  We present a systematic study of quantum phases in a one-dimensional
spin-polarized Fermi gas. Three comparative theoretical methods are used to
explore the phase diagram at zero temperature: the mean-field theory with
either an order parameter in a single-plane-wave form or a self-consistently
determined order parameter using the Bogoliubov-de Gennes equations, as well as
the exact soluble Bethe ansatz method. We find that a spatially inhomogeneous
Fulde-Ferrell-Larkin-Ovchinnikov phase, which lies between the fully paired BCS
state and the fully polarized normal state, dominates most of the phase diagram
of a uniform gas. The phase transition from the BCS state to the
Fulde-Ferrell-Larkin-Ovchinnikov phase is of second order, and therefore there
are no phase separation states in one-dimensional homogeneous polarized gases.
This is in sharp contrast to the three-dimensional situation, where a phase
separation regime is predicted to occupy a very large space in the phase
diagram. We conjecture that the prediction of the dominance of the phase
separation phases in three dimension could be an artifact of the
non-self-consistent mean-field approximation, which is heavily used in the
study of three-dimensional polarized Fermi gases. We consider also the effect
of a harmonic trapping potential on the phase diagram, and find that in this
case the trap generally leads to phase separation, in accord with the
experimental observations for a trapped gas in three dimension. We finally
investigate the local fermionic density of states of the
Fulde-Ferrell-Larkin-Ovchinnikov ansatz. A two-energy-gap structure is shown
up, which could be used as an experimental probe of the
Fulde-Ferrell-Larkin-Ovchinnikov states.
