  We study fractional boundary charges (FBCs) for two classes of strongly
interacting systems. First, we study strongly interacting nanowires subjected
to a periodic potential with a period that is a rational fraction of the Fermi
wavelength. For sufficiently strong interactions, the periodic potential leads
to the opening of a charge density wave gap at the Fermi level. The FBC then
depends linearly on the phase offset of the potential with a quantized slope
determined by the period. Furthermore, different possible values for the FBC at
a fixed phase offset label different degenerate ground states of the system
that cannot be connected adiabatically. Next, we turn to the fractional quantum
Hall effect (FQHE) at odd filling factors $\nu=1/(2l+1)$, where $l$ is an
integer. For a Corbino disk threaded by an external flux, we find that the FBC
depends linearly on the flux with a quantized slope that is determined by the
filling factor. Again, the FBC has $2l+1$ different branches that cannot be
connected adiabatically, reflecting the $(2l+1)$-fold degeneracy of the ground
state. These results allow for several promising and strikingly simple ways to
probe strongly interacting phases via boundary charge measurements.
