Ginzburg-Landau theory for bosonic gases in optical lattices

王波涛 WANG Bo-tao, 姜颖 JIANG Ying

哲 小鍋

Mohamed Mobarak

In recent years the quantum simulation of condensed-matter physics problems has resulted from ex- citing experimental progress in the realm of ultracold atoms and molecules in optical lattices. In this thesis we analyze theoretically a spinor Bose gas loaded into a three-dimensional cubic optical lattice. In order to account for different superfluid phases of spin-1 bosons with a linear Zeeman effect, we work out a Ginzburg-Landau theory for the underlying spin-1 Bose-Hubbard model. To this end we add artificial symmetry- breaking currents to the spin-1 Bose-Hubbard Hamiltonian in order to break the global U (1) symmetry. With this we determine a diagrammatic expansion of the grand-canonical free energy up to fourth order in the symmetry-breaking currents and up to the leading non-trivial order in the hopping strength which is of first order. As a cross-check we demonstrate that the resulting grand- canonical free energy allows to recover the mean-field theory. Applying a Legendre transfor- mation to the grand-canonical free energy, where the symmetry-breaking currents are transformed to order parameters, we obtain the effective Ginzburg-Landau action. With this we calculate in detail at zero temperature the Mott insulator-superfluid quantum phase boundary as well as condensate and particle number density in the superfluid phase. We find that both mean-field and Ginzburg-Landau theory yield the same quantum phase transition between the Mott insulator and superfluid phases, but the range of validity of the mean-field theory turns out to be smaller than that of the Ginzburg-Landau theory. Due to this finding we expect that the Ginzburg-Landau theory gives better results for the superfluid phase and, thus, we restrict ourselves to extremize only the effective Ginzburg-Landau action with respect to the order parameters. With- out external magnetic field the superfluid phase is a polar (ferromagnetic) state for anti-ferromagnetic (ferromagnetic) interactions, i.e. only the hyperfine spin 0 (1) is macroscopically occupied, in accor- dance with previous mean-field results. On the other hand, in the presence of the external magnetic field for ferromagnetic interaction, the superfluid phase does not change as the minimization of the energy implies the maximal spin value. However, when an anti-ferromagnetic interaction competes with the linear Zeeman effect, we can distinguish various ferromagnetic and anti-ferromagnetic su- perfluid phases within the range of validity of the Ginzburg-Landau theory. Increasing the external magnetic field yields a breaking of spin singlet pairs and a subsequent alignment of spins, thus anti- ferromagnetic phases decrease until only a ferromagnetic superfluid phase prevails. In addition, we find that the superfluid-Mott insulator phase transition is always of second order for both ferromag- netic and anti- ferromagnetic interactions. However, the transitions between different superfluid phases for an anti-ferromagnetic interaction can be both of first and second order depending on whether the respective macroscopic occupation of hyperfine spin states changes discontinuously or continuously. The established Ginzburg-Landau theory for spin-1 bosons in optical lattices will certainly be the basis for many further applications as, for instance, time-of-flight absorption pictures or collective excitations, which are of experimental importance.

Emek Yesilada

Masahiko Machida, T. Koyama

We construct a time-dependent Ginzburg-Landau (TDGL) theory for the superfluid atomic Fermi-gases near the Feshbach resonance from the fermion-boson model on the basis of the functional integral formalism. We show that the GL coefficients in the TDGL equation are complex numbers except in the BEC limit. The complex TDGL equation describes both damping and propagating dynamics, which leads to very rich superfluid dynamics near the Feshbach resonance. We predict multiple plane-wave modes corresponding to the in-phase and out-of-phase like oscillations between fermionic and bosonic superfluids and a galaxylike spiral pattern for a vortex.

Mohamed Mobarak, Axel Pelster

We analyze theoretically a spinor Bose gas loaded into a three-dimensional cubic optical lattice. In order to account for different superfluid phases of spin-1 bosons in the presence of an external magnetic field, we work out a Ginzburg-Landau theory for the underlying spin-1 Bose-Hubbard model. In particular at zero temperature, we determine both the Mott and the superfluid phases for the competition between the anti-ferromagnetic interaction and the linear Zeeman effect within the validity range of the Ginzburg-Landau theory. Moreover, we find that the phase transition between the superfluid and Mott insulator phases is of second order and that the transitions between the respective superfluid phases for anti-ferromagnetic interaction can be both of first and second order.