Topological obstructed atomic limit by annihilating Dirac fermions

We show that annihilating a pair of Dirac fermions implies a topological transition from the critical semi-metallic phase to an Obstructed Atomic Limit (OAL) insulator phase instead of a trivial insulator. This is shown to happen because of branch-cuts in the phase of the wave functions, leading to non trivial Zak phase along certain directions. To this end, we study their Z$_2$ invariant and also study the phase transition using Entanglement Entropy. We use low energy Hamiltonians and numerical result from model systems to show this effect. These transitions are observed in realistic materials including strained graphene and buckled honeycomb group-V (Sb/As).