Narrow Results

An algorithm and a computer program in Fortran 95 are presented which enumerate the Hugenholtz diagram representation of the many-body perturbation series for the ground state energy with a two-body interaction. The output is in a form suitable for post-processing such as automatic code generation. The result of a particular application, generation of LATEX code to draw the diagrams, is shown.

The many-body problem is solved for f-element complexes in solids. Multielectron orbitals are constructed from single-electron ones. Complexes of f-ions surrounded by sulphur, oxygen and fluoride ligands (atoms or ions) are employed as a model of radiating centres in order to describe the optoelectronic behavior of materials in the cluster approximation. Probabilities and intensities of luminescent- and laser-transition Stark components are calculated. Inhomogeneous broadening of line shapes in solids is studied for various excitation regimes.

Within the 1D Hubbard model, linear closed chains with various numbers of sites are considered in particle–particle self-consistent random phase approximation (SCRPA) for the T-matrix. Encouraging results with a minimal numerical effort are obtained, confirming earlier results with this theory for other models. SCRPA solves the two-site problem exactly. It therefore contains the two electrons and high density Fermi gas limits correctly.

The discovery of Pluto’s small moons in the last decade has brought attention to the dynamics of the dwarf planet’s satellites. With such systems in mind, we study a planar $N$-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar $N$-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid ‘Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits nonspherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in $N$-body systems such as the Pluto system where some of the bodies are highly nonspherical.

It is shown that the random-phase approximation (RPA) method with its nonlinear higher generalization, which was previously considered as approximation except for a very limited case, reproduces the exact solutions of the Lipkin model. The nonlinear higher RPA is based on an equation nonlinear on eigenvectors and includes many particle-many hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number $N=2$ and, numerically, for $N=8$. This finding indicates that the nonlinear higher RPA is equivalent to the exact Schrödinger equation, which opens up new possibilities for realistic calculations in many-body problems.