Effective field theory of boson-fermion mixtures and bound fermion states on a vortex of boson superfluid

We construct a Galilean invariant low-energy effective field theory of boson-fermion mixtures and study bound fermion states on a vortex of boson superfluid. We derive a simple criterion to determine for which values of the fermion angular momentum l there exist an infinite number of bound energy levels. We apply our formalism to two boson-fermion mixed systems: the dilute solution of He-3 in He-4 superfluid and the cold polarized Fermi gas on the BEC side of the "splitting point." For the He-3-He-4 mixture, we determine parameters of the effective theory from experimental data as functions of pressure. We predict that infinitely many bound He-3 states on a superfluid vortex with l=-2,-1,0 are realized in a whole range of pressure 0-20 atm, where experimental data are available. As for the cold polarized Fermi gas, while only S-wave (l=0) and P-wave (l=+/- 1) bound fermion states are possible in the BEC limit, those with higher negative angular momentum become available as one moves away from the BEC limit.