Hartree-Fock-Bogoliubov theory of a charged Bose gas

Thomas Chen, Michael Hott

Dániel Schumayer, Sam Cormack, Brandon P. van Zyl +4 more

B. Davoudi, Anna Minguzzi, M. P. Tosi

We critically examine the Hartree-Fock-Bogoliubov (HFB) solution of the equations of motion for condensate fluctuations in a weakly coupled plasma of charged bosons at finite temperature. Analytic expressions are derived for the first two infrared-divergent terms in both the momentum distribution of the noncondensate and the anomalous Bose correlation function at low momenta. Incorporation into the theory of the appropriate form of the Hugenholtz-Pines relation for the chemical potential is needed to cancel an unphysical divergence. Exact cancellation of infrared-divergent terms is demonstrated in the HFB shift of the single-particle excitation energy away from the Bogoliubov value at long wavelengths, with the residual terms raising it towards the plasma frequency at low temperature. Numerical illustrations are presented for a number of properties of the boson plasma as functions of temperature and density in the weak-coupling regime: these are the chemical potential, the condensate fraction, the normal and anomalous momentum distribution functions and the corresponding one-body density matrices, and the dispersion relation of single-particle excitations.

Jacky J. Chong

The thesis provides rigorous quantitative analyses for studying quantum fluctuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent years, the dynamics of the condensate and the excitations of a Bose gas was shown to be well approximated by the quasifree dynamics with governing equations given by a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-Fock-Bogoliubov (HFB) system (c.f. Grillakis and Machedon, J. Fix. Point Theory A., (2013), Grillakis and Machedon, Comm. PDEs, (2017), Bach et al. in arXiv:1602.05171v2, (2018), Benedikter et al., Ann. Henri Poincare, (2018)). However, both the quantitative and qualitative analysis of the time-dependent HFB system are still in their early stages of development. Thus, the primary purposes of the thesis are to further the development of some analytic tools necessary for studying the time-dependent HFB system and use these effective equations to provide quantitative estimates for the true dynamics of the Bose gas at absolute zero temperature in Fock space norm. The thesis comprises the entirety of the author's current and past projects on the time-dependent HFB system. Each project falls into one or more of two categories: studying the local or global well-posedness of the time-dependent HFB system, or obtaining global-in-time Fock space estimates for the error terms of the quasifree approximation to the dynamics of a system of interacting bosons. In the former category, the author employs techniques of dispersive PDE theory developed in the past three decades along with classical methods of harmonic analysis to study lower regularity solutions to the time-dependent HFB system. The lower regularity well-posedness of solutions for the time-dependent HFB system is necessary for studying norm approximation of the dynamics of a dilute Bose gas with strong interactions. In the latter category, we prove global a-priori estimates for the solutions to the HFB system and use them to obtain estimates for the error terms of the Fock space approximation.

Ka-Wo Poon