Topological phase transitions in superconductors with chiral symmetry

We study topological transitions in one-dimensional superconductors that can harbor multiple edge Majorana bound states protected by chiral symmetry. The chiral symmetry arises from the structure of the internal spin degrees of freedom of the superconductor and can be guided by the coupling of the superconductor with sources of time-reversal symmetry breaking. We then consider distinct regions of the phase diagram in the parameter space that are marked by gapless excitations in the spectrum and evaluate the conditions for inducing a topological transition. We show that for gapless chiral symmetric superconductors one can identify a class of physical perturbations that enable a gap opening in the spectrum, without breaking chirality, and turn the system into a topological state. This type of superconductor is dubbed a marginal topological superconductor because an infinitesimally small perturbation is able to induce a transition into a topological nontrivial phase. To explicitly demonstrate and evaluate the character of the transitions from gapless to topological gapfull phases, we explore different physical cases including a 𝑝-wave superconductor in the presence of an applied magnetic field or proximity-coupled to a ferromagnet, and an 𝑠-wave superconductor in a noncollinear magnetic ordering.