Renormalization approach to the analysis and design of Hermitian and non-Hermitian interfaces

I describe a concrete and efficient real-space renormalization approach that provides a unifying perspective on interface states in a wide class of Hermitian and non-Hermitian models, irrespective of whether they obey a traditional bulk-boundary principle or not. The emerging interface physics are governed by a flow of microscopic interface parameters, and the properties of interface states become linked to the fixed-point topology of this flow. In particular, the quantization condition of interface states converts identically into the question of the convergence to unstable fixed points. As its key merit, the approach can be directly applied to concrete models and utilized to design interfaces that induce states with desired properties, such as states with a predetermined and possibly symmetry-breaking energy. I develop the approach in general, and then demonstrate these features in various settings, including for the design of circular, triangular, and square-shaped complex dispersion bands and associated arcs at the edge of a two-dimensional system. Furthermore, I describe how this approach transfers to nonlinear settings, and demonstrate the efficiency, practicability, and consistency of this extension for a paradigmatic model of topological mode selection by distributed saturable gain and loss.