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Many-Body Green’s Functions

Green’s Functions Theory for Quantum Many-Body Systems


Contacts:

Carlo BarbieriTheoretical Nuclear physics LaboratoryRIKEN, Nishina Center

At RIKEN: RIBFビル, Room 405電話番号: 048-462-111 ext. 4324

Email: 名前@riken.jp, 名前=barbieri

Lectures website: http://ribf.riken.jp/~barbieri/mbgf.html

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mailto:名前@riken.jp�

mailto:名前@riken.jp�

http://ribf.riken.jp/~barbieri/mbgf.html�



Many-body Green's functions (MBGF) are a set of techniques that originated in quantum field theory but have then found wide applications to the many-body problem.

In this case, the focus are complex systems such as crystals, molecules, or atomic nuclei.

Development of formalism: late 1950s/ 1960s imported from quantum field theory

1970s – today applications and technical developments…


Purpose and organization

Many-body Green’s functions are a VAST formalism. They have a wide range of applications and contain a lot of information that is accessible from experiments.

Here we want to give an introduction:

Teach the basic definitions and results

Make connection with experimental quantities gives insight into physics

Discuss some specific application to many-bodies


Purpose and organization

Most of the material covered here is found on

W. H. Dickhoff and D. Van Neck, Many-Body Theory Exposed!,

(covers both formalism and recent applicationsvery large 700+ pages)

I will provide:

•notes on formalism discussed (partial)

•the slides of the lectures

Download from the website: http://ribf.riken.jp/~barbieri/mbgf.html


Literature

Books on many-body Green’s Functions:

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

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

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

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

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

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

• …


Literature

Recent reviews:

• F. Aryasetiawan and O. Gunnarsson, arXiv:cond-mat/9712013. GW method

• G. Onida, L. Reining and A. Rubio, Rev. Mod. Phys. 74, 601 (2002). comparison of TDDTF and GF

• H. Mϋther and A. Polls, Prog. Part. Nucl. Phys. 45, 243 (2000). Applications to • C.B. and W. H. Dickhoff, Prog. Part. Nucl. Phys. 52, 377 (2004). nuclear physics

(Some) classic papers on formalism:• G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961). • G. Baym, Phys. Rev. 127, 1391 (1962).

• L. Hedin, Phys. Rev. 139, A796 (1965).


Schedule (4 weeks)

Date Time Content (tentative)

4/6（月） 15:00-16:30 second quantization (review), definitions of GF

4/9（木） 14:00-15:30 Basic properties and sum rules

4/9（木） 16:00-17:30 Link to experimental quantities

4/13（月） 15:00-16:30 Equation of motion method, expansion of the self-energy

4/16（木） 13:30-15:00 Introduction to Feynman diagrams

4/16（木） 15:30-17:00 Self-consistency and RPA

week break

Basics and link tospectroscopy

Advanced formalism


Schedule (4 weeks)

Date Time Content (tentative)4/27（月） 15:00-16:30 RPA and GW method

4/27（木） 13:30-15:00 Particle-vibration coupling, applications for atoms and nuclei

4/27（木） 15:30-17:00 Systems of bosons

Golden week break

5/14（木） 13:30-15:00 Superfluidity, BCS/BEC cross over

5/14（木） 15:30-17:00 Cold atoms

5/18（月） 15:00-16:30 Finite temperature/nucleonic matter (time permitting)

Practical calculationsfor fermions

Bosons and other applications


• Green’s functions• Propagators names for the same objects• Correlation functions

• Many-body Green’s functions Green’s functions applied to the MB problem

• Self-consistent Green’s functions (SCGF) a particular approach to calculate GFs


GFMC と MBGF の違いは何ですか??

In Green’s Function Monte Carlo one starts with a “trial” wave function, and lets it propagate in time:

Better to break the time in many little intervals Δt,

For t-i∞, this goes to the gs wave function!

Green’s function (GF)

Monte Carlo integral (MC)

GFMC is a method to compute the exact wave function.(typically works for few bodies, A ≤ 12 in nuclei).



MBGF is a method that DO NOT compute the wave function:It assumes that the system is in its ground state and attempts at calculating simple excitation on from it directly

•Large N (number of particles)

•The N-body ground state plays the role of vacuum (of excitations)

•Degrees of freedom are a few particles (or holes) on top of this vacuum

•It is a microscopic method (and capable of “ab-initio” calculations)



Don’t get confused:

Green’s function Monte Carlo (GFMC) and

Many-body Green’s Functions

are NOT the same method!!!!!!


