Explicit diagrammatic solution of normalised, nondegenerate Rayleigh-Schr\"odinger perturbation theory

TL;DR

This paper presents a diagrammatic method for solving normalized, nondegenerate Rayleigh-Schrödinger perturbation theory, extending Bloch's diagrams to include normalization effects and their topological dependencies.

Contribution

It introduces a diagrammatic approach for normalized perturbation series, accounting for normalization constraints and diagram topology in nondegenerate Rayleigh-Schrödinger theory.

Findings

Derived explicit diagrammatic solutions for normalized eigenstates.

Extended Bloch diagrams to include normalization effects.

Revealed the influence of diagram topology on coefficients.

Abstract

We solve the coupled recurrence relations for eigenenergies and -vectors in nondegenerate Rayleigh-Schr\"odinger perturbation theory under the constraint that the approximate eigenvector be normalised to $1$ in every order. The series can be expressed in terms of diagrams that were first introduced by C. Bloch [Nucl. Phys. 6, 329 (1958)] for the degenerate, unnormalised case. Normalisation increases the number of terms and introduces a nontrivial dependence on the diagrams' topology to the coefficients.