Abstract:
We discuss the “Higgs” amplitude mode in the relativistic quantum O($N$) model in two space dimensions. This model describes the critical regime of many condensed-matter systems, including antiferromagnets and the superfluid–Mott-insulator transition of bosons in an optical lattice. We explain why the Higgs resonance does not show up in the longitudinal correlation function but might be observed in the O($N$)-invariant scalar susceptibility. After a brief introduction to the nonperturbative renormalization group, we show how this approach can be used to compute the scalar susceptibility in the vicinity of the $T=0$ quantum critical point. In the $T=0$ ordered phase, we find a well defined Higgs resonance for $N=2$ and $N=3$ and determine its universal properties. The resonance does not exist in the $T=0$ disordered phase but persists at finite temperature. For all values of $N$ larger than 3, i.e. $Ngeq 4$, we find that the resonance is suppressed. Our results are compared with quantum Monte Carlo simulations and $epsilon=4-(d+1)$ expansion about $d=3$.