A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions
d>2. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions
d>1, in agreement with other recent field-theoretical treatments of clean Fermi liquids.
The disorder averaged single-particle Green's function of electrons subject to a time-dependent random potential with long-range spatial correlations is calculated by means of bosonization in arbitrary dimensions. For static disorder our method is equivalent with conventional perturbation theory based on the lowest order Born approximation. For dynamic disorder, however, we obtain a new non-perturbative expression for the average Green's function. Bosonization also provides a solid microscopic basis for the description of the quantum dynamics of an interacting many-body system via an effective stochastic model with Gaussian probability distribution.
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.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (
qc/
kF)dF0 [1+
F0${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here
qc is the cutoff of the interaction in wave-vector space,
kF is the Fermi wave vector, \ensuremath{\mu} is the chemical potential,
F0 is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff
qc is small compared with
kF, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.