The spectral theorem of many-body Green's function theory when there are zero eigenvalues of the matrix governing the equations of motion

TL;DR

This paper introduces a singular-value decomposition approach to simplify the spectral theorem in many-body Green's function theory, removing the need for anti-commutator Green's functions when zero eigenvalues are present.

Contribution

It presents a general, easy-to-apply reformulation using singular-value decomposition to handle zero eigenvalues in the equations of motion matrix, improving the spectral theorem's applicability.

Findings

Eliminates the need for anti-commutator Green's functions in certain cases.

Demonstrates the method on a ferromagnetic monolayer example.

Applicable to more complex many-body problems.

Abstract

In using the spectral theorem of many-body Green's function theory in order to relate correlations to commutator Green's functions, it is necessary in the standard procedure to consider the anti-commutator Green's functions as well whenever the matrix governing the equations of motion for the commutator Green's functions has zero eigenvalues. We show that a singular-value decomposition of this matrix allows one to reformulate the problem in terms of a smaller set of Green's functions with an associated matrix having no zero eigenvalues, thus eliminating the need for the anti-commutator Green's functions. The procedure is quite general and easy to apply. It is illustrated for the field-induced reorientation of the magnetization of a ferromagnetic Heisenberg monolayer and it is expected to work for more complicated cases as well.