Unique one-body position operator for periodic systems

In this work we prove that the one-body position operator for periodic systems that we have recently proposed [Phys. Rev. B 99, 205144 (2019)] is unique modulo a phase factor and an additive constant. The proof uses several general physical constraints that a periodic one-body position operator should satisfy. We show that these constraints are sufficient to uniquely define a position operator that is compatible with periodic boundary conditions.