In the past few decades, research into electromagnetic properties of topological quantum materials has been one of the most active research areas in the field of condensed matter physics. Physicists have discovered a large class of materials, e.g. Weyl semimetals, topological insulators, and topological superconductors that can host a plethora of interesting topological properties. In addition to their theoretical value as novel and exotic phases of quantum matter, topological quantum materials provide a promising platform for an array of technological applications, particularly as building blocks of topological quantum computers. Unfortunately, despite great progress in the theoretical understanding of topological phases of matter, practical problems have made it difficult to: (i) identify unambiguous examples of topological quantum material and (ii) harness their potential for technological applications. The overarching goal of this thesis is to understand such difficulties and to find ways to overcome them by studying specific problems.

This thesis is divided into four independent parts, each of which is dedicated to a particular problem: In the first part, we study chiral magnetic effect in Weyl semimetals and discuss whether it can be used to probe topological properties of Weyl semimetals in real experiments.

In the second part, we propose an experimental setup to realize a certain type of topological excitation called $\mathbb{Z}_3$ parafermionic zero mode using a quantum dot array structure from the $2/3$ fractional quantum Hall state. Importantly, our proposal does not rely on Andreev backscattering. We argue that this feature makes our proposal suitable for experimental realization.

In the third part, we provide a quantitative analysis of supercurrent in superconductor/quantum Hall/superconductor junctions and show that by making critical assumptions about the interface, it is possible to obtain a quantitative agreement between theory and the magnitude of the observed supercurrent.

In the fourth part, we study quantum anomalous Hall effect and flavor ferromagnetism in twisted bilayer graphene and argue that the one-magnon spectrum can be used as a numerically accessible tool to study the stability of the quantum anomalous Hall phase in twisted bilayer graphene.