Theoretical description of interacting topological condensed matter systems

Abstract

This thesis aims to review the theory of non-interacting topological insulators and to develop a path towards the treatment of interacting systems. Starting from how Berry phase arises in a system with non-trivial topology, we will develop the underlying mathematical description of topology and talk about the topological invariant that characterizes the topology. We will then study the integer quantum Hall effect, which is an experimentally well-established topological system. We further the discussion to non-interacting systems with time reversal symmetry that gives rise to (2+1)D and (3+1)D topological insulators with another experimentally verified model – the BHZ model. We will also show how topological band theory and field theory can arrive at the same result and briefly discuss the role of the topological invariant in the context of differential geometry. Next, we will prepare the techniques to work with interacting system by constructing the framework of finite temperature many- body condensed matter field theory using Matsubara Green’s function. We will calculate the one-loop electron-phonon self energy in electron-phonon interaction to investigate how interaction change the behaviour of electrons. Finally, we will apply the dimensional reduction method to time reversal invariant integer quantum Hall effect in (4+1)D, which enables us to obtain the (3+1)D topological insulator.