Em [MeV]

σred ≈ S(h)

10-50

0p1/20p3/2

0s1/2

One-hole spectral function -- example

correlations

distribution of momentum (pm) and energies (Em)

independentparticle picture

Saclay data for 16O(e,e’p)[Mougey et al., Nucl. Phys. A335, 35 (1980)]

∑ −− −−⟩ΨΨ⟨=n

An

Am

Ap

Anmm

h EEEcEpSm

))((||),( 10

20

1)( || δ


Examples of quasiparticles – Nuclei-I

The nuclear force has strong repulsive behavior at short distances

The short range core is:•required by elastic NN

scattering•supported by high-energy

electron scattering (Jlab)•and supported by Lattice-QCD

(Ishii now in 東大)

Repulsive core: 500 - 600 MeVAttractive pocket: about 30 MeV

Yukawa tail ∝ e-mr/r[From N. Ishii et al. Phys. Rev. Lett. 99, 022001 (2007)]


Examples of quasiparticles – Nuclei-II

Nucleons attract themselves at intermediate distances and scatter like billiard balls: Naively, nuclei cannot be treated as orbits structures

“BAD” model of a nucleus

“GOOD” model of a nucleus


Examples of quasiparticles – Nuclei-III

…BUT, understanding binding energies and magic number DOES require a shell structure!!!

Single particle orbits?

• M. G. Mayer, Phys. Rev. 75, 1969 (1949)• O. Haxel, J. H. D. Jensen and

H. E. Suess, Phys. Rev. 75, 1766 (1949)

Nobel prize (1 963)!


Examples of quasiparticles – ions in liquid

Ions in a liquid screen each other’s charge and interact weakly

[Picture adapted form Mattuck]

+

+

++

++

++

+ +

+ ---

-

--

--

--

-

--

-- -

--+

-

+- -

-++


Examples of quasiparticles – electron in gas

[Picture adapted form Mattuck]


Second quantization

Choose an orthonormal single-particle basis {α} and useit to build bases for the many-body states.E.g.,

Need states of different particle number N use the Fock space:

It must include the vacuum state:

=1 for fermions= ∞ for bosons


Second quantization

Basis states for bosons are constructed as

creation and annihilation operators give

Commutation rules:


Second quantization

Basis states for fermions are constructed as

creation and annihilation operators give

with:

Commutation rules:


Pictures in quantum mechanics

Consider an N-body system in a state at time t=t0.The time evolution operator is

Schrödinger pict. Heisenberg pict.timeevolutionequation:

solutions:

does not evolve

for a time-indep. OS

same time!


Propagating a free particle

Consider a free particle with Hamiltonian

h1 = t + U(r)the eigenstates and egienenergies are

The time evolution is

with: wave fnct. at t=0wave fnct. at time t


Green’s function (=propagator) for a free particle:

Propagating a free particlepo

siti

on

time

r1

r1’r2’ r3’

r2

r3


Green’s function (=propagator) for a free particle:

states

energies

Propagating a free particle

Fourier transform of the eigenspectrum!

The spectrum of the Hamiltonian is separated by the FT because the time evolution is driven by H:


Definitions of Green‘s functions

Take a generic the Hamiltonian H and its static Schrödinger equation

We evolve in time the field operators instead of the wave function by using the Heisenberg picture

( creation/annihilation of a particle in r at time t)



The one body propagator (≡Green’s function) associated to the ground state is defined as

with the time ordering operator

Expand t-dep in operators:

(+ for bosons, - for fermions)

aadds a particleremoves a particle



With explicit time dependence:

r r’

t’ t

rr’

t t

adds a particle

removesa particle



Green’s function can be defined in any single-particle basis (not just r or k space). So let’s call {α} a general orthonormal basis with wave functions {uα(r)}

The Heisenberg operators are:

and



In general it is possible to define propagators for more particles and different times:

…


Definitions of Green‘s functionsGraphic conventions:

tim

e

≡ gαβ(t>t’)(quasi)particle

≡ gαβ(t’>t)(quasi)hole

α

βα

β

α

gαβ,γδ4-pt

δγ

β

t1 ’

t2’

t1

t2

p; (N+1)-body

p; (N+1)-body

2p (N+2)-body


Definitions of Green‘s functionsGraphic conventions:

tim

e

≡ gαβ(t>t’)(quasi)particle

≡ gαβ(t’>t)(quasi)hole

α

βα

β

α

gαβ,γδ4-pt

δ

γ

β

t1 ’

t2’

t1

t2

p; (N+1)-body

h; (N-1)-body

ph N-body



With explicit time dependence:

r‘ r

t’ t

r’r

t t

adds a particle

removesa particle


Lehmann representation and spectral function

Expand on the eigenstates of N±1

Fourier transform to energy representation…

(- bosons, + fermions)



The Lehman representation of gαβ(ω) is:

Poles energy absorbed/released in particle transfer

Residues: particle addition

particle ejected

(quasi)particles

(quasi)holes



The Lehman representation of gαβ(ω) is:

To extract the imaginary part:

(quasi)particles

(quasi)holes(- bosons, + fermions)



The spectral function is the Im part of gαβ(ω)

Contains the same information as the Lehmann rep.

(quasi)particles

(quasi)holes

(- bosons, + fermions)



gαβ(ω) is fully constrained by its imaginary part:

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