  Topological materials are often characterized by unique edge states which are
in turn used to detect 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 is often
broken in genuine materials, leading to vulnerable corner states even when the
higher-order topological numbers remain quantized and invariant. Here, we show
that a local artificial gauge flux can serve as a robust probe of the Wannier
type higher-order topological insulators which is effective even when the
chiral symmetry is broken. The resultant observable signature is the emergence
of the cyclic spectral flows traversing one or multiple band gaps. These
spectral flows are associated with the local modes bound to the artificial
gauge flux. This phenomenon is essentially due to the cyclic transformation of
the Wannier orbitals when the local gauge flux acts on them. We extend
topological Wannier cycles to systems with C2 and C3 symmetries and show that
they can probe both the bulk and the edge Wannier centers, yielding rich
topological phenomena.
