Exploring the interplay between topological order, magnetism and superconductivity

This thesis presents a theoretical study of topological insulators coupled with superconductor and magnet. We discuss different physics due to these novel couplings and the topological properties. Chapter 2 describes the background of the research projects. We present different Quantum Hall systems, discuss their topological properties. Also we provide some basic formulas in Luttinger liquid theory, which will be used heavily in this thesis. Chapter 3 presents different phases we discovered in 2D topological insulators. We explore the phases exhibited by an interacting quantum spin Hall edge state in the presence of finite chemical potential (applied gate voltage) and spin imbalance (applied magnetic field). We find that the helical nature of the edge state gives rise to orders that are expected to be absent in non-chiral one-dimensional electronic systems. For repulsive interactions, the ordered state has an oscillatory spin texture whose ordering wavevector is controlled by the chemical potential. We analyze the manner in which a magnetic impurity provides signatures of such oscillations. We find that finite spin imbalance favors a finite current carrying groundstate that is not condensed in the absence of interactions and is superconducting for attractive interactions. This state is characterized by FFLO-type oscillations where the Cooper pairs obtain a finite center of mass momentum. Chapter 4 describes the new spin Josephson effect. We explore a spin Josephson effect in a system of two ferromagnets coupled by a tunnel junction formed of 2D time-reversal invariant topological insulators. In analogy with the more commonly studied instance of the Josephson effect for charge in superconductors, we investigate properties of the phase-coherent {\it spin} current resulting from the misalignment of the in-plane magnetization angles of the two ferromagnets. We show that the topological insulating barrier offers the exciting prospect of hosting a {\it fractional} spin Josephson effect mediated by bound states at the ferromagnet-topological insulator interface. We provide multiple perspectives to understand the $4\pi$ periodic nature of this effect. We discuss several measurable consequences, such as, the generation of a transverse voltage signal which allows for purely electrical measurements, an inverse of this effect where an applied voltage gives rise to a transverse spin-current, and a fractional AC spin-Josephson effect. Chapter 5 presents the inverse spin pumping effect. We study the dynamics of a quantum spin Hall edge coupled to a magnet with its own dynamics. Using spin transfer torque principles, we analyze the interplay between spin currents in the edge state and dynamics of the axis of the magnet, and draw parallels with circuit analogies. As a highlighting feature, we show that while coupling to a magnet typically renders the edge state insulating by opening a gap, in the presence of a small potential bias, spin-transfer torque can restore perfect conductance by transferring angular momentum to the magnet. In the presence of interactions within the edge state, we employ a Luttinger liquid treatment to show that the edge, when subject to a small voltage bias, tends to form a unique dynamic rotating spin wave state that naturally couples into the dynamics of the magnet. We briefly discuss realistic physical parameters and constraints for observing this interplay between quantum spin Hall and spin-transfer torque physics. Chapter 6 discusses possible mass terms in 3D topological insulators. We provide a characterization of tunneling between coupled topological insulators in 2D and 3D under the influence of a ferromagnetic layer. We explore conditions for such systems to exhibit integer quantum Hall physics and localized fractional charge, also taking into account interaction effects for the 2D case. We show that the effects of tunneling are topologically equivalent to a certain deformation or folding of the sample geometry. Our key advance is the realization that the quantum Hall or fractional charge physics can appear in the presence of only a \emph{single} magnet unlike previous proposals which involve magnetic domain walls on the surface or edges of topological insulators respectively. We give illustrative topological folding arguments to prove our results and show that for the 2D case our results are robust even in the presence of interactions