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.
Highlights
Perturbation theory isn’t usually considered in the complex plane
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
We have seen that the success and failure of perturbation-based methods are directly connected to the position of exceptional point singularities in the complex plane
Summary
Perturbation theory isn’t usually considered in the complex plane. Normally, it is applied using real numbers as one of very few available tools for describing realistic quantum systems. An entirely different perspective on quantisation can be found by analytically continuing quantum mechanics into the complex domain In this inherently non-Hermitian framework, the energy levels emerge as individual sheets of a complex multi-valued function and can be connected as one continuous Riemann surface.[23] This connection is possible because the orderability of real numbers is lost when energies are extended to the complex domain. Throughout this review, we present illustrative and pedagogical examples based on the ubiquitous Hubbard dimer, reinforcing the amazing versatility of this powerful simplistic model
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