Band structure and the exceptional ring in a two-dimensional superconducting circuit lattice

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.