  We use the density matrix renormalization group method(DMRG) and the infinite
time evolved block decimation method(iTEBD) to investigate the ground states of
the spin-orbit coupled Fermi gas in a one dimensional optical lattice with a
transverse magnetic field. We discover that the system with attractive
interaction can have a polarized insulator(PI), a superconducting phase(SC), a
Luther-Emery(LE) phase and a band insulator(BI) phase as we vary the chemical
potential and the strength of magnetic field. We find that spin-orbit coupling
induces a triplet pairing order at zero momentum with the same critical
exponent as that of the singlet pairing one in both the SC and the LE phase. In
contrast to the FFLO phase found in the spin imbalanced system without
spin-orbit coupling, pairings at finite momentum in these two phases have a
larger exponent hence do not dictate the long range behavior. We also find good
agreements of the dominant correlations between numerical results and the
prediction from the bosonization method. The presence of Majorana fermions is
tested. However, unlike results from the mean field study, we do not find
positive evidence of Majorana fermions in our system.
