Edge states across the topological phase transition due to approximate chiral symmetry in quantum anomalous and spin Hall systems

In this work we demonstrate that a quantum anomalous Hall system (Chern
insulator) always exhibits edge states right at the phase transition (except when the system becomes charge-conjugation symmetric), where the Hamiltonian is gapless, and in the vicinity of the topologically trivial and non-trivial gapped phases. We show that the origin of these edge states can be attributed to the approximate chiral symmetry of the Hamiltonian. Additionally, we demonstrate that while the non-trivial quantum Hall topology is responsible only for the existence of these edge states within the energy gap region, the approximate chiral symmetry is the ingredient responsible for the edge state existence in the remaining energy range in both topological and trivial phases. This behavior is described by the realistic one block of the BHZ model of a quantum spin Hall system, which was already noticed but remained unexplained.