  The (one-dimensional) Su-Schrieffer-Heeger Hamiltonian, augmented by
spin-orbit coupling and longer-range hopping, is studied at half filling for an
even number of sites. The ground-state phase diagram depends sensitively on the
symmetry of the model. Charge-conjugation (particle-hole) symmetry is conserved
if hopping is only allowed between the two sublattices of even and odd sites.
In this case (of BDI symmetry) we find a variety of topologically non-trivial
phases, characterized by different numbers of edge states (or, equivalently,
different quantized Zak phases). The transitions between these phases are
clearly signalled by the entanglement entropy. Charge-conjugation symmetry is
broken if hopping within the sublattices is admitted (driving the system into
the AI symmetry class). We study specifically next-nearest-neighbor hopping
with amplitudes $t_a$ and $t_b$ for the $A$ and $B$ sublattices, respectively.
For $t_a=t_b$ parity is conserved, and also the quantized Zak phases remain
unchanged in the gapped regions of the phase diagram. However, metallic patches
appear due to the overlap between conduction and valence bands in some regions
of parameter space. The case of alternating next-nearest neighbor hopping,
$t_a=-t_b$, is also remarkable, as it breaks both charge-conjugation $C$ and
parity $P$ but conserves the product $CP$. Both the Zak phase and the
entanglement spectrum still provide relevant information, in particular about
the broken parity. Thus the Zak phase for small values of $t_a$ measures the
disparity between bond strengths on $A$ and $B$ sublattices, in close analogy
to the proportionality between the Zak phase and the polarization in the case
of the related Aubry-Andr\'e model.
