  In a superconductor with a sublattice degree of freedom, we find
unconventional Andreev bound states whose charge density oscillates in sign
between the two sublattices. The appearance of these oscillating-charged
Andreev bound states is characterized by a Zak phase, rather than a
conventional topological invariant. In contrast to conventional Andreev bound
states, for oscillating-charged Andreev bound states the proportionality
between the electron-like spectral function, the local density of states and
the tunneling conductance is broken. We examine the possible appearance of
these novel Andreev bound states in UTe$_2$ and locally noncentrosymmetric
superconductors.
