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Proceedings of the Physical Society

A new method of expansion in the quantum many-body problem: II: Thermodynamic Green functions

D Sherrington

Published Feb 1, 1967• 5 citations
Publisher

Abstract

The procedure of Edwards and Sherrington for expanding the zero-temperature Green functions of quantum many-body theory in terms of correlations about maximal quantum randomness is extended to cover themodynamic Green functions
DOI
10.1088/0370-1328/90/2/131

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In a discussion of the quantum dynamics of a non-relativistic many-particle system it is shown how a two-body interaction, quartic in the field operators, may be replaced by an interaction with an auxliary density-like field. An extension of the procedure of Edwards and Sherrington for expanding in terms of correlations about the state of maximal quantum randomness is applied to the coupled system to give the density correltions and excitations as well as the corresponding single-particle quantities. As an example, an explicit expression for the density excitations is obtained in terms of the quasi-particle energies, for the case in which quasi-particles exist as well-defined modes of single-particle excitation.
The spectral theorem of many-body Green function theory relates thermodynamic correlations to Green functions. More often than not, the matrix governing the equations of motion has zero eigenvalues. In this case, the standard textbook approach requires both commutator and anti-commutator Green 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, k . 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 of in order to project out the null space, thus reducing the size of the matrix and eliminating the need for the anti-commutator Green function. We extend our previous work (Fröbrich and Kuntz 2003 Phys. Rev. B 68 014410), dealing with a ferromagnetic Heisenberg monolayer and a momentum-independent projector onto the null space, to models 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 algorithm that should be applicable to any similar problem arising in Green function theory.
In a preceding paper [J. Math. Phys. 3, 475 (1962)], some model Hamiltonians were proposed for quantum-mechanical many-body systems with pair forces. For infinite systems in thermal equilibrium, they led to temperature-domain propagator expansions which were formally summable and expressible by closed equations. These expansions were identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models corresponded to ring- and ladder-diagram summations from the true propagator expansion, augmented by infinite classes of self-energy corrections. The model Hamiltonians were called stochastic because they contained parameters whose phases were fixed by random choices. In the present paper, more general models are formulated which yield formally summable propagator expansions for finite systems. The analysis is extended to correlation and Green's functions defined for nonequilibrium ensembles. The nonequilibrium treatment is developed in the Heisenberg representation in such a way that unlinked diagrams do not arise. A basic convergence question associated with the formal closed equations for the model propagators and correlation functions is examined by means of finite-difference integration of the Heisenberg equations of motion. This procedure appears to converge independently of whether the perturbation expansions for the propagators and correlation functions converge. It yields substantial support for the validity of the formal closed model equations.