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Bound states in two-dimensional Fermi systems with quadratic band touching

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.