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The non-Hermitian skin effect dramatically reshapes the energy bands ofnon-Hermitian systems, meaning that the usual Bloch band theory isfundamentally inadequate as their characterization. The non-Bloch band theory,in which the concept of Brillouin zone is generalized, has been widely appliedto investigate non-Hermitian systems in one spatial dimension. However, itsgeneralization to higher dimensions has been challenging. Here, we develop aformulation of the non-Hermitian skin effect and non-Bloch band theory inarbitrary spatial dimensions, which is based on a natural geometrical objectknown as the amoeba. Our theory provides a general framework for studyingnon-Hermitian bands beyond one dimension. Key quantities of non-Hermitianbands, including the energy spectrum, eigenstates profiles, and the generalizedBrillouin zone, can be efficiently obtained from this approach.

Amoeba formulation of the non-Hermitian skin effect in higher dimensions