Normal ordered exponential approach to thermal properties and time-correlation functions: General theory and simple examples

TL;DR

This paper introduces a normal ordered exponential method for deriving thermal properties and correlation functions in many-body quantum systems, providing a first-principles, non-perturbative approach with exact results for simple models.

Contribution

It presents a novel, first-principles differential equation approach for thermal density matrices that avoids perturbation theory and integral formulations, applicable to Fermionic and Bosonic Hamiltonians.

Findings

Exact results for Fermi-Dirac density matrices in one-body Fermionic systems

Numerically exact results for multidimensional harmonic oscillators

Time-autocorrelation functions and spectra for Franck Condon problems

Abstract

A normal ordered exponential parametrization is used to obtain equations for thermal one-and two-particle reduced density matrices, as well as free energies, partition functions and entropy for both Fermionic (electronic) and Bosonic (vibrational) Hamiltonians. A first principles derivation of the equations, relying only on a simple Wick's theorem and starting from the differential equation $\frac{d \hat{D}}{d \beta}= - (\hat{H}-\mu \hat{N})\hat{D}$, is presented that yields a differential equation for the amplitudes representing density cumulants, as well as the grand potential. In contrast to other approaches reported in the literature the theory does not use perturbation theory in the interaction picture and an integral formulation as a starting point, but rather requires a propagation of the resulting differential equation for the amplitudes. While the theory is applicable to general classes of many-body problems in principle, here, the theory is illustrated using simple model systems. For one-body Fermionic Hamiltonians, Fermi-Dirac one-body reduced density matrices are recovered for the grand-canonical formulation. For multidimensional harmonic oscillators numerically exact results are obtained using the thermal normal ordered exponential (TNOE) approach. As an application of the related time-dependent formulation numerically exact time-autocorrelation functions and absorption spectra are obtained for harmonic Franck Condon problems. These examples illustrate the basic soundness of the scheme and are used for pedagogical purposes. Other approaches in the literature are only discussed briefly and no detailed comparative discussion is attempted.