1. Functional Integration in many-body systems: application to ultracold gases Klaus Ziegler, Institut für Physik, Universität Augsburg in collaboration with Oleksandr Fialko (Augsburg) Cenap Ates (MPIPKS Dresden) Path Integrals, Dresden, September 2007

2. Outline: 1)Mixtures of atoms with elastic scattering: Competition between thermal and quantum fluctuations a) asymmetric fermion/bosonic Hubbard model b) distribution of heavy fermions c) density of states of light fermions d) diffusion and localization of light fermions 2) Mixtures of atoms with inelastic scattering a) Holstein-Hubbard model b) phases: Neel state, density wave, alternating dimer state

3. Functional Integrals for many-body systems    Partition function: Green’s function:

4. Calculational Methods Integral (Hubbard-Stratonovich) transform Grassmann fields vs. complex fields Generalization (e.g. to N components) Approximations: Saddle-point integration Gaussian fluctuations Monte-Carlo simulation

5. Ultracold gases in an optical lattice counterpropagating Laser fields: “Computer” for 1D, 2D or 3D many-body systems: Tunable tunneling rate via amplitude of the Laser field Tunable interaction via magnetic field (near Feshbach resonance) Tunable density of particles Free choice of particle statistics (bosonic or fermionic atoms) Free choice of spin Free choice of lattice type

6. Key experiment: BEC-Mott transition M.Greiner et al. 2002

7. Key experiment: BEC-Mott transition M.Greiner et al. 2002

8. Bose gas in an optical lattice Bose-Einstein Condensate: kinetic energy wins tunneling Mott insulator: repulsion wins NO phase coherence but constant local particle density (n=1) Phase coherence but fluctuating particle density Competition between kinetic and repulsive energy

9. Phase diagram for T=0 Bose-Einstein Condensate Mott insulator

10. Mixture of two atomic species in an optical lattice Two types of atoms with different mass a minimal model: consider two types of fermions (e.g. 6 Li and 40 K) - both species are subject to thermal fluctuations - only light atoms can tunnel - magnetic trap: atoms are spin polarized (Pauli exclusion!) - different species are subject to a local repulsive interaction

11. Mixture of Heavy and Light Particles Adding heavy particles heavy particles are thermally distributed time Light particles follow random walks “tunneling”

12. Mixtures in an optical lattice: elastic scattering Asymmetric (Fermion or Boson) Model light atoms:heavy atoms: Local repulsive interaction: asymmetric Hubbard Model

13. Single-site approximation: Mott state Fermionic Mixtures Bosonic Mixtures Fermionic-Bosonic Mixtures

14. Ising representation of heavy atoms Green’s function of light atoms: quenched average(!): with respect to the distribution self-organized disorder

15. Schematic phase diagrams distribution of heavy atoms P(n) fully occupied empty  =U/2

16. Distribution of heavy atoms Monte-Carlo simulation for decreasing temperature T Phase separation T=1/3 T=1/7 T=1/14 near phase transition

17. Phase separation density of light atoms in a harmonic trap

18. Density of States for Light Atoms  =U/2: increasing U

19. localization propagation Propagation vs.localization >c>c <c<c

20. Localization transition Finite size scaling

21. Mixtures in an optical lattice: Inelastic Scattering Heavy atoms in a Mott state

22. Locally oscillating heavy particles Adding oscillating particles heavy particles can not tunnel but are local harmonic oscillators heavy particles in Mott-insulating state

23. Single-Site Approximation Lang-Firsov transformation: effective fermionic interaction:

24. Ground states Neel and density wave U eff >0 U eff <0 First order phase transition

25. Alternating dimer states two types of dimers: Degeneracy under  rotation of dimers! dimer liquid on frustrated lattices?

26. Conclusions Results: New quantum states due to competing atomic species Degeneracy can lead to complex states Heavy atoms represent correlated random potential for light atoms heavy fermions ->Ising spins correlation effect: opening of a gap disorder effect: localization of atoms Open Questions: Can exotic states (e.g. spin liquids) be realized experimentally? Is there a glass phase (frustration)?