Topological Wannier cycles

Topological materials are often characterized by unique edge states which are in turn used to discern different topological phases in experiments. Recently, with the discovery of various higher-order topological insulators, such spectral topological characteristics are extended from edge states to corner states. However, the chiral symmetry protecting the corner states are often broken in most materials, leading to vulnerable corner states even when the higher-order topological numbers (such as fractional corner charges) remain quantized and invariant. Here, we propose a single-plaquette artificial gauge flux insertion as a robust spectral probe of the Wannier type higher-order topological insulators which is effective even when the chiral symmetry is broken. Remarkably, the single-plaquette artificial gauge flux leads to the topological Wannier cycles that emerge essentially due to the cyclic transformation of the Wannier orbitals composing the energy bands. Topological Wannier cycles are cyclic spectral flows that may appear in a single or multiple band gaps when the single-plaquette gauge flux is acting on the Wannier orbitals. Therefore, in real space topological Wannier cycles probe the location of the Wannier orbitals and distinguishes various higher-order topological phases. In finite systems, topological Wannier cycles probe the filling anomaly and fractional corner charges which are the two key features of higher-order topological insulators. Finally, we illustrate how such scenario can be implemented in phononic systems in sonic crystals via step screw dislocations.