  In this work, we present a nonlocal expansion scheme to study correlated
electron systems aiming at a better description of its spatial fluctuations at
all length scales. Taking the nonlocal coupling as a perturbation to the local
degrees of freedom, we show that the nonlocality in the self-energy function
can be efficiently constructed from the coupling between local fluctuations. It
can provide one unified framework to incorporate nonlocality to both ordered
and disordered correlated many-body fermion systems. In this application, we
prove that the dual-fermion approach can be understood as a special case of
this nonlocal expansion scheme. The scheme presented in this work is
constructed without introducing any dual variable, in which the interacting
nature and the correlated behaviors of the lattice fermions have a clear
physics correspondence. Thus, in this special case, the equivalence of the
dual-fermion approach to the nonlocal expansion scheme beautifully reveals the
physics origin of the dual variables. We show that the noninteracting
dual-fermion Green's function corresponds exactly to a nonlocal coupling of the
lattice fermion renormalized by the local single-particle charge fluctuations,
and the dual-fermion self-energy behaves as the one-particle fully irreducible
components of the lattice Green's function. Not only limited to this specific
example, the nonlocal expansion scheme presented in this work can also be
applied to other problems depending on the choice of the local degrees of
freedom.
