energies relatives to the |N0 ground state.Hence they give the energies actually relased in a capture reaction experimentto a bound state of |N+1n . The residues are transition amplitudes for theaddition of a particle and take the name of spectroscopic amplitudes. Theyplay the same role of the

n|

r

wave function in Eq. (2.3). In fact theseenergies and amplitudes are solutions of a Schrodinger-like equation: theDyson equation. The hole part of the propagator gives instead informationon the process of particle emission, the poles being the exact energy absorbedin the process. For example, in the single particle Greens function of amolecule, the quasiparticle and quasihole poles are respectively the electronaffinities an ionization energies.

We will look at the physical significance of spectroscopic amplitudes inthe next Chapter and derive the Dyson equation (which is the fundamentalequation in many-body Greens function theory) only later on, when devel-opin the formalism.

2.2.3 Spectral function and dispersive relation

As a last definition, we rewrite the contents of Eq. (2.20) in a form that cancompared more easily to experiments. By using the relation

1

x i = P1

x i(x) , (2.21)

1 H. Lehmann, Nuovo Cimento 11, 324 (1954).

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it is immediate to extract the one-body spectral function

S() = Sp() + S

h() , (2.22)

where the partcle and hole components are

Sp() = 1

Im gp()

=n

N0 |c|N+1n N+1n |c|N0

h (EN+1n EN0 )

, (2.23)

Sh() =1

Im gh()

= k

N0 |c|N1k N1k |c|N0 h (EN0 EN1k ) . (2.24)The diagonal part of the spectral function is interpreted as the probabilityof adding [Sp()] or removing [S

h()] one particle in the state leaving

the residual system in a state of energy .By comparing Eqs. (2.23) and (2.24) to the Lehmann representation (2.20),

it is seen that the propagator is completely constrained by its imaginary part.Indeed,

g() = d

Sp()

+ i+ d

Sh(

)

i

. (2.25)

In general the single particle propagator of a finite system has isolatedpoles in correspondence to the bound eigenstates of the (N+1)-body system.For larger enegies, where |N+1n are states in the continuum, it develops abranch cut. The particle propagator gp() is analytic in the upper half ofthe complex plane, and so is the full propagator (2.16) for EN+10 EN0 .Analogously, the hole propagator has poles for EN0 EN10 and is analyticin the lower complex plane. Note that high excitation energies in the (N-1)-body system correspond to negative values of the poles EN0 EN1k , so gh()develops a branch cut for large negative energies.

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2.3 Observables from g

2.3.1 Calculation of Expectation Values

The one-body density matrix (1.40) can be obtained from the one-body prop-agator. One simply chooses the appropriate time ordering in Eq. (2.17)

= N0 |cc|N0 = ih limtt+ g(t, t) (2.26)(where the upper sign is for bosons and the lower one is for fermions). Al-ternatively, the hole spectral function can be used

= d Sh

() . (2.27)

Thus, the expectation value of a one-body operator, Eq. (1.41), on the groundstates |N0 is usually written in one the following ways

N0 |O|N0 =

d o S

h()

= ih limtt+

o g(t, t) (2.28)

which are equivalent.From the particle spectral function, one can extract the quantity

d = N0 |cc|N0 =

d Sp() (2.29)

which leads to the following sum ruledS() = d = N0 |[c, c]|N0 = . (2.30)

2.3.2 Sum Rule for the Energy

For the case of an Hamiltonian containing only two-body interactions,

H = U + V

=

t cc +

1

4

v, cccc , (2.31)

there exist an important sum rule that relates the total energy of the state|N0 to its one-body Greens function. To derive this, one makes use of theequation of motion for Heisenberg operators (1.51), which gives

ihd

dtc(t) = e

iHt/h [c, H] eiHt/h , (2.32)

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with2

[c, H] =

t c + 12

vccc , (2.33)

which is valid for both fermions and bosons.If one uses Eq. (2.33) and derives the propagator (2.17) with respect to

time,

ih

tg(t t) = (t t) +

tg(t t)

ih

1

2v, N0 | T[c(t)c(t)c(t)c(t)]|N0 . (2.34)

The braket in the last line contains the four points Greens function [seeEq. (2.38)], which can describe the simultaneous propagation of two parti-cles. Thus, one sees that applying the equation of motion to a propagatorleads to relations which contain Greens functions of higher order. This resultis particularly important because it shows there exist a hierarchy betweenpropagators, so that the exact equations that determine the one-body func-tion will depend on the two-body one, the two-body function will containcontributions from three-body propagators, and so on.

