Keeping it Real: An Alternative Picture for Symmetry and Topology in Condensed Matter Systems

Abstract

While physical systems are typically defined in real space, oftentimes the origins of intriguing phenomena are understood only after one passes into descriptions in the momentum space or, more abstractly, in the parameter space of all possible systems sharing certain common traits. This dissertation asserts that a direct analysis on the symmetry and topology of the real-space description can serve as a powerful alternative. The assertion is supported by new results obtained in three rather distinct classes of problems. First, for noninteracting electrons in a crystal symmetric under any one of the 1,651 magnetic space groups, we will prove that, insofar as symmetry representations are concerned, a comprehensive analysis on all real-space insulators leads to the full set of possible band structures, including those which are topological. Second, in the context of strongly correlated quantum magnets, we will argue that nontrivial topology of the spin lattice can necessitate the development of either spontaneous symmetry breaking or intrinsic topological orders. Finally, we will turn our attention to the study of quantum-information flow in the real space, and show that it leads to an extension of chiral topological phases to the reign of strongly out-of-equilibrium quantum dynamics.