Geometric phases lead to a nontrivial interference result when an electron's different quantum mechanical paths choices encircle a magnetic coil in an Aharonov-Bohm experiment. They are also... Show moreGeometric phases lead to a nontrivial interference result when an electron's different quantum mechanical paths choices encircle a magnetic coil in an Aharonov-Bohm experiment. They are also responsible for the daily precession of a Foucault pendulum in Paris. A dynamical shape change induces a geometric phase, which, for instance, cats use to rotate when falling and swimmers use to swim forward.A modern application of such geometric phases has led to the notion of topological phases, which are described by a global property of the system. These phases are very different from the classical phases of matter, which are characterized by a local order parameter. A topological phase transition is therefore a fundamentally different process compared to a classical one as in a liquid-gas transition, because the former requires a change of a global topological index of the system. Topological phases can, for example, lead to the presence of traveling electronic modes which are robust against being backscattered by obstacles at the boundary of an insulator.This thesis describes some applications of geometric and topological phases in soft-matter systems. Show less
