Geometric criterion of topological phase transition for non-Hermitian systems

We propose a geometric criterion of the topological phase transition for non-Hermitian systems. We define the length of the boundary of the bulk band in the complex energy plane for non-Hermitian systems. For one-dimensional systems, we find that the topological phase transition occurs when the derivatives of the length with respect to parameters are discontinuous. For two-dimensional systems, when the length is discontinuous, the topological phase transitions between the gapped and gapless phases occurs. When the derivatives of the length with respect to parameters are discontinuous, the topological phase transition between the gapless and gapless phases occurs. These nonanalytic behaviors of the length in the complex energy plane provide a signal to detect the topological phase transitions. We demonstrate this geometric criterion by the one-dimensional non-Hermitian Su-Schieffer-Heeger model and the two-dimensional non-Hermitian Chern insulator model. This geometric criterion provides an efficient insight to the global topological invariant from a geometric local object in the complex energy plane for non-Hermitian systems