Functional renormalization for quantum phase transitions with nonrelativistic bosons

Abstract

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 ${k}_{\mathit{ph}}$, which is characteristic for a given experiment. For the dilute regime ${k}_{\mathit{ph}}\ensuremath{\gtrsim}{n}^{1∕d}$, 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 ${k}_{\mathit{ph}}⪡{n}^{1∕d}$ reveals a rather rich spectrum of features, depending on $d$ and $M$. In this regime, one observes for $d\ensuremath{\leqslant}3$ a crossover to a relativistic action with second time derivatives. This admits order for $dg1$, whereas $d=1$ shows a behavior similar to the low temperature phase of the classical two-dimensional $\mathrm{O}(2M)$ models.