Electron fractionalization in two-dimensional quantum systems: majorana fermions and fractional topological insulators

In condensed matter physics, two distinct notions of topological phases exist, both of which assume a ground state that is separated from all excited states by an energy gap. On one hand, topological band insulators are characterized by gapless excitations that propagate at the system boundary, are universal and robust to localization by disorder. The existence of these boundary degrees of freedom can be predicted from bulk topological invariants of the system, giving rise to a bulk-boundary correspondence. On the other hand, incompressible and featureless many-body phases possess non-trivial topological order, if they support deconfined particle-like excitations with quantum numbers that are a fraction of the quantum numbers of the constituent particles (electrons) of the state. This fractionalization is in one-to-one correspondence with a degeneracy of the groundstate that depends on the genus of the configuration space manifold on which the state is defined. The fractionalized excitations are generically anyons, which in two-dimensional space can have either Abelian or non-Abelian exchange statistics. In this Thesis, di↵erent phases of matter with topological order are studied. Microscopic models in which these phases emerge are provided and characterized. The first Part is concerned with topologically ordered two-dimensional superconductors. The fractionalized excitations of these systems are non-Abelian Majorana states that are bound to vortices in the superconducting order parameter. A Majorana state is a superposition of an electron and a hole and possesses half the degrees of freedom of an electron. We find that whether or not a vortex binds an isolated Majorana state depends sensitively on the normal state band structure of the two-dimensional electron gas. We contrast the situation where the dispersion is that of a massive fermion subject to a Rashba spin-orbit coupling that breaks the SU(2) spin-rotation symmetry with the situation where the dispersion is that of massless Dirac fermions. The former case pertains to two-dimensional electron gases that emerge at interfaces between di↵erent crystals, while the latter case is realized at the surface of three-dimensional topological band insulators. We find that isolated Majorana states can only exist in vortices of the Dirac electron superconductor, regardless whether the superconducting order parameter is dominantly of s-wave or p-wave type. Furthermore, we discuss the peculiar properties of the two-dimensional superconductors without spin-rotation symmetry. They lead us to proposing a setup for a heterostructure that binds a chain of equidistantly spaced Majorana states. We proceed with a second type of topologically ordered states, the so-called fractional Chern insulators and fractional topological insulators. These states emerge in two-dimensional lattice models of repulsively interacting electrons, if the band structure of the model in absence of electron-electron interactions satisfies two prerequisites: First, the electronic band structure must have a nontrivial topological property. We consider the case where a band of either a Chern insulator or a Z2 topological insulator is partially filled. After interactions are switched on, it then yields a fractional Chern insulator and a fractional topological insulator, respectively. Second, the electronic band structure must have a specific energetical property, namely the band of interest must be reasonably flat as compared to the energy gap and the interaction energy scale. We devise lattice models that have these properties and show with exact numerical diagonalization studies of small systems that they support topologically ordered ground states. This way, we find fractional Chern insulators in Chern bands with Chern number one and two, as well as their spontaneous formation in spectrally flattened Z2 topological band insulators, if the interaction breaks time-reversal symmetry by the Stoner mechanism. The fractional Chern insulators resemble in their universal topological properties known fractional quantum Hall states that appear in Landau levels. For the Laughlin series of filling fractions ⌫ = 1/m, m 2 Z, of the flat band, fractional Chern insulators support chiral edge states of anyons with the m-th fraction of the electron charge. These edge states cannot be localized by local perturbations at the edge. In contrast, fractional topological insulators with time-reversal