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Many ny-Body
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Greens Functions Theory for Quantum Many - Body Systems Many - - PowerPoint PPT Presentation
Greens Functions Theory for Quantum Many - Body Systems Many - - PowerPoint PPT Presentation
Greens Functions Theory for Quantum Many - Body Systems Many ny-Body ody Green ens Function ons Contacts: Carlo Barbieri Theoretical Nuclear physics Laboratory RIKEN, Nishina Center At RIKEN: RIBF , Room 405
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Many-Body Green’s Functions
Many-body Green's functions (MBGF) are a set of techniques that
- riginated 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…
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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
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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 applications very 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
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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)
- …
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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).
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Schedule (4 weeks)
Da Date Time me Conte tent ( t (te tenta tativ tive) 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 to spectroscopy Advanced formalism
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Schedule (4 weeks)
Da Date Time me Conte tent ( t (te tenta tativ tive) 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 calculations for fermions Bosons and
- ther
applications
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- 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
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GFM GFMC と MBGF GF 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).
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GFM GFMC と MBGF GF 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)
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GFM GFMC と MBGF GF 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!!!!!!
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Em [MeV]
σred ≈ S(h)
10-50
0p1/2 0p3/2 0s1/2
One-hole spectral function -- example
correlations distribution of momentum (pm) and energies (Em)
independent particle picture
Saclay data for 16O(e,e’p) [Mougey et al., Nucl. Phys. A335, 35 (1980)]
∑
− −
− − 〉 Ψ Ψ 〈 =
n A n A m A p A n m m h
E E E c E p S
m
)) ( ( | | ) , (
1 2 1 ) (
| |
δ
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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 MeV Attractive pocket: about 30 MeV Yukawa tail ∝ e-mr/r
[From N. Ishii et al. Phys. Rev. Lett. 99, 022001 (2007)]
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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
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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 p prize ( (1 963) 963)!
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Examples of quasiparticles – ions in liquid
Ions in a liquid screen each other’s charge and interact weakly
[Picture adapted form Mattuck]
+ + + + + + + + + + +
- +
- +
- +
+
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Examples of quasiparticles – ions in liquid
Ions in a liquid screen each other’s charge and interact weakly
[Picture adapted form Mattuck]
+ + + + + + + + + + +
- +
- +
- +
+
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Examples of quasiparticles – ions in liquid
Ions in a liquid screen each other’s charge and interact weakly
[Picture adapted form Mattuck]
+ + + + + + + + + + +
- +
- +
- +
+
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Examples of quasiparticles – electron in gas
[Picture adapted form Mattuck]
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Second quantization
Choose an orthonormal single-particle basis {α} and use it 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
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Second quantization
Basis states for bosons are constructed as creation and annihilation operators give Commutation rules:
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Second quantization
Basis states for fermions are constructed as creation and annihilation operators give with: Commutation rules:
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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.
time evolution equation: solutions: does not evolve for a time-indep. OS same time!
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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=0 wave fnct. at time t
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Green’s function (=propagator) for a free particle:
Propagating a free particle
position time r1 r1’ r2’ r3’ r2 r3
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Green’s function (=propagator) for a free particle: states energies
Propagating a free particle
Fourier transform
- f the eigenspectrum!
The spectrum of the Hamiltonian is separated by the FT because the time evolution is driven by H:
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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)
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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 particle removes a particle
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Definitions of Green‘s functions
With explicit time dependence: r r’ t’ t r r’ t t adds a particle removes a particle
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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
- rthonormal basis with wave functions {uα(r)}
The Heisenberg operators are: and
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Definitions of Green‘s functions
In general it is possible to define propagators for more particles and different times:
…
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Definitions of Green‘s functions
Graphic conventions:
time
≡ gαβ(t>t’)
(quasi)particle
≡ gαβ(t’>t)
(quasi)hole
α β α
β
α
gαβ,γδ
4-pt
δ γ β
t1’ t2’ t1 t2
p; (N+1)-body p; (N+1)-body 2p (N+2)-body
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Definitions of Green‘s functions
Graphic conventions:
time
≡ gαβ(t>t’)
(quasi)particle
≡ gαβ(t’>t)
(quasi)hole
α β α
β
α
gαβ,γδ
4-pt
δ γ β
t1’ t2’ t1 t2
p; (N+1)-body h; (N-1)-body ph N-body
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Definitions of Green‘s functions
With explicit time dependence: r‘ r t’ t r’ r t t adds a particle removes a particle
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Lehmann representation and spectral function
Expand on the eigenstates of N±1 Fourier transform to energy representation…
(- bosons, + fermions)
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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
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Lehmann representation and spectral function
The Lehman representation of gαβ(ω) is: To extract the imaginary part:
(quasi)particles (quasi)holes
(- bosons, + fermions)
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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)
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