We demonstrate that two-dimensional chiral superconductors on curved surfaces spontaneously develop magnetic flux. This geometric Meissner effect provides an unequivocal signature of chiral superconductivity, which could be observed in layered materials under stress. We also employ the effect to explain some puzzling questions related to the location of zero-energy Majorana modes.

With the discovery of the quantum Hall effect more than thirty years ago, a whole new field emerged—that of topological quantum matter. This field is now a very mature one, and many different aspects are covered in the literature. The main text of this thesis introduces the field and gives a background to topological quantum matter, as well as topological aspects of superconductivity and the Abelian fractional quantum Hall (FQH) states.

 Together with the main text there are five articles that address five different questions, all connected to topological quantum matter.

In the first article, representative wave functions for the Abelian FQH states are calculated using conformal field theory methods. Before this paper was published, similar constructions had been restricted to flat geometries, but in this paper we generalize the analysis to the simplest curved geometry, namely the sphere. On top of being of interest for numerical studies (which usually are performed on a sphere), the response of the FQH liquids to curvature can be used to detect a topological quantity, the shift, which is the average orbital spin of the constituent electrons.

In the second article, we construct an effective field theory for the two-dimensional spinless, chiral p-wave superconductor that faithfully describes the topological properties of the bulk state, and also provides a model for the subgap states at vortex cores and edges. In particular, it captures the topologically protected zero-modes and has the correct ground state degeneracy on the torus.

In the third paper, tools for a hydrodynamic theory for insulators in three dimensions are derived. Specifically, we use functional bosonization to write insulators as a condensation phase of the U(1) gauge theory obtained in the functional bosonization language.

In the fourth paper, we investigate the edge Majorana modes in the two-dimensional chiral p-wave superconductor. We define the model on surfaces with different geometries—the annulus, the cylinder, the Möbius band, and a cone—and with different configurations of magnetic flux threading holes in these surfaces. In particular, we address the following question: Given that, in the absence of magnetic flux, the ground state on the annulus does not support Majorana modes, while the one on the cylinder does, how is it possible that the conical geometry can interpolate smoothly between the two?

In the fifth and last article, we demonstrate that two-dimensional chiral superconductors on curved surfaces spontaneously develop magnetic flux. We propose this geo-Meissner effect as an unequivocal signature of chiral superconductivity that could be observed in layered materials under stress. We also employ the effect to explain some puzzling questions related to the location of Majorana modes.

In this paper, we investigate the edge Majorana modes in the simplest possible p(x) + ip(y) superconductor defined on surfaces with different geometries, the annulus, the cylinder, the Mobius band, and a cone (by cone we mean a cone with the tip cut away so it is topologically equivalent to the annulus and cylinder), and with different configurations of magnetic fluxes threading holes in these surfaces. In particular, we shall address two questions: Given that, in the absence of any flux, the ground state on the annulus does not support Majorana modes while the one on the cylinder does, how is it possible that the conical geometry can interpolate smoothly between the two? Given that in finite geometries edge Majorana modes have to come in pairs, how can a p(x) + ip(y) state be defined on a Mobius band, which has only one edge? We show that the key to answering these questions is that the ground state depends on the geometry, even though all the surfaces are locally flat. In the case of the truncated cone, there is a nontrivial holonomy, while the nonorientable Mobius band must necessarily support a domain wall.

While many features of topological band insulators are commonly discussed at the level of single-particle electron wave functions, such as the gapless Dirac boundary spectrum, it remains elusive to develop a hydrodynamic or collective description of fermionic topological band insulators in 3+1 dimensions. As the Chern-Simons theory for the 2+1-dimensional quantum Hall effect, such a hydrodynamic effective field theory provides a universal description of topological band insulators, even in the presence of interactions, and that of putative fractional topological insulators. In this paper, we undertake this task by using the functional bosonization. The effective field theory in the functional bosonization is written in terms of a two-form gauge field, which couples to a U(1) gauge field that arises by gauging the continuous symmetry of the target system [the U(1) particle number conservation]. Integrating over the U(1) gauge field by using the electromagnetic duality, the resulting theory describes topological band insulators as a condensation phase of the U(1) gauge theory (or as a monopole condensation phase of the dual gauge field). The hydrodynamic description of the surface of topological insulators and the implication of its duality are also discussed. We also touch upon the hydrodynamic theory of fractional topological insulators by using the parton construction.

We construct an effective field theory for the 2d spinless p-wave paired superconductor that faithfully describes the topological properties of the bulk state, and also provides a model for the subgap states at vortex cores and edges. In particular, it captures the topologically protected zero modes and has the correct ground-state degeneracy on the torus. We also show that our effective field theory becomes a topological field theory in a well defined scaling limit and that the vortices have the expected non-Abelian braiding statistics.

States of matter with quasi-particle excitations that exhibit anyonic statisticsare of great theoretical interest, especially in those cases where the anyons arenon-Abelian. Two of the most promising states that could support non-Abeliananyons are the ν = 5/2 quantum Hall state and the two dimensional p-wavesuperconductor, and they can both be understood as p-wave paired states. Themotivation for the present thesis is to get a better understanding of the theo-retical description of p-wave paired states.In the accompanying paper we construct an effective field theory for the 2Dspin-less p-wave paired superconductor that faithfully describes the topologicalproperties of the bulk state, and also provides a model for the subgap statesat vortex cores and edges. In particular it captures the topologically protectedzero modes and has the correct ground state degeneracy on the torus. Wealso show that our effective field theory becomes a topological field theory in awell defined scaling limit and that the vortices have the expected non-Abelianbraiding statistics.

Representative wave functions, which encode the topological properties of the spin polarized fractional quantum Hall states in the lowest Landau level, can be expressed in terms of correlation functions in conformal field theories. Until now, the constructions have been restricted to flat geometries, but in this paper we generalize to the simplest curved geometry, namely that of a sphere. Except for being of interest for numerical studies, that usually are performed on a sphere, the response of the fractional quantum Hall (FQH) liquids to curvature can be used to detect a topological quantity, the shift S which is the average orbital spin of the constituent electrons. We give explicit expressions for representative wave functions on the sphere, for the full Abelian FQH hierarchy, and calculate the corresponding shifts. These microscopic results, based on wave functions, agree with the predictions from the effective Chern-Simons field theory. The methods we develop can also be applied to the planar case. It gives simpler expressions for states with both quasiparticle and quasihole condensates, and allows us to give closed form expressions for a general state in the hierarchy, rather than finding the wave function on a case by case basis.

We demonstrate that two-dimensional chiral superconductors on curved surfaces spontaneously develop magnetic flux. We propose this geo-Meissner effect as an unequivocal signature of chiral superconductivity that could be observed in layered materials under stress. We also employ the effect to explain some puzzling questions related to the location of Majorana modes.