  We systematically study the BCS-BEC crossover and the quantum phase
transition in ultracold 6Li and 40K atoms across a wide Feshbach resonance. The
background scattering lengths for 6Li and 40K have opposite signs, which lead
to very different behaviors for these two types of atoms. For 40K, both the
two-body and the many-body calculations show that the system always has two
branches of solutions: one corresponds to a deeply bound molecule state; and
the other, the one accessed by the current experiments, corresponds to a weakly
bound state with population always dominantly in the open channel. For 6Li,
there is only a unique solution with the standard crossover from the weakly
bound Cooper pairs to the deeply bound molecules as one sweeps the magnetic
field through the crossover region. Because of this difference, for the
experimentally accessible state of 40K, there is a quantum phase transition at
zero temperature from the superfluid to the normal fermi gas at the positive
detuning of the magnetic field where the s-wave scattering length passes its
zero point. For 6Li, however, the system changes continuously across the zero
point of the scattering length. For both types of atoms, we also give detailed
comparison between the results from the two-channel and the single-channel
model over the whole region of the magnetic field detuning.
