  The formation of bound states between mobile impurity particles and fermionic
atoms has been demonstrated in spin-polarized Fermi gases with attractive
interspecies interaction. We investigate bound states of mobile impurities
immersed in a two-dimensional system with a symmetry-protected quadratic band
touching. In addition to the standard s-wave interaction, we consider an
anisotropic dipolar exchange interaction that locally breaks point group
symmetries. Using a weak-coupling renormalization group approach and a ladder
approximation for the impurity-fermion propagator, we establish that the number
of bound states can be controlled by varying the anisotropy of the exchange
interaction. Our results show that the degeneracy and momentum dependence of
the binding energies reflect some distinctive properties of the quadratic band
touching.
