We consider a number of different models in this chapter, but they share some important unifying characteristics. They all have a global U(1) symmetry. We are particularly interested in the behavior of the conserved density, generically denoted as Q, associated with this symmetry. All the models exhibit a quantum phase transition between two phases with the a specific T = 0 behavior in the expectation value of Q. In one of the phases áQ ñ is pinned precisely at a quantized value, and does not vary as microscopic parameters are varied. This quantization ends at the quantum critical point with a discontinuity in the derivative of áQ ñ with respect to the tuning parameter, and áQ ñ varies smoothly in the other phase; there is no discontinuity in the value of áQ ñ, however.
We have already met a transition of the above type in the previous Chapter 10: the Mott insulator to superfluid transition. The finite temperature crossovers near this transition are discussed in some detail, including their exact determination in d = 1. We also discuss a closely related transition in a dilute Fermi gas: exact crossovers can be determined here in general d, and this provides a simple and instructive example of the physics near a quantum critical point.