Analysis and applications of the generalised Dyson mapping

[摘要] ENGLISH ABSTRACT:In this thesis, generalized Dyson boson-fermion mappings are considered. These aretechniques used in the analysis of the quantum many-body problem, and are instancesof so-called boson expansion methods. A generalized Dyson boson-fermion mapping, ora Dyson mapping for short, is a one-to-one linear but non-unitary operator that can beapplied to vectors representing the states of a many-fermion system. A vector representinga fermion system maps onto a vector that is most naturally interpreted as representinga state of a many-body system that contains both bosons and fermions. The motivationfor doing such a mapping is the hope that the mapping will reveal some property of thesystem that simplifies its analysis and that was hidden in the original form. The aims ofthis thesis are1. to review the theory of generalized Dyson boson-fermion mappings,2. by considering a tutorial example, to demonstrate that it is feasible to implementthe theory and3. to find a useful application for a generalized Dyson boson-fermion mapping, byconsidering a non-trivial model, namely the Richardson model for superconductivity.The realization of the first two aims mainly involve the collecting together of ideas thathave already appeared in the literature, into one coherent text. Some subtle points thatwere treated only briefly due to space restrictions in the journal publications where thetheory was first expounded, are elaborated on in the present work. On the other hand, theanalysis of the Richardson Hamiltonian that uses a Dyson mapping, goes beyond what hasalready appeared in the literature. It is the first time that a boson expansion techniqueis implemented for a system where the roles of both collective and non-collective fermionpairs are important. (The Dyson mapping associates bosons with Cooper pairs, whilethe fermions not bound in Cooper pairs result in fermions being present in the mappedsystem as well.) What is found is that the Dyson mapping uncovers non-trivial propertiesof the system. These properties aid the construction of time-independent perturbationexpansions for the stationary states of the system, as well as time-dependent expansionsfor transition amplitudes between states. The time-independent expansions agree withresults that other authors obtained through methods other than boson expansions. Thetime-dependent expansions, that one would be hard-pressed to develop without a Dysonmapping, might in future prove useful in understanding aspects of the dynamics of ultracoldfermi gases, when time-dependent magnetic fields are used to vary the atom-atominteraction strenght.

[发布机构] Stellenbosch University