Topological phase transitions with zero indirect band gaps

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 trimer 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.