Green’s Functions Theory for Quantum Many - Body Systems Many ny-Body ody Green’ en’s Function ons
Contacts: Carlo Barbieri Theoretical Nuclear physics Laboratory RIKEN, 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 ny-Body ody Green’ en’s Function ons ＝＝
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 ny-Body ody Green’ en’s Function ons
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 ny-Body ody Green’ en’s Function ons
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 Many ny-Body ody Green’ en’s Function ons
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 ny-Body ody Green’ en’s Function ons
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. M ϋ ther 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 ny-Body ody Green’ en’s Function ons
Schedule (4 weeks) Da Date Time me Conte tent ( t (te tenta tativ tive) 4/6 （月） 15:00-16:30 second quantization (review), Basics and definitions of GF link to 4/9 （木） spectroscopy 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 Advanced formalism 4/16 （木） 13:30-15:00 Introduction to Feynman diagrams 4/16 （木） 15:30-17:00 Self-consistency and RPA week break Many ny-Body ody Green’ en’s Function ons
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 Practical calculations 4/27 （木） 13:30-15:00 Particle-vibration coupling, for fermions applications for atoms and nuclei 4/27 （木） 15:30-17:00 Systems of bosons Golden week break Bosons and 5/14 （木） 13:30-15:00 Superfluidity, other BCS/BEC cross over applications 5/14 （木） 15:30-17:00 Cold atoms 5/18 （月） 15:00-16:30 Finite temperature/nucleonic matter (time permitting) Many ny-Body ody Green’ en’s Function ons
• 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 ny-Body ody Green’ en’s Function ons
GFMC と GFM 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: For t -i ∞ , this goes to the gs wave function ! Better to break the time in many little intervals Δ t, 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 ny-Body ody Green’ en’s Function ons
GFMC と GFM 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) Many ny-Body ody Green’ en’s Function ons
GFMC と GFM 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 !!!!!! Many ny-Body ody Green’ en’s Function ons
One- hole spectral function -- example σ red ≈ S (h) independent particle picture 10-50 correlations 0p 1/2 0p 3/2 0s 1/2 Saclay data for 16 O(e,e’p) E m [MeV] [Mougey et al., Nucl. Phys. A335, 35 (1980)] ∑ − − = | 〈 Ψ Ψ 〉 | δ − − ( ) 1 2 1 h A A A A ( , ) | | ( ( )) S p E c E E E 0 0 m m n m n p m n distribution of momentum ( p m ) and energies ( E m ) Many ny-Body ody Green’ en’s Function ons
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)] Many ny-Body ody Green’ en’s Function ons
Examples of quasiparticles – Nuclei -II Nucleons attract themselves at intermediate distances and scatter like billiard balls: Naively, nuclei cannot be treated as orbits structures “GOOD” model of a nucleus “BAD” model of a nucleus Many ny-Body ody Green’ en’s Function ons
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)! Many ny-Body ody Green’ en’s Function ons
Examples of quasiparticles – ions in liquid + - - + - - + - + - + - - + + - + - - + + - - + - - - + - - + - + - + - - Ions in a liquid screen each other’s charge and interact weakly [Picture adapted form Mattuck] Many ny-Body ody Green’ en’s Function ons
Examples of quasiparticles – ions in liquid + - - + - - + - + - + - - + + - + - - + + - - + - - - + - - + - + - + - - Ions in a liquid screen each other’s charge and interact weakly [Picture adapted form Mattuck] Many ny-Body ody Green’ en’s Function ons
Examples of quasiparticles – ions in liquid + - - + - - + - + - + - - + + - + - - + + - - + - - - + - - + - + - + - - Ions in a liquid screen each other’s charge and interact weakly [Picture adapted form Mattuck] Many ny-Body ody Green’ en’s Function ons
Examples of quasiparticles – electron in gas [Picture adapted form Mattuck] Many ny-Body ody Green’ en’s Function ons
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: =1 for fermions = ∞ for bosons It must include the vacuum state: Many ny-Body ody Green’ en’s Function ons
Second quantization Basis states for bosons are constructed as creation and annihilation operators give Commutation rules: Many ny-Body ody Green’ en’s Function ons
Second quantization Basis states for fermions are constructed as creation and annihilation operators give with: Commutation rules: Many ny-Body ody Green’ en’s Function ons
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