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

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.

arXiv comment16 pages, 7 figures

References in corpus (7)

• Many-body Green's function theory for the magnetic reorientation of thin ferromagnetic films
• Quantum Monte Carlo simulation of thin magnetic films
• Many-body Green's function theory for thin ferromagnetic anisotropic Heisenberg films: treatment of the exchange anisotropy
• Many-body Green's function theory for thin ferromagnetic films: exact treatment of the single-ion anisotropy
• The spectral theorem of many-body Green's function theory when there are zero eigenvalues of the matrix governing the equations of motion
• Single-ion versus exchange anisotropy in calculating anisotropic susceptibilities of thin ferromagnetic Heisenberg films within many-body Green's function theory
• Anisotropic susceptibilities of thin ferromagnetic films within many-body Green's function theory

Cited by in corpus (5)

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• The internal energies of Heisenberg magnetic systems
• Coupled ferro-antiferromagnetic Heisenberg bilayers investigated by many-body Green's function theory
• Projective-truncation-approximation study of the one-dimensional ${\varphi }^4$ lattice model
• The magnetic reorientation transition in thin ferromagnetic films treated by many-body Green's function theory