Convergence of many-body perturbation methods with lattice summations in extended systems

Abstract

Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations.