  Spectral measurements of boundary localized in-gap modes are commonly used to
identify topological insulators via the bulk-boundary correspondence. This can
be extended to high-order topological insulators for which the most striking
feature is in-gap modes at boundaries of higher co-dimension, e.g. the corners
of a 2D material. Unfortunately, this spectroscopic approach is not always
viable since the energies of the topological modes are not protected and they
can often overlap the bulk bands, leading to potential misidentification. Since
the topology of a material is a collective product of all its eigenmodes, any
conclusive indicator of topology must instead be a feature of its bulk band
structure, and should not rely on specific eigen-energies. For many topological
crystalline insulators the key topological feature is fractional charge density
arising from the filled bulk bands, but measurements of charge distributions
have not been accessible to date. In this work, we experimentally measure
boundary-localized fractional charge density of two distinct 2D
rotationally-symmetric metamaterials, finding 1/4 and 1/3 fractionalization. We
then introduce a new topological indicator based on collective phenomenology
that allows unambiguous identification of higher-order topology, even in the
absence of in-gap states. Finally, we demonstrate the higher-order
bulk-boundary correspondence associated with this fractional feature by using
boundary deformations to spectrally isolate localized corner modes where they
were previously unobservable.
