  In this work, we explore the polaron and molecule physics by utilizing a
unified variational ansatz with up to two particle-hole(p-h)
excitations(V-2ph). We confirm the existence of a first-order transition in 3D
and 2D Fermi polarons, and show that the nature of such transition lies in an
energy competition between systems with different total momenta $Q=0$ and
$|Q|=k_F$, where $k_F$ is the Fermi momentum of majority fermions. The
literally proposed molecule ansatz is identified as an asymptotic limit of
$|Q|=k_F$ state in strong coupling regime, which also implies a huge SO(3) (for
3D) or SO(2) (for 2D) ground state degeneracy in this regime. The recognization
of such degeneracy is crucially important for evaluating the molecule
occupation in realistic experiment with finite impurity density and at finite
temperature. To compare with recent experiment of 3D Fermi polarons, we have
calculated various physical quantities under the V-2ph framework and obtained
results that are in good agreements with experimental data, especially in the
unitary regime. Further, we have adopted a different variational method in 2D
based on the Gaussian sample of high-order p-h excitations, and found the same
conclusion on the nature of polaron-molecule transition therein. For 1D system,
the V-2ph method predicts no sharp transition and the ground state is always at
$Q=0$ sector, consistent with exact Bethe ansatz solution. The presence/absence
of polaron-molecule transition is analyzed to be closely related to the
interplay effect of Pauli-blocking and p-h excitations in different dimensions.
