Metastable quantum phase transitions in a periodic one-dimensional Bose gas: II. Many-body theory

We show that quantum solitons in the Lieb-Liniger Hamiltonian are precisely the yrast states. We identify such solutions clearly with Lieb's type II excitations from weak to strong interactions, clarifying a long-standing question of the physical meaning of this excitation branch. We demonstrate that the metastable quantum phase transition previously found in mean field analysis of the weakly-interacting Lieb-Liniger Hamiltonian [Phys. Rev. A {\bf 79}, 063616 (2009)] extends into the medium- to strongly-interacting regime of a periodic one-dimensional Bose gas. Our methods are exact diagonalization, finite-size Bethe ansatz, and the boson-fermion mapping in the Tonks-Girardeau limit.