Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them

TL;DR

This paper investigates the role of exceptional points in non-Hermitian quantum chemistry, analyzing how complex energy singularities influence perturbation series convergence and exploring resummation techniques with the Hubbard dimer model.

Contribution

It introduces the complex-plane perspective to quantum chemistry, linking exceptional points with perturbation theory convergence and demonstrating resummation methods for improved accuracy.

Findings

Exceptional points are crucial in understanding perturbation series behavior.

Resummation techniques like Padé improve series convergence.

The Hubbard dimer effectively illustrates complex-plane perturbation phenomena.

Abstract

We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schr\"odinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Pad\'e and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.