Unveiling non-Hermitian band structures with non-Bloch supercells.

Real-valued band structures underpin the analysis of periodic Hermitian systems and are well established experimentally. By contrast, non-Hermitian systems feature complex band structures, in which energy and momentum acquire imaginary parts, enabling phenomena that defy conventional Bloch theory. Mapping these complex bands-relating complex momentum to complex energy-and identifying their associated eigenstates is essential yet remains challenging. Here, we introduce a non-Bloch supercell framework that addresses this challenge by treating non-Bloch Fermi points (NBFPs) as key observables and disentangling the roles of real and imaginary parts of momentum. As the non-Hermitian counterparts of Fermi surfaces, NBFPs provide experimentally accessible markers of point‑gap topology and open‑boundary behavior. Experimentally, our approach combines an exponent-flattening protocol with twisted boundary conditions, enabling system-size-independent control of imaginary momentum and preserving high-resolution sampling in real momentum. Implemented in programmable one- and two-dimensional acoustic crystals, our framework acquires momentum-resolved complex energy surfaces and biorthogonal eigenstates by Green's function measurements. The resulting non‑Hermitian band structures, including complex momentum and energy and Berry curvature distributions, accurately predict open‑boundary spectra and eigenstates, validated in independent open‑geometry experiments. Our work delivers a broadly applicable experimental toolkit for exploring non-Hermitian band geometry and topology in diverse platforms with programmable couplings.