1. Hartree-Fock approximation for BEC revisited Hartree-Fock approximation for BEC revisited Jürgen Bosse Freie Universität Berlin Panjab University, Chandigarh 3 rd March, 2014

2. Overview Introduction Thermodynamic Variational Principle Review: HFA for T >T c T ≤ T c : E xc modified by HFA including ground-state fluctuations T ≤ T c : chemical potential

5. grand-canonical potential A = -  ( H r -  N op )A+B = -  ( H -  N op ) e.g., interacting bosons in a trap Variational Principle 1.Calculate GCP-upper bound using reference hamiltonian of single-particle type 2.Find effective hamiltonian (HF) from extremum conditions for upper-bound

6. grand-canonical potential A = -  ( H r -  N op )A+B = -  ( H -  N op ) e.g., interacting bosons in a trap Variational Principle 1.Calculate GCP-upper bound using reference hamiltonian of single-particle type 2.Find effective hamiltonian (HF) from extremum conditions for upper-bound

7. average occupation number of state |  k > inadequate for bosons in condensed phase (T ≤ T c ) Calculation of GCP-upper bound

8. average occupation number of state |  k > inadequate for bosons in condensed phase (T ≤ T c ) Calculation of GCP-upper bound

9. normal system HFA HF hamiltonian from extremum conditions

10. normal system HFA HF hamiltonian from extremum conditions

14. average occupation number of state |  k > Calculation of GCP-upper bound (1-c lk ) ground-state fluctuations modify E xc [{  j },{  j }, N 0, f 0 ]

15. Huse & Siggia (1982)

16. Fluctuation effect on chemical potential appears to be small ng 2ng Uniform Gas of s=0 Bosons Interacting via Repulsive Contact idealinteracting

17. N=200 bosons in isotropic harmonic trap

18. Summary and Outlook HF-hamiltonian for BEC phase derived by accounting not only for exchange but also for (ground-state) correlation in E xc N 0 and  N 0 identified as unknowns to be determined from additional sources