One-dimensional multicomponent fermions with δ-function interaction in strong- and weak-coupling limits: κ-component Fermi gas

We derive the first few terms of the asymptotic expansion of the Fredholm equations for one-dimensional kappa-component fermions with repulsive and attractive delta-function interaction in strong- and weak-coupling regimes. We thus obtain a highly accurate result for the ground-state energy of a multicomponent Fermi gas with polarization for these regimes. This result provides a unified description of the ground-state properties of the Fermi gas with higher spin symmetries. However, in contrast to the two-component Fermi gas, there does not exist a mapping that can unify the two sets of Fredholm equations as the interacting strength vanishes. Moreover, we find that the local pair-correlation functions shed light on the quantum statistic effects of the kappa-component interacting fermions. For the balanced spin case with repulsive interaction, the ground-state energy obtained confirms Yang and You's result [Chin. Phys. Lett. 28, 020503 (2011)] that the energy per particle as kappa -> infinity is the same as for spinless Bosons.