Exact analysis of a δ-function spin-1/2 attractive Fermi gas with arbitrary polarization

The ground state of a one-dimensional delta-function attractive spin-1/2 fermions is studied by the nested Bethe ansatz method. Explicitly, the Gaudin integral equation for the system is solved in the form of power series with arbitrary spin polarization. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities, such as the ground state energy and the chemical potentials, are expressed in terms of the dimensionless coupling constant gamma = c/(D up arrow + D down arrow) and the polarization P = (D up arrow - D down arrow)/(D up arrow + D down arrow), where c is the coupling constant and D up arrow(down arrow) are the number densities of the spin-up (down) particles. While in the limiting case P = 1, the system consists of spinless free fermions, the other limit P = 0 describes the BCS-BEC crossover in one dimension. As a function of P, the evolution is continuous and smooth.