Spin-orbit coupling, edge states and quantum spin Hall criticality due to Dirac fermion confinement: the case study of graphene

We propose a generalized Dirac fermion description for the electronic state of graphene terminated by a zigzag edge. This description admits a specific spin-orbit coupling needed to preserve time-reversal invariance of the zigzag confinement, otherwise, for spinless particles, showing the parity anomaly typical of quantum electrodynamics in (2+1) dimensions. At a certain critical strength the spin-orbit coupling induces a phase transition of the quantum-spin-Hall type. It is manifested by a novel type of the edge states consisting of a Kramers' pair of counter-propagating modes with opposite spin orientations. Such edge states are capable of accumulating an integer spin in response to a transverse electric field in the absence of a magnetic one. They exist without any excitation gap in the bulk, due to which our system stands out among other quantum spin Hall systems studied earlier. We show that at the transition the local density of states is discontinuous and its energy dependence reflects the phase diagram of the system.