Shift and Polarization of Excitons from Quantum Geometry

Despite a long history, certain aspects of excitons - the bound inter-band states which form when a valence band hole and a conduction band electron pair - have remained relatively unexplored. This holds particularly true for the wavefunction of an exciton, for which few properties have been explored theoretically in various limiting cases. An intuitive language robustly characterizing the topology of bound electron-hole states is lacking, but needed in order to address the global features of the charge distribution of the excitonic state, to properly understand their transport theory, and to supplement the numerical investigation of excitons in ab-initio approaches. Here, we address these gaps by developing a comprehensive framework for the quantum geometry and topology of two-dimensional exciton states in terms of the exact connections which describe the interaction-renormalized exciton bundle in a periodic lattice. Based on this description, we derive two gauge-invariant quantities, which we identify as the exciton shift vector and the exciton dipole vector. Using the shift vector, we elucidate the topology of exciton bands compared to the topology of the parent electronic band structure, pinpointing precisely how interactions can introduce nontrivial topology to the exciton bands beyond the topology which is contained in the single-particle bands. We further elucidate how shift and polarizations enter into the semiclassical equations of motion for the exciton center of mass coordinates.