The discovery of topological states of matters has sparked intense interests amongresearchers in the past decade. Topologically non-trivial band structure in thesequantum states can give rise to a variety of topological phenomena, the experimentaldemonstration of which can have a huge impact on our understandingof fundamental states of matter. Transport measurement is one of the majorexperimental techniques to probe these topological phenomena. This dissertationis devoted to theoretical and numerical studies of quantum transport phenomenain a variety of topological materials, including magnetic topological insulator films,the quantum anomalous Hall insulator/superconductor hetero-structures, the kinkstates in bilayer graphene and the photonic crystal of topological mirror insulatorphase in the optical regime. The numerical simulations of transport phenomenaand the analytical understanding of the underlying physical mechanism in thisdissertation will provide guidance for the future transport measurements.The numerical methods to simulate quantum transport in this dissertation arebased on Landauer-Bttiker formalism and Greens function method, which willbe introduced in Chapter 2. The transmission through certain sample regionscan be extracted from the Greens function method and serves as the input forthe Landauer-Bttiker formalism to compute conductance tensor that is directlymeasured in transport experiments. Physical understanding of the transportmechanism can be provided by analyzing different components of the transmissionmatrix, in combination with other analytical methods for transport phenomena.Defects and impurities can be incorporated in numerical simulations by includingrandom potentials into the model Hamiltonian, and thus this method can be appliedin different transport regimes, from ballistic to diffusive transport.Chapter 3 to 5 of the dissertation is to apply the above numerical methodsto three different topological mesoscopic systems: magnetic topological insulator(MTI) films, quantum anomalous Hall insulator (QAHI) - superconductor (SC)junctions, and bilayer graphene devices.Chapter 3 is dedicated to the study of quantum transport through magnetictextures in a thin film of MTI. We focus on both the longitudinal and Hall transports,which reveal complicated features due to the coexistence of strong spin-orbit couplingfrom TI materials and magnetic non-colinearity from magnetic textures in thissystem. The manifested Hall transport can be induced by different topologicalmechanisms, including the intrinsic anomalous Hall effect from strong SOC and thetopological Hall effect (or known as geometric Hall effect) from magnetic textures.Thus, this system provides a nice platform to understand the interplay betweenspin-orbit coupling and real-space magnetic texture, as well as disorder scatterings.Our numerical simulations have shown different roles of spin-orbit coupling in theclean and disordered limits for this system. In the clean limit when SOC strengthis increased, the topological Hall conductance (THC) almost remains constant butthe topological Hall resistance (THR) can increase by an order of magnitude dueto the reduction of longitudinal conductance, caused by SOC-induced spin flips.However, in the disordered limit, both the THC and THR increase with increasingSOC, while longitudinal conductance is not influenced much by SOC.In Chapter 4, we study the transport of chiral edge channels in a QAHI/superconductorjunction. This type of hetero-junction has been recently fabricated andmeasured in experiments, in pursue of topological superconductivity and Majoranafermions. We focus on the disorder effect in the weak superconductor proximitylimit. Our results show that the quantized valued of conductance remains robustfor a single chiral edge channel even in the presence of disorder in the zero-biaslimit. However, such quantization is broken down for a finite bias, or for multiplechiral edge modes, or for the coexistence of a single chiral edge mode with othertrivial metallic modes, when disorders are present. Our theory provides guidanceto understand transport phenomena in these systems for future experiments.Chapter 5 is a simulation of transport behaviors through the so-called kinkstates in a bilayer graphene device under external electric and magnetic fields. Thedevice, known as a valley valve and electron beam splitter, has been fabricatedby our experimental collaborators and its unusual transport properties have beenmeasured experimentally. Our numerical simulations provide a justification of theguiding center physical picture for topological transport through this device.Chapter 6 goes beyond electronic systems and concerns topological phase inphotonic systems. We utilize a method of dynamic evolution of states to studya topological crystalline insulator phase in a photonic system. The crystallineprotection, achieved by the fine manufacturing of emulated atoms in a photoniclattice, selectively pumps incident states with a certain parity while reflects theother.The studies in the dissertation are in close collaboration with experimentalgroups, including Prof. Moses Chans and Prof. Cui-zu Changs group for the transportmeasurements in MTI films and QAHI/SC junctions, Prof. Jun Zhus groupfor the experiments on the bilayer graphene device, and Prof. Mikael Rechtsmansgroup for the photonic topological systems.

Topological insulators are materials that are insulating in the bulk but that conduct via topologically protected states on the boundary. The concept of topology in condensed matter physics was first introduced to explain the integer quantum Hall (IQH) effect. The perfect quantization of these topologically protected edge states, insensitive to sample geometry and disorder, stimulated an extensive search for many exciting new topological materials. One of the milestones along the journey was the theoretical prediction and experimental discovery of Z2 topological insulators.The first class of Z2 topological insulators discovered was the 2-dimensional topological insulator (2D TI), also known as the quantum spin Hall (QSH) insulator. The 2D TI can be viewed as a variation of the IQH system but with time-reversal-symmetry (TRS). The topological invariant for a 2D TI is the Z2 number, defined by its nontrivial band structure instead of the Chern number in the IQH case. Generalizing this idea to 3 dimensions led to the discovery of the 3D TI with four Z2 invariants. Both the 2D and 3D TIs are of interest as model platforms for testing theoretical problems of fundamental interest. For instance, they allow us to realize artificial condensed matter analogs of fundamental particles such as Majorana fermions and axions that have yet to be observed in nature. They are also of interest for potential technological applications, principally spintronics and quantum computing.This dissertation focuses on the synthesis, characterization, and transport properties of both 2D and 3D TIs. We first discuss the 2D TI candidate material system, type II InAs/GaSb quantum wells, which exhibits a rich topological phase diagram that can be tuned by several parameters such as sample geometry or electrostatic gating. By changing the thicknesses of relevant layers, we are able to enter a new insulating regime where unexpected high-density quantum oscillations are observed. We elucidate this phenomenon through theoretical calculation and through control experiments. The seemingly controversial coexistence of high density states and the insulating regime can be explained by the effect of the attractive Coulomb interaction, which was not considered in earlier theories.The second topic we address is quantum transport in 3D TI systems. Breaking the TRS of the 3D TI surface states leads to many exotic phenomena, including the quantum anomalous Hall (QAH) effect and the axion insulator state. By constructing a sandwich heterostructure that has different magnetic coercive fields in the top and bottom magnetic layers, while keeping the center layer free from magnetic impurities, both the QAH and the axion insulator state can be observed in low-temperature transport measurements, when the magnetization alignment of the top and bottom layers is parallel and antiparallel, respectively. We also discuss the scaling behavior of the topological quantum phase transition between these two states.

This book is the result of dynamic developments that have occurred in condensed matter physics after the recent discovery of a new class of electronic materials: topological insulators. A topological insulator is a material that behaves as a band insulator in its interior, while acting as a metallic conductor at its surface. The surface current car