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Christopher Moseley
Phases and phase transitions in an interacting Bose gas
Supervisor: Prof. Dr. Klaus Ziegler [Theoretical physics II]
Date of oral examination: 02/12/2007
153 pages, english
The many-particle problem of strongly interaction bosons in a lattice potential is investigated. Motivated by recent experiments on Bose-Einstein condensates in optical lattices which showed the existence of a Mott-insulator, four different models are presented, which allow the calculation of the phase diagram, and experimentally observable physical quantities like the total density and the condensate density, the quasiparticle spectrum, and the static structure factor. All these models have in common that they simulate a strong repulsive interaction by imposing a hard-core condition on the bosons, which prohibits a multiple occupation of lattice sites. They are defined by means of the functional integral method.

The first model describes non-interacting fermions in a one-dimensional lattice. We exploited the well-known fact that such a fermionic system is equivalent to impenetrable bosons in one dimension, and that the static structure factors of the fermionic and the bosonic system are identical. At zero temperature, we find a phase diagram with three phases, a phase where the lattice is empty, a phase with an incommensurate filling of the lattice, and a Mott-insulator. We calculate the local particle density, the density-density correlation function and the static structure factor in a translational invariant system as well as in a system with a harmonic trap potential.

The other three models were applied on a Bose gas in a three dimensional lattice. The first two are constructed as fields of pairs of Grassmann variables. They can be seen as interacting fermionic models. The third one was based on a slave boson approach. A Hubbard-Stratonovich transformation allows to integrate out the original fields in all three models. A saddle point approximation provides both a mean-field solution and Gaussian fluctuations. The latter contain the information about quasiparticle excitations. The total particle density and the condensate density are calculated in mean-field theory, and the quasiparticle spectrum and the static structure factor is calculated on the level of Gaussian fluctuations. We find a Bose-Einstein condensate for all three models, but only one of the two fermionic models and the slave boson model, reveal a Mott-insulating phase on the mean-field level, because the mean-field solution of other fermionic model is valid only in the dilute regime. At higher temperatures, we have shown that the slave boson model leads to a "renormalised" Gross-Pitaevskii equation with temperature dependent coefficients.

The quasiparticle spectrum E(q) which was found for all three-dimensional models, is gapless (Goldstone mode) in the Bose-Einstein condensate due to a broken U(1) symmetry. In the dilute regime, it agrees with the well-known Bogoliubov spectrum. In the Mott-insulator, the quasiparticle spectrum is gapped. For the static structure factor we find relation S(q)=e(q)/E(q), where e(q) is the free-particle dispersion relation. Our results agree with results which were derived for the Bose-Hubbard model, if the repulsive on-site interaction is very large.