The treatment of zero eigenvalues of the matrix governing the equations of motion in many-body Green's function theory

TL;DR

This paper introduces a new method for handling zero eigenvalues in the equations of motion matrix in many-body Green's function theory, simplifying calculations by using singular value decomposition to project out null spaces.

Contribution

The paper proposes an alternative formulation that avoids the need for anti-commutator Green's functions by employing SVD to project out null spaces, applicable to momentum-dependent cases.

Findings

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

Develops numerical algorithms for momentum-dependent null space projection

Extends previous work to multilayer films and complex cases

Abstract

The spectral theorem of many-body Green's function theory relates thermodynamic correlations to Green's functions. More often than not, the matrix governing the equations of motion has zero eigenvalues. In this case, the standard text-book approach requires both commutator and anti-commutator Green's functions to obtain equations for that part of the correlation which does not lie in the null space of the matrix. In this paper, we show that this procedure fails if the projector onto the null space is dependent on the momentum vector. We propose an alternative formulation of the theory in terms of the non-null space alone and we show that a solution is possible if one can find a momentum-independent projector onto some subspace of the non-null space. To do this, we enlist the aid of the singular value decomposition (SVD) of the equation of motion matrix in order to project out the null space, thus reducing the size of the matrix and eliminating the need for the anti-commutator Green's function. We extend our previous work, dealing with a ferromagnetic Heisenberg monolayer and a momentum-independent projector onto the null space, where both multilayer films and a momentum-dependent projector are considered. We develop the numerical methods capable of handling these cases and offer a computational algorithmus that should be applicable to any similar problem arising in Green's function theory.