Hartree-Fock approximation for BEC Hartree-Fock approximation for BEC revisitedrevisited

Jürgen BosseJürgen BosseFreie Universität BerlinFreie Universität Berlin

Panjab University, Chandigarh 3rd March, 2014

Overview

Introduction Thermodynamic Variational Principle Review: HFA for T >Tc

T ≤ Tc : Exc modified by <N02>

HFA including ground-state fluctuations T ≤ Tc : chemical potential

grand-canonical potential

A = - ( H r - Nop )A+B = - ( H - Nop )

e.g., interacting bosons in a trap

Variational Principle

1. Calculate GCP-upper bound using reference hamiltonian of single-particle type2. Find effective hamiltonian (HF) from extremum conditions for upper-bound

grand-canonical potential

A = - ( H r - Nop )A+B = - ( H - Nop )

e.g., interacting bosons in a trap

Variational Principle

1. Calculate GCP-upper bound using reference hamiltonian of single-particle type2. Find effective hamiltonian (HF) from extremum conditions for upper-bound

average occupation number of state | k >

inadequate for bosonsincondensed phase(T ≤ Tc)

Calculation of GCP-upper bound

average occupation number of state | k >

inadequate for bosonsincondensed phase(T ≤ Tc)

Calculation of GCP-upper bound

normal systemHFA

HF hamiltonian from extremum conditions

normal systemHFA

HF hamiltonian from extremum conditions

average occupation number of state | k >

Calculation of GCP-upper bound

<Nkk>

(1-clk)

ground-state fluctuationsmodify Exc

<Nkk> <Nll>

[{j},{j}, N0, f0]

Huse & Siggia (1982)

Fluctuation effect on chemical potential appears to be small

ng

2ng

Uniform Gas of s=0 Bosons Interacting via Repulsive Contact

ideal interacting

N=200 bosons in isotropic harmonic trap

Summary and Outlook

HF-hamiltonian for BEC phase derived by accounting not only for exchange but also for (ground-state) correlation in Exc

N0 and N0 identified as unknowns to be determined from additional sources