Field theory of disordered bosons
Sebastian E. Schmittner
The goal of this diploma thesis is to develop a field theory, for a specific class of disordered bosonic systems, in order to study, whether disordered bosons generically \nhave universal statistical properties, similar to fermions. \nNow, if disordered bosons show system independent statistical features, the evident \nquestion is whether the bosonic universality classes are different from the fermionic \nones. For fermions, the classification in terms of symmetric spaces is a largely algebraical one, which happens in a complexified setting before compact or non-compact \nreal forms are specified. Hence one can rightly state that all effective bosonic non-linear \nσ-models are already classified within the ten fold way together with the fermionic ones. \nAnd indeed, in the system studied in [LSZ06], well known universal GUE statistics \nwere found in the bulk and Lück finds GOE statistics in [Lü09], both for systems of \ndisordered bosons. \nBut for bosons non-linear σ-models are not everything. A subtle point about noninteracting, i.e. quadratic, bosonic systems is stability. One single particle state \nof negative energy for a bosonic system immediately leads to an unbounded many \nparticle Hamiltonian. For fermionic systems, a finite number of such single particle \nstates is of no concern, due to the Pauli principle. This important difference leads \nto the well studied Gaussian ensembles not providing feasible distributions of bosonic \nHamiltonians. If, however, more complicated probability distributions are taken into \naccount, one has to be prepared to face a more complicated effective model than a \npure non-linear σ-model in the end. In this sense, an interesting question is whether \nbosonic systems if they show universal behaviour lead to novel universality classes. \nA hint to those might be the unusual density of states near zero energy that was \nfound in [LSZ06] or [GC02] and [GA04]. On physical grounds, one can expect the \ncommutation relations of bosons to have important effects at low frequencies, which \nfits to the observations of [GC02] and [GA05]. \nIn this thesis, we will develop and study a specific model, which is the next step \nafter the work of Lück, Sommers and Zirnbauer, [LSZ06], towards a more realistic \ndescription of a disordered system of bosonic degrees of freedom. As in [LSZ06], our \nmodel is still purely random, i.e. there is no limit of an underlying pure system and \nwe also stay with a model without global symmetries, such as time reversal or charge \nconjugation. But we turn from the homogeneous, zero-dimensional model to a spatially \nextended one and implement the physically important feature of locality. Furthermore, \nwe add an additional parameter to tune the number of modes per volume. In particular \nwe can hereby enforce a macroscopical number of zero modes. The goal is to study the \ndensity of states of the new model and therefore to develop an effective field theory. \nThroughout we will pay special attention to the region near zero frequency and watch \nout for unusual, possibly universal features.
•Nov 1, 2010•0 citations