Hybrid Phase Space Topological Insulators

Topological insulators are usually studied in the context of translationally invariant systems for which the crystal momentum of electronic states is a well defined quantum number. In any real experimental system the translational symmetry of the system is destroyed either by defects or disorder in the crystal structure or by the system's edge. However, experimental topological markers persist, either in the observation of robust edge states or anomalous quantized transport coefficients. In contrast, some systems possess topological phases deriving from properties of the system that manifest not in a momentum space description, but instead most readily in a position space description, or basis, of the system's Hamiltonian.

Here we study the effects of systems that maintain both types of quantum geometry. The simplest such system is a Chern insulator coupled to a skyrmion magnetic texture. The Chern insulator results from non-trivial momentum space geometry through its momentum space Berry curvature, while the skyrmion magnetic texture alone manifests in a non-trivial real space Berry curvature whose integral across the system is equal to a real space Chern number. Other systems include Moire materials where long wavelengths potentials and correlations can drive position space ordering of the degrees of freedom and topological order.