We investigate the energy band structure and exceptional ring in a two-dimensional superconducting circuit lattice. In the case of a Hermitian Hamiltonian, by modulating the on-site potential, we find that the single Dirac point splits into four degenerate points. For the non-Hermitian situation, we find that the flat band is significantly destroyed by the presence of gain and loss and the introduction of long-range coupling makes the zero-energy flat band survive because of the protection of chiral symmetry. Meanwhile, the purely real spectrum can be transformed into the purely imaginary spectrum via modulating the parameters appropriately. Furthermore, when the nonreciprocal next-nearest-neighbor couplings are continuously modulated, the two exceptional rings become one and a Dirac-like point occurs inside the exceptional ring. Our scheme opens the promising possibilities for exploring the topological property of two-dimensional superconducting circuit lattice system.

中文翻译：

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二维超导电路晶格中的能带结构和异常环

我们研究了二维超导电路晶格中的能带结构和异常环。对于Hermitian哈密顿量，通过调制现场电势，我们发现单个Dirac点分裂为四个退化点。对于非Hermitian情况，我们发现平坦带被增益和损耗严重破坏，并且引入长程耦合使零能量平坦带得以生存，因为它具有手性对称性。同时，通过适当地调整参数，可以将纯实谱变换为纯虚谱。此外，当不可逆的最近邻耦合被连续调制时，两个例外环成为一个，并且在例外环内部出现类似狄拉克的点。