For the moment we just want to select a particular order of the operatorsin Eq. (2.34) in order to extract the one- and two-body density matrices. To

do this, we chose t to be a later time than t and take its limit to the latterfrom above. This yields

ih limtt+

tg(t t) = T + 2V (2.35)

(note that for t = t, the term (t t)=0 and it does not contribute to thelimit). This result can also be expressed in energy representation by invertingthe Fourier transformation (2.14), which gives

lim0

g() =

d Sh() (2.36)

By combining (2.35) with Eq. (2.28) one finally obtains

H = U + V = ih 12

limtt+

t+ t

g(t t)

= 12

d { + t} Sh() . (2.37)

2We use the relation [A,BC] = [A,B]C B[C,A] = {A,B}C B{C,A} which isvalid for both commutators and anticommutators

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Surprisingly, for an Hamiltonian containing only two-body forces it is possible

to extract the ground state energy by knowing only the one-body propaga-tor. This result was derived independently by Galitski and Migdal3 and byKolutn4. When interactions among three or more particles are present, thisrelation has to be augmented to include additional terms. In these caseshigher order Greens functions will appear explicitly.

2.4 Higher Order Greens Functions

The definition (2.17) can be extended to Greens functions for the propa-gation of more than one particle. In general, for each additional particle itwill be necessary to introduce one additional creation and one annihilationoperator. Thus a 2n-points Greens function will propagate a maximum ofn quasiparticles. The explicit definition of the 4-points propagator is

g4pt,(t1, t2; t1, t

2) =

i

hN0 |T[c(t2)c(t1)c(t1)c(t2)]|N0 , (2.38)

while the 6-point case is

g6pt,(t1, t2, t3; t1, t

2, t

3) =

i

hN0 |T[c(t3)c(t2)c(t1)c

(t

1)c

(t

2)c

(t

3)]|N0 , (2.39)

It should be noted that the actual number of particles that are propagatedby these objects depends on the ordering of the time variables. Therefore theinformation on transitions between eigenstates of the systems with N, N 1and N 2 bodies are all encoded in Eq. (2.38), while additional states ofN 3-body states are included in Eq. (2.38). Obviously, the presence of somany time variables makes the use of these functions extremely difficult (andeven impossible, in many cases). However, it is still useful to consider onlycertain time orderings which allow to extract the information not included

in the 2-point propagator.

2.4.1 Two-particlestwo-holes Propagator

The Two-particletwo-hole propagator is a two-times Greens function de-fined as

gII,(t, t) = i

hN0 |T[c(t)c(t)c(t)c(t)]|N0 , (2.40)

3V. M. Galitski and A. B. Migdal, Sov. Phys.-JEPT 7, 96 (1958).4D. S. Koltun, Phys. Rev. Lett. 28, 182 (1972); Phys. Rev. C 9, 484 (1974)

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which corresponds to the limit t1 = t+2 and t2 = t

+1 of g

4pt.

As for the case ofg(t, t), if the Hamiltonian is time-independent, Eq. (2.40)is a function of the time difference only. Therefore it has a Lehmann repre-sentation containing the exact spectrum of the (N 2)-body systems

gII,() =n

N0 |cc|N+2n N+2n|cc|N0 (EN+2n EN0 ) + i

k

N0 |cc|N2k N2k |cc|N0

EN0 EN2k

i

. (2.41)

Similarly one defines the two-particle and two-hole spectral functions

SII, () = Spp,() + S

hh,() , (2.42)

and

Spp,() = 1

Im gpp,()

=n

N0 |cc|N+2n N+2n |cc|N0

h (EN+2n EN0 )

, (2.43)

Shh,() =1

Im ghh,()

= k

N0 |cc|N2k N2k |cc|N0

h (EN0 EN2k )

. (2.44)

Following the demonstration of Sec. 2.3.1, it is immediate to obtain rela-tions for the two-body density matrix (1.44)

, = N|cccc|N =

dShh,() (2.45)

and, hence, for the expectation value of any two-body operator

N0 |V|N0 =

dv,S

h,()

= +ih limtt+1

4

v,gII,(t, t

) . (2.46)

2.4.2 Polarization Propagator

The polarization propagator , corresponds to the time ordering of g4pt

in which a particle-hole excitation is created at one single time. Therefore,

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no process involving particle transfer in included. However it describes tran-

sition to the excitations of the system, as long as they can be reached with aone-body operator. For example, this includes collective modes of a nucleus.This is defined as

