  Topological phase transitions in band models are usually associated to the
gap closing between the highest valance band and the lowest conduction band,
which can give rise to different types of nodal structures, such as Dirac/Weyl
points, lines and surfaces. In this work, we show the existence of a different
kind of topological phase transitions in one-dimensional systems, which are
instead characterized by the presence of a robust zero indirect gap, which
occurs when the top of the valence band coincides with the bottom of the
conduction band in energy but not in momentum. More specifically, we consider
an one-dimensional model on a diamond-like chain that is protected by both
particle-hole and chiral-inversion symmetries. At the critical point, the
system supports a Dirac-like point. After introducing a deforming parameter
that breaks both inversion and chiral symmetries but preserves their
combination, we observe the emergence of a zero indirect band gap, which
results to be related to the persymmetry of our Hamiltonian. Importantly, the
zero indirect gap holds for a range of values of the deforming parameter. We
finally discuss the implementation of the deforming parameter in our
tight-binding model through time-periodic (Floquet) driving.
