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

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-Schrödinger 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 Møller-Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Padé and quadratic approximants) that can improve the overall accuracy of the Møller-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.

Hartree-Fock Critical Nuclear Charge in Two-Electron Atoms

“…We have previously observed this behaviour in analytic models, but our current results suggest that this phenomenon extends to the complete-basis-set limit. 24,27 Furthermore, Ref. 27 shows that these complex branch points can allow a ground-state wave function to be smoothly "morphed" into an excited-state wave function by following a continuous complex contour.…”

“…The critical nuclear charge Z c can then be identified from the radius of convergence of these series, 2,4,5,23 defined by the distance of the closest singularity to the origin in the complex λ plane. 24 Both E(λ) and |Ψ(λ)| 2 have complicated singularities on the positive real axis at λ c = 1/Z c , 5 which have been interpreted as a quantum phase transition in the complete-basis-set limit. 12 The HF wave function is a single Slater determinant Ψ HF (x 1 , x 2 ) built from the antisymmetrised product of the occupied spin-orbitals ψ i (x).…”

“…26 By analytically continuing an equivalent two-electron coupling parameter to complex values, we have recently shown that the UHF wave functions form a non-Hermitian square-root branch point at the symmetry-breaking threshold. 24,27 Remarkably, following a complex-valued contour around this point leads to the interconversion of the degenerate solutions, and allows a ground-state wave function to be smoothly evolved into an excited-state wave function. 27…”

“…The two degenerate UHF wave functions are therefore connected as a square-root branch point in the complex-Z plane, in agreement with our previous observations in analytically solvable models. 24,27 Furthermore, the branch point behaves as a quasi-exceptional point, where the two solutions become identical but remain normalised (see Ref. 27), providing the first example of this type of non-Hermitian HF degeneracy in the complete-basis-set limit.…”