Correlation effects in Low-Dimensional Fermion Systems

1. Correlation effects in Low-Dimensional Fermion Systems Reza Asgari

3. Some examples 1)Correlations in Electron Liquid - Low density - Low Dimensionality - Transports effects 2) Bose Einstein Condensation in Low dimension - 1D cold atom in optical lattice - BEC-BSC Crossover 3) Localization in correlated disorder Low dimension - disorder/interaction? - dimensionality and Anderson impurity 4) Strongly correlated in electronic structure - L(S)DFT ( Weakly interaction) - DMFT , LDA+DMFT ( Mediate interaction) - LDA+U ( Insulator system)

4. Outline 1)Correlations in Electron Liquid - Introduction - Luttinger Liquid, Bosonization 2) Bose Einstein Condensation in Atomic Fermi Gases - Introduction - BEC-BSC Crossover - Pairing Gap in Strongly Interacting Fermi Gas - Pairing Without Superfluidity - HTSC and Ultracold Atomic Fermi Gases

5. General Properties Total Hamiltonian Electron-electron interaction Fourier transformation of the Coulomb potentials

6. Density in D dimension Or , more explicitly Fermi wave vector G. Giuliani and G. Vignale “ Quantum theory of the Electron Liquid” Cambridge 2005

7. Ground state energy: Kinetic, Exchange and Correlation K.E : XC. E :

8. Jellium Model: High density region, Correlation Ceperley & Alder, PRL 45, 566 (1980)

9. paramagnetic to fully spin-polarized quantum phase transition of a 2D EL TC: B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 (1989) RS: F. Rapisarda and G. Senatore, Aust. J. Phys. 49, 161 (1996) AMGB : C. Attaccalite, et al., Phys. Rev. Lett 88, 256601 (2002) R. Asgari, B. Davoudi and M. P. Tosi, Solid State Communication 131,1(2004)

10. One Dimensional Electron System • Fermi Liquid Theory? • Divergent behavior ? • Perturbation Theory? • How to solve? • Bosonization

11. Strongly interacting system Weakly interacting system

12. Fermi Liquid versus Non-Fermi Liquid

13. Particle-Hole Excitations • 2D system • 1D system

14. Most efficient processes in the interaction For spinless fermions Corresponding to spins directions :

15. Vertex Interactions and renormalization velocity Solyom 1979

16. Bosonization

17. Density fluctuation and boson fields Y is the step function

18. Fermion fields in terms of boson operators

20. Spinless Fermion: Interactions

22. Spinless 1D net Hamiltonian

23. Thermodynamics

24. Compressibility

25. Correlations

26. Green’s Function

27. Occupation number

28. Many Body effects 1D system arises from the variation of the Bose expression of the ground state energy respect to g(r) The Induced exchange potential arises from the Fermi part The exact ground state wave-function of a Bose system The four- and five- body elementary diagrams (1): L. J. Lantto and P. J. Siemens, Nucl. Phys. A 317, 55 (1979)(2): A. Kallio and J. Piilo, Phys. Rev. Lett. 77, 4237 (1996) (3): B. Davoudi et al., Phys. Rev. B 68, 155112 (2003) and series of our works

29. Numerical Results R. Asgari SSC 141, 563 (2007)

30. Correlation energy in comparison with DMC R. Asgari SSC 141, 563 (2007)

31. End of 2th Lecture Thank you for your attention