Quantum phase transitions and quantum transport in low-dimensional topological systems

In this thesis, we focus on quantum phase transitions that change the topological index of topological insulators and superconductors, which are states of matter featuring topologically protected edge states and insulating bulk, and on transport of charge and spin in topological insulator nanostructures. We consider topological phases in disordered quasi- 1D topological superconductors. The Majorana edge states on topologically nontrivial nanowires were previously found to be protected from disorder as long as the localization length is larger than the coherence length, after which the wire transitions to a trivial state. We find that changing disorder can push the system back into a topological state in multichanneled nanowires, creating previously unreported fragmentation of the topological phase diagram. We next discuss arbitrarily-shaped and/or disordered topological superconductors and their ground state fermion parity. As external parameters are varied, even and odd parity ground states cross, causing quantum phase transitions. We find that the statistics of parity-crossings are universal and described by normal-state properties and determine the shape dependence of the parity crossings. Finally, we consider edge state quantum transport in quantum spin Hall insulators in the presence of nuclear spins. We find that a properly initialized nuclear spin bath can be used as a non-energetic resource to induce charge current in the device, providing power an external load using heat from electrical reservoirs. Resetting the spin-resource requires dissipation of heat in agreement with the Landauer's principle. Our calculations show that the equivalent energy/power density stored in the device exceeds existing supercapacitors