Hidden physics in the dual-fermion approach: A special case of a nonlocal expansion scheme

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.