Low-energy theories for strongly correlated spin-orbit electron systems in one and two spatial dimensions
The main goal of our SFB 1170 project is to investigate the interplay of interactions, spin-orbit cou- pling, lattice degrees of freedom, and topology in strongly correlated electron systems in one and two spatial dimensions. The previous SFB 1170 funding period led to several pioneering findings within our project, which we wish to continue and complement with new pursuits. As the first principal di- rection, we intend to study the pivotal role of electronic interactions on topological edge modes. This particularly applies to one-dimensional edge modes in novel high-temperature quantum spin Hall ma- terial candidates such as bismuthene on SiC where, in contrast to previous instances, phonons need to be taken into consideration due to the elevated temperatures. We plan to employ the equations-of- motion approach we have developed for interacting helical edge states, and extend it in order to allow for the inclusion of dissipation, spatially dependent Luttinger parameters, and multi-gate settings on a universal footing. We further wish to analyze the dimensional edge state hierarchies discovered at step edges of three-dimensional crystalline topoloigcal insulators, where the inclusion of interactions suggests intricate connections to interacting higher-order topological insulators. As the second prin- cipal direction, we intend to extend insights we have derived from interactions on topological band structures to novel avenues of topologically ordered many-body ground states. Specifically, inspired by the fractional quantum Hall effect, spin liquids, and fractional Chern insulators, we wish to develop a wave function approach to generic topological order in a two-dimensional crystal of spin-orbit cou- pled electrons. This involves fractional topological insulators and generalizations thereof. In terms of model systems, we plan to use a toroidal geometry as well as the spherical geometry we have re- cently developed to describe a 3D topological insulator within a numerically amenable environment.