Green's Function-Free Formalism of Projective Truncation Approximation

TL;DR

This paper introduces a Green's function-free, self-consistent formalism for the projective truncation approximation, connecting it to variational reduced density matrix theory and addressing key computational issues.

Contribution

It reformulates PTA as a self-consistent RDM theory, clarifies properties of the dynamical matrix, and casts the solution as an over-constrained optimization problem.

Findings

Connection of PTA to variational RDM theory established.

Solution of PTA equations formulated as an optimization problem.

Discussion of alternative inner product and generalized theorems included.

Abstract

In previous works, the projected truncation approximation (PTA) was developed as a systematic and controlled method to truncate the equation of motion of Green's functions (GFs) for a given quantum or classical many-body Hamiltonian. The static averages are obtained self-consistently with the GF through the spectral theorem. In this work, PTA is reformulated as a self-consistent theory for the reduced density matrices (RDMs) without reference to GF. We separately discuss the issues of determining the dynamical matrix ${\bf M}$ and solving the physical quantities from it. The properties of ${\bf M}$ is clarified and the solution of PTA equations is cast into an over-constrained optimization problem. This makes connection of the present theory to the variational RDM theory. We discuss various issues of PTA under this formalism, including the scheme of alternative inner product, the generalized virial theorem, the generalized Wick's theorem, and the static component problem of PTA.