Square-root topology is a recently emerged subfield describing a class of insulators and superconductors whose topological nature is only revealed upon squaring their Hamiltonians, i.e., the finite energy edge states of the starting square-root model inherit their topological features from the zero-energy edge states of a known topological insulator/superconductor present in the squared model. Focusing on one-dimensional models, we show how this concept can be generalized to $2^n$-root topological insulators and superconductors, with $n$ any positive integer, whose rules of construction are systematized here. Borrowing from graph theory, we introduce the concept of arborescence of $2^n$-root topological insulators/superconductors which connects the Hamiltonian of the starting model for any $n$ through a series of squaring operations followed by constant energy shifts to the Hamiltonian of the known topological insulator/superconductor, identified as the source of its topological features. Our work paves the way for an extension of $2^n$-root topology to higher-dimensional systems.

中文翻译：

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一维2n根拓扑绝缘体和超导体

平方根拓扑是最近出现的一个子领域，描述了一类绝缘体和超导体，它们的拓扑性质只有在对其哈密顿量求平方时才能揭示，即起始平方根模型的有限能量边缘状态从零能量继承了它们的拓扑特征。存在于平方模型中的已知拓扑绝缘体/超导体的边缘状态。专注于一维模型，我们展示了如何将这个概念推广到$2^n$-根拓扑绝缘体和超导体，与 $n$任何正整数，其构造规则在这里被系统化。借用图论，我们引入了树状图的概念$2^n$-root 拓扑绝缘体/超导体，它连接任何起始模型的哈密顿量 $n$通过一系列平方运算，然后将恒定能量转移到已知拓扑绝缘体/超导体的哈密顿量，确定为其拓扑特征的来源。我们的工作为扩展$2^n$-root 拓扑到更高维的系统。