  Andreev bound states are low energy excitations appearing below the
particle-hole gap of superconductors, and are expected to be topologically
trivial. Here, we report the theoretical prediction of topologically
$nontrivial$ Andreev bound states in one-dimensional superconductors. These
states correspond to another topological invariant defined in a synthetic
two-dimensional space, the particle-hole Chern number, which we construct in
analogy to the spin Chern number in quantum spin Hall systems. Nontrivial
Andreev bound states have distinct features and are topologically nonequivalent
to Majorana bound states. Yet, they can coexist in the same system, have
similar spectral signatures, and materialize with the concomitant opening of
the particle-hole gap. The coexistence of Majorana and nontrivial Andreev bound
state is the direct consequence of "double dimensionality", i.e., the
dimensional embedding of the one-dimensional system in a synthetic
two-dimensional space, which allows the definition of two distinct topological
invariants ($\mathbb{Z}_2$ and $\mathbb{Z}$) in different dimensionalities.
