Convergent perturbation theory for a

Jun-Qiang Sun, Rodney J. Bartlett

The convergence behavior of many-body perturbation theory (MBPT) with lattice summations directly reflects the spatial property of electron correlation in extended systems. A knowledge of such properties is essential for both theory and numerical calculations. It is shown solely from theory that the second-order MBPT contribution to the total energy per unit cell in polymers converges with lattice size $N$ as $1/{N}^{3}$ while the second-order MBPT contribution to the band energy converges as $1/{N}^{2}$. These convergence behaviors are demonstrated numerically for polyacetylene.

Jochen Pade

Alto Osada, Rekishu Yamazaki, Atsushi Noguchi

Gerald Speisman

A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first $n$ terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.

Michele Cini

Vasily Sazonov

The standard perturbation theory in QFT and lattice models leads to the asymptotic expansions. However, an appropriate regularization of the path or lattice integrals allows one to construct convergent series with an infinite radius of the convergence. In the earlier studies, this approach was applied to the purely bosonic systems. Here, using bosonization, we develop the convergent perturbation theory for a toy lattice model with interacting fermionic and bosonic fields.