Crystalline phases of bosons in rotating traps: Tonks-Girardeau gas on a ring.

We analyze the systems of strongly repelling bosons in two-dimensional harmonic and ring-shaped traps as a function of the rotational frequency of the trap for neutral atoms (and of an applied magnetic field for charged bosons). Our two-step approach consists of breaking the rotational symmetry at the Hartree-Fock level and of subsequent symmetry restoration via projection techniques, \footnote{Phys. Rev. Lett.{\bf 93}, 230405 (2004)} thus taking into account correlations beyond the Gross-Pitaevskii (GP) solution. The bosons localize and form crystalline patterns both for a repulsive contact potential and a Coulomb interaction, as revealed via conditional probability distribution (CPD) analysis. This behavior of the bosons in the ring-shaped traps in the strong repulsion limit is similar to the behavior of fermions and is a manifestation of the fermionization phenomenon. We present calculations for the ground state energies as a function of the rotational frequency (or the strength of the magnetic field) and as a function of the repulsion strength.