  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 momenta ${\mathbf Q}=0$ and
$|{\mathbf Q}|=k_F$, here ${\mathbf Q}$ is defined as the momentum of Fermi
polaron system with respect to the Fermi sea of majority fermions (with Fermi
momentum $k_F$). The literally proposed molecule ansatz is identified as an
asymptotic limit of $|{\mathbf Q}|=k_F$ state in strong coupling regime, which
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 systems 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 in the weak coupling and near resonance regime. Further, to check the
validity of our conclusion in 2D, we have adopted a different variational
method based on the Gaussian sample of high-order p-h excitations(V-Gph), 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 ${\mathbf Q}=0$ 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.
