Functional renormalization for quantum phase transitions with nonrelativistic bosons

Functional renormalization yields a simple unified description of bosons at zero temperature, in arbitrary space dimension d and for M complex fields. We concentrate on nonrelativistic bosons and an action with a linear time derivative. The ordered phase can be associated with a nonzero density of (quasi)particles n. The behavior of observables and correlation functions in the ordered phase depends crucially on the momentum kₚₕ, which is characteristic for a given experiment. For the dilute regime kₚₕn1∕d, the quantum phase transition is simple, with the same ``mean field'' critical exponents for all d and M. On the other hand, the dense regime kₚₕ⪡n1∕d reveals a rather rich spectrum of features, depending on d and M. In this regime, one observes for d3 a crossover to a relativistic action with second time derivatives. This admits order for $d>1$, whereas $d=1$ shows a behavior similar to the low temperature phase of the classical two-dimensional O(2M) models.