Topological states in aperiodic, non Hermitian and electronically correlated systems

Abstract
This thesis explores how topological concepts, usually associated with abstract mathematics, provide insights into complex physical systems. Physicists use topology to study systems that exhibit robust properties unaffected by minor changes. For example, topological insulators are materials that conduct electricity only at their boundaries and could improve the efficiency of current electronics due to their dissipationless transport properties. Similarly, topological superconductors might enable scalable quantum computing by protecting delicate quantum states from environmental interference. This thesis focuses on multiple aspects where topology can arise. One of those is on non-Hermitian systems, which typically describe systems coupled to environmental interactions. Using graph theory and the concept of latent symmetries, we identified new topological phases within seemingly complicated systems. We also constructed a continuum approximation framework to study disorder at phase transitions in non-Hermitian systems, providing insights into topological stability. Beyond non-Hermitian systems, we also developed, along similar lines, continuum approximations to study a topological insulator in the presence of electron-electron interactions, showing that they do not alter the principal mechanism behind a topological phase. A second focus is on aperiodic structures like quasicrystals, which lack traditional periodic order but exhibit long-range correlations. Quasicrystals can be constructed by projecting higher-dimensional crystals into lower dimensions, revealing hidden fractal structures. The thesis explores how impurities disrupt quasicrystalline order and examines topological phenomena in these aperiodic systems, such as topological charge pumping and boundary states protected by inversion symmetry. To summarize, this work reveals hidden topological structures in non-traditional systems, offering a foundation for future experimental and theoretical research in topological phases and aperiodic materials.