The hierarchy of Davydov’s Ansätze: from guesswork to numerically “exact” many-body wave functions

Abstract

This perspective presents an overview of the development of the hierarchy of Davydov’s Ansätze and a few of its applications in many-body problems in computational chemical physics. Davydov’s solitons originated in the investigations of vibrational energy transport in protein s in the 1970s. Momentum-space projection of these solitary waves turned up to be accurate variational ground-state wave functions for the extended Holstein molecular crystal model, lend ing unambiguous evidence to the absence of formal quantum phase transitions in the Holstein systems. The multiple Davydov Ansätze have been proposed, with the increasing Ansatz multiplicity, as incremental improvements of their single-Ansatz parents. For a given Hamiltonian, the time-dependent variational formalism is utilized to extract accurate dynamic and spectroscopic properties using Davydov’s Ansätze as its trial states. A quantity proven to disappear for large multiplicities, the Ansatz relative deviation is introduced to quantify how closely the Schrödinger equation is obeyed. Three finite-temperature extensions to the time-dependent variation scheme are elaborated, i.e., the Monte Carlo importance sampling, the method of thermofield dynamics, and the method of displaced number states. To demonstrate the versatility of the methodology, applications of Davydov’s Ansätze are made to the generalized Holstein Hamiltonian, variants of the spin-boson model, and systems of cavity-assisted singlet fission, yielding accurate dynamic and spectroscopic properties of the many-body systems.