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Many-Body Green’s Functions
Green’s Functions Theory for Quantum Many-Body Systems
Many-Body Green’s Functions
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
==
Many-Body Green’s Functions
Many-Body Green’s Functions
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…
Many-Body Green’s Functions
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
Many-Body Green’s Functions
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
Many-Body Green’s Functions
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)
• …
Many-Body Green’s Functions
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).
Many-Body Green’s Functions
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
Many-Body Green’s Functions
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
Many-Body Green’s Functions
• 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
Many-Body Green’s Functions
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).
Many-Body Green’s Functions
GFMC と MBGF の違いは何ですか??
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)
Many-Body Green’s Functions
GFMC と MBGF の違いは何ですか??
Don’t get confused:
Green’s function Monte Carlo (GFMC) and
Many-body Green’s Functions
are NOT the same method!!!!!!
Many-Body Green’s Functions
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
Many-Body Green’s Functions
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)]
Many-Body Green’s Functions
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
Many-Body Green’s Functions
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)!
Many-Body Green’s Functions
Examples of quasiparticles – ions in liquid
Ions in a liquid screen each other’s charge and interact weakly
[Picture adapted form Mattuck]
+
+
++
++
++
+ +
+ ---
-
--
--
--
-
--
-- -
--+
-
+- -
-++
Many-Body Green’s Functions
Examples of quasiparticles – ions in liquid
Ions in a liquid screen each other’s charge and interact weakly
[Picture adapted form Mattuck]
+
+
++
++
++
+ +
+ ---
-
--
--
--
-
--
-- -
--+
-
+- -
-++
Many-Body Green’s Functions
Examples of quasiparticles – ions in liquid
Ions in a liquid screen each other’s charge and interact weakly
[Picture adapted form Mattuck]
+
+
++
++
++
+ +
+ ---
-
--
--
--
-
--
-- -
--+
-
+- -
-++
Many-Body Green’s Functions
Examples of quasiparticles – electron in gas
[Picture adapted form Mattuck]
Many-Body Green’s Functions
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
Many-Body Green’s Functions
Second quantization
Basis states for bosons are constructed as
creation and annihilation operators give
Commutation rules:
Many-Body Green’s Functions
Second quantization
Basis states for fermions are constructed as
creation and annihilation operators give
with:
Commutation rules:
Many-Body Green’s Functions
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!
Many-Body Green’s Functions
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
Many-Body Green’s Functions
Green’s function (=propagator) for a free particle:
Propagating a free particlepo
siti
on
time
r1
r1’r2’ r3’
r2
r3
Many-Body Green’s Functions
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:
Many-Body Green’s Functions
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)
Many-Body Green’s Functions
Definitions of Green‘s functions
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
Many-Body Green’s Functions
Definitions of Green‘s functions
With explicit time dependence:
r r’
t’ t
rr’
t t
adds a particle
removesa particle
Many-Body Green’s Functions
Definitions of Green‘s functions
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
Many-Body Green’s Functions
Definitions of Green‘s functions
In general it is possible to define propagators for more particles and different times:
…
Many-Body Green’s Functions
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
Many-Body Green’s Functions
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
Many-Body Green’s Functions
Definitions of Green‘s functions
With explicit time dependence:
r‘ r
t’ t
r’r
t t
adds a particle
removesa particle
Many-Body Green’s Functions
Lehmann representation and spectral function
Expand on the eigenstates of N±1
Fourier transform to energy representation…
(- bosons, + fermions)
Many-Body Green’s Functions
Lehmann representation and spectral function
The Lehman representation of gαβ(ω) is:
Poles energy absorbed/released in particle transfer
Residues: particle addition
particle ejected
(quasi)particles
(quasi)holes
Many-Body Green’s Functions
Lehmann representation and spectral function
The Lehman representation of gαβ(ω) is:
To extract the imaginary part:
(quasi)particles
(quasi)holes(- bosons, + fermions)
Many-Body Green’s Functions
Lehmann representation and spectral function
The spectral function is the Im part of gαβ(ω)
Contains the same information as the Lehmann rep.
(quasi)particles
(quasi)holes
(- bosons, + fermions)
Many-Body Green’s Functions
Lehmann representation and spectral function
gαβ(ω) is fully constrained by its imaginary part: