Quantum thermodynamics, spintronics and zero mode statistics in topological matter

The aim of this thesis is to explore the charge and spin dynamics in topological systems from the point of view of quantum thermodynamics, spintronics and quantum chaos. In particular, we propose and investigate a novel way of implementing Maxwell’s demon engines by using topological systems that feature spin-momentum locking. As a memory resource for Maxwell’s demon, we utilize a "spin bath" that usually includes the nuclear spins which are naturally present in the device and/or magnetic impurities introduced via doping. The significance of this Maxwell’s demon implementation is its scalability of the memory size. We introduce two novel Maxwell’s demon implementations on two different setups. The first setup is a quantum spin Hall insulator in the presence of nuclear spins, which we call a quantum information engine, alluding to conversion of information entropy of the Maxwell’s demon memory into usable free energy. We investigate heat and charge transport in a quantum information engine and describe the protocol for memory erasure and work extraction. The second setup is a hybrid system, composed of a quantum spin Hall insulator connected to two quantum anomalous Hall insulators with different spins and chiralities. The distinctive feature of this setup is that it not only operates at the Landauer’s limit, providing higher efficiency and denser energy storage capability at weak couplings, but it also provides additional functionality via its memristive response. Furthermore, we extend the discussion to three dimensional topological insulators, featuring spin-momentum locked surface states that allow dissipative nonmagnetic impurity scattering. Finally, we study the spectra of fermion-parity switches in topological superconductors that host Majorana zero modes from the perspective of spectral geometry and random matrix theory and show that these spectra depend on the geometrical shape of the topological superconductor and can distinguish chaotic shapes from integrable shapes