,(t, t) = i

hN0 |T[c(t)c(t)c(t)c(t)]|N0

+i

hN0 |cc|N0 N0 |cc|N0 . (2.47)

After including a completeness of |Nn states in (2.47), the contributionof to the ground states (at zero energy) is cancelled by the last term in the

equation. Thus one can Fourier transform to the Lehmann representation

,() =n=0

N0 |cc|Nn Nn |cc|N0 (ENn EN0 ) + i

n=0

N0 |cc|Nn Nn |cc|N0 + (ENn EN0 ) i

, (2.48)

Note that ,() = ,() due to time reversal symmetry. Also theforward and backward parts carry the same information.

Once again, the residues of the propagator (2.48) can be used to calculateexpectation values. In this case, given a one-body operator (1.30) on obtainsthe transition matrix elements to any excited state

Nn |O|N0 =

oNn |cc|N0 . (2.49)

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Chapter 3

Relation to Experimental Data

In this chapter we will partially explore the connection between the informa-tion contained in various propagators and experimental data. The focus ison the experimental properties that are probed by the removal of particles.Also, from now on, we will only consider fermionic systems.

An important case is when the spectrum for the N 1-particle systemnear the Fermi energy involves discrete bound states. This happens in finitesystem like nuclei or molecules. In these cases the main quantity of interestis the overlap wave function, which appears in the residues of Eq. (2.20) andin Eq. (2.24). This is

overlapk (r) = N1k |s(r)|N0 =

N

dr2

dr3

drN (3.1)

[N1k (r2, r3, . . . rN)]N0 (r1, r2, r3, . . . rN) .The second line in Eq. (3.1) can be proved by using relations (1.19) and (1.20).This integral comes out in the description of most particle knock out processesbecause it represents the matrix element between the initial and final states,in the case when the emitted particle is ejected with energy large enough theit interacts only weakly with the residual system. The quantity of interesthere is the so called spectroscopic factor to the final state k,

Sk =

dr|overlapk (r)|2 . (3.2)

When the system is made of completely non interacting particles, Sk is unity.In real cases however, correlations among the constituents reduce this value.The possibility of extracting this quantity from experimental data gives usinformation on the spectral function and therefore on the structure of thecorrelated system.

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3.0.3 Spectroscopic strength from particle emission

In order to make the connection with experimental data obtained from knock-out reactions, it is useful to consider the response of a system to a weak probe.The hole spectral function introduced in Eq. (2.24) can be substantially ob-served these reactions. The general idea is to transfer a large amount ofmomentum and energy to a particle of a bound system in the ground state.This is then ejected from the system, and one ends up with a fast-movingparticle and a bound (N 1)-particle system. By observing the momentumof the ejected particle it is then possible the reconstruct the spectral functionof the system, provided that the interaction between the ejected particle andthe remainder is sufficiently weak or treated in a controlled fashion, e. g. byconstraining this treatment with information from other experimental data.

We assume that the N-particle system is initially in its ground state,

|i = |N0 , (3.3)and makes a transition to a final N-particle eigenstate

|f = ap|N1n , (3.4)composed of a bound (N1)-particle eigenstate, |N1n , and a particle withmomentum p.

For simplicity we consider the transition matrix elements for a scalarexternal probe

(q) =N

j=1

exp(iq rj), (3.5)

which transfers momentum q to a particle. Suppressing other possible spquantum numbers, like e.g. spin, the second-quantized form of this operatoris given by

(q) =p,p

p| exp(iq r)|pap

ap =p

ap

apq. (3.6)

The transition matrix element now becomes

f|(q)|i =p

N1n |apapapq|N0

=p

N1n |p,papq + apapqap|N0

N1n |apq|N0 . (3.7)The last line is obtained in the so-called Impulse Approximation (or SuddenApproximation), where it is assumed that the ejected particle is the one that

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has absorbed the momentum from the external field. This is a very good

approximation whenever the momentum p of the ejectile is much larger thantypical momenta for the particles in the bound states; the neglected termin Eq. (3.7) is then very small, as it involves the removal of a particle withmomentum p from |N0 .

