The non-Hermitian skin effect dramatically reshapes the energy bands of non-Hermitian systems, meaning that the usual Bloch band theory is fundamentally inadequate as their characterization. The non-Bloch band theory, in which the concept of Brillouin zone is generalized, has been widely applied to investigate non-Hermitian systems in one spatial dimension. However, its generalization to higher dimensions has been challenging. Here, we develop a formulation of the non-Hermitian skin effect and non-Bloch band theory in arbitrary spatial dimensions, which is based on a natural geometrical object known as the amoeba. Our theory provides a general framework for studying non-Hermitian bands beyond one dimension. Key quantities of non-Hermitian bands, including the energy spectrum, eigenstates profiles, and the generalized Brillouin zone, can be efficiently obtained from this approach.

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任意维度非布洛赫能带理论的阿米巴公式

非厄米集肤效应极大地重塑了非厄米系统的能带，这意味着通常的布洛赫能带理论根本不足以描述它们。非布洛赫能带理论推广了布里渊区的概念，已被广泛应用于研究一维空间中的非厄米系统。然而，将其推广到更高维度一直具有挑战性。在这里，我们开发了任意空间维度的非厄米集肤效应和非布洛赫能带理论的公式，该公式基于一种称为阿米巴的自然几何物体。我们的理论为研究一维之外的非厄米能带提供了一个通用框架。通过这种方法可以有效地获得非厄米能带的关键量，包括能谱、本征态分布和广义布里渊区。