We propose a general scheme to construct a Hamiltonian $H_{\text{root}}$ describing a square root of an original Hamiltonian $H_{\text{original}}$ based on the graph theory. The square-root Hamiltonian is defined on the subdivided graph of the original graph of $H_{\text{original}}$, where the subdivided graph is obtained by putting one vertex on each link in the original graph. When $H_{\text{original}}$ describes a topological system, there emerge in-gap edge states at nonzero energy in the spectrum of $H_{\text{root}}$, which are the inherence of the topological edge states at zero energy in $H_{\text{original}}$. In this case, $H_{\text{root}}$ describes a square-root topological insulator or superconductor. Typical examples are square roots of the Su-Schrieffer-Heeger (SSH) model, the Kitaev topological superconductor model, and the Haldane model. Our scheme is also applicable to non-Hermitian topological systems, where we study an example of a nonreciprocal non-Hermitian SSH model.

中文翻译：

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平方根拓扑绝缘子和超导体的系统构造

我们提出了构造哈密顿量的一般方案 $H_{\text{根}}$ 描述原始哈密顿量的平方根 $H_{\text{原版的}}$基于图论。平方根哈密顿量定义在原始图的细分图上$H_{\text{原版的}}$，其中细分图是通过在原始图的每个链接上放置一个顶点获得的。什么时候$H_{\text{原版的}}$ 描述了一个拓扑系统，在光谱的非零能量处出现了能隙边缘状态 $H_{\text{根}}$，这是在能量为零时拓扑边缘状态的固有 $H_{\text{原版的}}$。在这种情况下，$H_{\text{根}}$描述了平方根拓扑绝缘体或超导体。典型示例是Su-Schrieffer-Heeger（SSH）模型，Kitaev拓扑超导体模型和Haldane模型的平方根。我们的方案也适用于非Hermitian拓扑系统，在该系统中我们研究了不可逆的非Hermitian SSH模型的示例。