  We investigate the topological properties of a two-chain quantum ladder with
uneven legs, i.e. the two chains differ in their periods by a factor of two.
Such an uneven ladder presents rich band structures classified by the closure
of either direct or indirect bandgaps. It also provides opportunities to
explore fundamental concepts concerning band topology and edge modes, including
the difference of intracellular and intercellular Zak phases, and the role of
the inversion symmetry (IS). We calculate the Zak phases of the two kinds and
find excellent agreement with the dipole moment and extra charge accumulation,
respectively. We also find that configurations with IS feature a pair of
degenerate two-side edge modes emerging as the closure of the direct bandgap,
while configurations without IS feature one-side edge modes emerging as not
only the closure of both direct and indirect bandgap but also within the band
continuum. Furthermore, by projecting to the two sublattices, we find that the
effective Bloch Hamiltonian corresponds to that of a generalized
Su-Schrieffer-Heeger model or Rice-Mele model whose hopping amplitudes depend
on the quasimomentum. In this way, the topological phases can be efficiently
extracted through winding numbers. We propose that uneven ladders can be
realized by spin-dependent optical lattices and their rich topological
characteristics can be examined by near future experiments.
