The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k4 at large momentum k, as pointed out by Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.

We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two-and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to 4 He atoms and calculate the binding energy of the 4 He tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to 4 He atoms, the correlation is approximately linear.

Few-body systems with large scattering length have universal properties that do not depend on the details of their interactions at short distances. We study the universal bound state properties of the four-boson system with large scattering length in an effective quantum mechanics approach. We compute the four-body binding energies using the Yakubovsky equations for positive and negative scattering length. Moreover, we study the correlation between three-and four-body energies and present a generalized Efimov plot for the four-body system. These results are useful for understanding the cluster structure of nuclei and for the creation of weakly-bound tetramers with cold atoms close to a Feshbach resonance.

We consider the correlation between the binding energies of the triton and the α-particle which is empirically observed in calculations employing different phenomenological nucleon-nucleon interactions. Using an effective quantum mechanics approach for short-range interactions with large scattering length |a| ≫ ℓ, where ℓ is the natural low-energy length scale, we construct the effective interaction potential at leading order in ℓ/|a|. In order to renormalize the four-nucleon system, it is sufficient to include a SU (4)-symmetric one-parameter three-nucleon interaction in addition to the S-wave nucleon-nucleon interactions. The absence of a four-nucleon force at this order explains the empirically observed correlation between the binding energies of the triton and the α-particle. We calculate this correlation and obtain a prediction for the α-particle binding energy. Corrections to our results are suppressed by ℓ/|a|.