We consider the problem of a single particle interacting with
N identical fermions, at zero temperature and in one dimension. We calculate the binding energy as well as the effective mass of the single particle. We use an approximate method developed in the three-dimensional case, where the Hilbert space for the excited states of the
N fermions is restricted to have at most two particle-hole pairs. When the mass of the single particle is equal to the fermion mass, we find excellent agreement with the exact results of McGuire. When the mass of the single particle is infinite, we solve exactly the problem and find again excellent agreement between approximate results and exact ones. This overall agreement in one dimension gives a strong validation for the approximate method applied in three dimensions. Moreover it shows that our approximate treatment is excellent for the one-dimensional problem in the general case with respect to the mass of the single particle.
We develop an effective field theory to describe the superfluid pairing in strongly interacting fermions with arbitrary short-range attractions by extending Kaplan's idea of coupling fermions to a fictitious boson state [presented in Nucl. Phys. B 494, 471 (1997)]. This boson field is assigned with an unconventional kinetic term to recover the exact scattering phase shift obtained either from scattering data or model calculations. The theory works even if the explicit form of the interaction potential has not been constructed from scattering data. The contact boson-fermion coupling allows us to go beyond mean-field to include Gaussian pair fluctuations, yielding reliable predictions on equations of state. As an application, we use our theory to address the nonuniversal ground-state energy of strongly paired fermions due to the nontrivial momentum dependence of the phase shift characterized, for example, by effective range. We find a good agreement between our predictions and recent quantum Monte Carlo simulations on the effective-range dependence in both three and two spatial dimensions. We propose that in cold-atom experiments, the nonuniversal dependence in thermodynamics can be probed using dark-state optical control of Feshbach resonances.