Real Space Invariants: Spectral Flow of Fragile Topological State under Twisted Boundary Conditions

In this paper, we propose the twisted boundary condition as a generic approach to theoretically and experimentally detect the fragile topological state. We prove that, when the fragile phase can be written as a difference of a trivial atomic insulator and the so-called obstructed atomic insulator, the gap between the fragile phase and other bands must close under a specific twist of
the boundary condition of the system. We explicitly work out all the twisted boundary conditions that can detect all the 2D fragile phases implied by symmetry eigenvalues in all wallpaper groups. We develop the concept of real space invariants, which are local good quantum numbers in real space, and which also fully characterize the eigenvalue fragile phases. We show that the number of unavoidable level crossings under the twisted boundary condition is completely determined by the real space invariants. Possible realizations of the twisted boundary condition of the fragile band in metamaterial systems are also discussed.