  This is a chapter for a book. The first paragraph of this chapter is as
follows: "Ultracold quantum gases offer a wonderful playground for quantum many
body physics, as experimental systems are widely controllable, both statically
and dynamically. One such system is the one-dimensional (1D) Bose gas on a
ring. In this system binary contact interactions between the constituent
bosonic atoms, usually alkali metals, can be controlled in both sign and
magnitude; a recent experiment has tuned interactions over seven orders of
magnitude, using an atom-molecule resonance called a Feshbach resonance. Thus
one can directly realize the Lieb-Liniger Hamiltonian, from the weakly- to the
strongly-interacting regime. At the same time there are a number of experiments
utilizing ring traps. The ring geometry affords us the opportunity to study
topological properties of this system as well; one of the main properties of a
superfluid is the quantized circulation in which the average angular momentum
per particle, L/N, is quantized under rotation. Thus we focus on a tunable 1D
Bose system for which the main control parameters are interaction and rotation.
We will show that there is a critical boundary in the interaction-rotation
control-parameter plane over which the topological properties of the system
change. This is the basis of our concept of \textit{metastable quantum phase
transitions} (QPTs). Moreover, we will show that the finite domain of the ring
is necessary for the QPT to occur at all because the zero-point kinetic
pressure can induce QPTs, i.e., the system must be finite; we thus seek to
generalize the concept of QPTs to inherently finite, mesoscopic or nanoscopic
systems."
