Unconventional quantum criticality in a two-dimensional fermion-boson coupled system

We study a two-dimensional two-band Fermi system with finite Fermi surfaces that couples with a $phi^4$ action of O(3) vector boson field with dynamical exponent $z=1$ through Yukawa-type coupling. By using RG analysis, we show that quantum criticality associated with an ordering of the O(3) vector field is controlled by a new saddle-point fixed point instead of the Wilson-Fisher fixed point. The new fixed point has a finite fermion-boson coupling and the boson velocity is strongly renormalized by the Yukawa coupling. We discuss possible superconducting instability near such a quantum critical point.