Interplay of Interaction and Topology From Topological Band Theory to Topological Field Theory

Abstract

Classification of phases of matter has long been a central point of interest in the research community in condensed matter physics. Over the past several decades mainly two kinds of classification schemes have emerged, namely, one related to spontaneous symmetry breaking and the other to symmetry invariant topological classification. Although the latter one is fairly new, a lot of progress has been made over the past couple of decades, but still these two classification schemes are generally treated separately. In this thesis, I discuss systems where both the classification schemes need to be invoked due to the interplay of interaction and topology and their effect on each other. In the first piece of work, I present the theory of a new type of topological quantum order which is driven by the spin-orbit density wave order parameter and distinguished by Z2 topological invariant. The resulting quantum order parameters break translational symmetry but preserve time-reversal symmetry. Consequently, the system is inherently associated with a Z2 topological invariant along each density wave propagation direction which makes it a weak topological insulator in two dimensions, with an even number of spin-polarized boundary states. In the second work, I discuss the effect of the parent topological ground state on the local order. In particular, I focus on a well-studied (experimentally) material TlCuCl3 and show that it has unique unexplored topological properties which arise when a time-reversal breaking antiferromagnetic order parameter sets into the system and how they can explain the uncanny properties of this material such as unconventional paramagnon lifetime, finite Higgs mass across the phase transition, among other. In the last work, I discuss our attempt to confirm the presence of bosonic integer and fractional quantum hall effect in an interacting lattice model. The model consists of bosons spread over the honeycomb lattice with the nearest neighbour and next nearest correlated hopping with flux per hexagon. I provide evidence for the presence of integer as well as fractional quantum Hall states and also a superfluid state, for different fillings and tuning parameters. I have used mean-field theory and path integral methods as the theoretical tools to study the above problems. Furthermore, I have also used numerical methods such as Density Functional Theory (DFT, as implemented in VASP) and exact diagonalization (using Lanczos algorithm) where appropriate. As I have shown in the thesis, a large number of interesting results emerge from these studies, leading to a better understanding of the problems and uncovering some interesting underlying physics. Overall, this thesis brings forth some notable and interesting possibilities in understanding the physics of topological state of mater. An idea that stands out is that effect of topological phase onto the symmetry breaking phase and vice-versa. This will have an important impact on the future studies in the related area and to find a unified theory of phase transitions which incorporate both symmetry breaking and topological transition together along with more sophisticated like inclusion of Gaussian fluctuations on the top of the mean-field.