The Hopf insulator is a three-dimensional topological insulator outside the standard classification of topological insulators. Here we consider two types of non-Hermitian Hopf insulators, one without and one with the non-Hermitian skin effect. The isolated gapless points of the Hermitian model are broadened into finite regimes in the non-Hermitian models. However, the modulus of the Hopf index remains quantized in the gapped regimes. The model without the non-Hermitian skin effect allows an accurate evaluation of its generalized Hopf index and energy spectrum, showing an agreement between the gapless-regime estimations from the systems with periodic and open boundary conditions. Near the zero-energy plane, Fermi rings can be observed whenever the Hopf index is quantized at nonzero values, and there is a bulk-boundary correspondence between the modulus of the Hopf index and the number of Fermi rings. The other model manifests the non-Hermitian skin effect in the generalized Brillouin zone and shows the skewed profiles of the bulk states. The Hopf index and energy spectrum are shown to be sensitive to the boundary condition in the presence of the non-Hermitian skin effect.

中文翻译：

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非赫米特三维二维带霍夫绝缘子

Hopf绝缘子是在拓扑绝缘子的标准分类之外的三维拓扑绝缘子。在这里，我们考虑两种类型的非Hermitian Hopf绝缘子，一种不带绝缘子，另一种具有非Hermitian趋肤效应。埃尔米特模型的孤立无间隙点在非埃尔米特模型中扩展为有限状态。但是，霍普夫指数的模量仍然在有缺口的制度中进行量化。该模型没有非Hermitian趋肤效应，可以对其广义Hopf指数和能谱进行准确评估，显示出具有周期性和开放边界条件的系统的无间隙状态估计之间的一致性。在零能平面附近，只要将Hopf指数量化为非零值，就可以观察到费米环。Hopf指数的模量与费米环数之间存在体边界的对应关系。另一个模型在广义布里渊区中表现出非埃尔米特集肤效应，并显示了整体状态的偏斜轮廓。霍普夫指数和能谱显示出在非赫米特皮肤效应的情况下对边界条件敏感。