The synergy of topology and physics has revolutionized how we understand and classify matter in recent decades. By bringing concepts from an abstract mathematical realm of topology, it has been shown that quantum mechanical wave functions can tie knots and do twists in the abstract spaces they live in. Most importantly, these abstract knots and twists come to life as observables in the form of perfectly quantized integer or fractional responses. These topological properties are extremely robust, and hence, even constitute promising candidates for advanced electronics and fault-tolerant quantum computation schemes. Topological systems involving non-Abelian braiding offer more exotic properties, where doing two chosen operations in different orders result in different effects. Characterization and experimental observation of such topologies are active fields of research in conventional materials as well as in state-of-the-art quantum simulators which are artificial systems cleverly designed to simulate these quantum phenomena. At this junction, this project has considered settings beyond equilibrium since life is dynamic and our technology relies on non-equilibrium physics. We have investigated out-of-equilibrium dynamics and classification of topological systems, unearthed novel robust responses far from equilibrium and explored how to harness these properties in laboratories. We have discovered new topological phases where electrons perform a precise dance to realize these non-Abelian properties which cannot exists in a static context.