There is one other assumption in the derivation: the fact that the finaleigenstate of the N-particle system was written in the form of Eq. (3.4), i.e.a plane-wave state for the ejectile on top of an (N 1)-particle eigenstate.This is again a good approximation if the ejectile momentum is large enough,as can be understood by rewriting the Hamiltonian in the N-particle systemas

HN =Ni=1

p2i

2m+

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respectively, the PWIA knock-out cross section can be rewritten as

d n

(Emiss EN0 + EN1n )|N1n |apmiss|N0 |2

= Sh(pmiss, Emiss). (3.12)

The PWIA cross section is therefore exactly proportional to the diagonalpart of the hole spectral function defined in Eq. (2.24). This is of course onlytrue in the PWIA, but when the deviations of the impulse approximationand the effects of the final state interaction are under control, it is possibleto obtain precise experimental information on the hole spectral function ofthe system under study.

3.0.4 An example: the (e, ep) reaction

Several studies of reacton theory have been done in past years constrain andimprove the analysis of electron scattering reactions2. Although the actual(e,ep) experiments involve more complicated one-body excitation operatorsthan the one considered in the simple simple example above, the basic con-clusions are not altered3.

In the practical analysis of an (e,ep) experiment it is conventional to finda local potential well (mostly of Woods-Saxon type) which will generate a sp

state at the removal energy for the transition that is studied. This state isfurther required to provide the best possible fit to the experimental momen-tum dependence of the cross section (with proper inclusion of complicationsdue to electron and proton distortion). The overall factor necessary to bringthe resulting calculated cross section into agreement with the experimentaldata, can then be interpreted as the spectroscopic factor corresponding tothe experimental quasihole wave function according to Eq.(3.2).

The resulting cross sections obtained at the NIKHEF facility are shownfor four different nuclei in Fig. 3.14 It is important to realize that the shapesof the wave functions in momentum space correspond closely to the onesexpected on the basis of a standard Woods-Saxon potential well (or moreinvolved mf wave functions). This is itself an important observation sincethe (e,ep) reaction probes the interior of the nucleus, a feat not availablewith hadronically induced reactions.

While the shapes of the valence nucleon wave functions correspond to thebasic ingredients expected on the basis of years of nuclear structure physics

2See for example, S. Boffi, C. Giusti, F. D. Pacati, and M. Radici, ElectromagneticResponse of Atomic nuclei, Oxford Studies in Nuclear Physics (Clarendon, Oxford, 1996).

3S. Frullani and J. Mougey, Adv. Nucl. Phys. 14, 1 (1984).4L. Lapikas, Nucl. Phys. A553, 297c (1993).

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Figure 3.1: Momentum distributions for various nuclei obtained from the(e,ep) reaction performed at NIKHEF (Note n. 4, p. 28).

experience, there is a significant departure with regard to the integral of thesquare of these wave functions. This quantity is of course the spectroscopicfactor and is shown in Fig. 3.2 for the data obtained at NIKHEF. The resultsshown in Fig. 3.2 indicate that there is an essentially global reduction of thesp strength of about 35 % which needs to be explained by the theoreticalcalculations. This depletion is somewhat less for the strength associatedwith slightly more bound levels. An additional feature obtained in the (e,ep)reaction is the fragmentation pattern of these more deeply bound orbitals innuclei. This pattern is such that single isolated peaks are obtained only in theimmediate vicinity of the Fermi energy whereas for more deeply bound statesa stronger fragmentation of the strength is obtained with larger distance fromF. This is beautifully illustrated by the (e,e

p) data from Quint5. Whereasthe 3s1/2 orbit exhibits a single peak, there is a substantial fragmentation ofthe 1f strength as indicated in this figure. Additional information about theoccupation number of the former orbit is also available and can be obtainedby analyzing elastic electron scattering cross sections of neighboring nuclei.

5E. N. M. Quint, Ph.D. thesis, University of Amsterdam (1988).

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101 1020.0

0.2

0.4

0.6

0.8

target mass

S/(2j+1)

7Li

12C

16O 31P

40Ca

48Ca 90Zr

208Pb

1.0

Mean Field Theory

VALENCE PROTONS

Figure 3.2: Spectroscopic factors from the (e,ep) reaction as a function oftarget mass. Data have been obtained at the NIKHEF facility (Note n. 4,p. 28).

The actual occupation number for the 3s1/2 proton orbit obtained from thisanalysis is about 10% larger than the quasihole spectroscopic factor 6 and

therefore corresponds to 0.75. All these features of the strength need to beexplained theoretically. This will be attempted in the material covered inlater sections.

6P. Grabmayr et al., Phys. Lett. B164, 15 (1985). P. Grabmayr, Prog. Part. Nucl.Phys. 29, 251 (1992).

30