Non-Hermitian topology and flat bands via an exact real-space decimation scheme

In recent years, non-Hermitian phases in classical and quantum systems have garnered significant attention. In particular, their intriguing band geometry offers a platform for exploring unique topological states and unconventional quantum dynamics. However, their topological characterization becomes particularly interesting and challenging in complex multiband systems. Here we propose a decimation framework, which leverages real-space renormalization group to streamline the analysis of complex multiband non-Hermitian systems. Our systematic approach allows us to probe different phases and transitions, analyze bulk-boundary correspondence, formulate generalized Brillouin zones, investigate open boundary spectra, survey non-Bloch van Hove singularities, study disorder-induced effects, and explore tunable non-Hermitian flat-band physics. Additionally, our framework allows proposing a hypothesis about quasi-one-dimensional bipartite non-Hermitian systems with flat bands, demonstrating their decoupling into Su-Schrieffer-Heeger chains and compact localized states across various models. Our paper presents a powerful and comprehensive framework for understanding the intricate properties of non-Hermitian multiband systems, offering insights into the evolving landscape of non-Hermitian topological physics.