Relation between the areal rate of Zitterbewegung and Berry curvature in two-band systems

We introduce the Zitterbewegung areal rate operator to study the relation between the orientation of trembling motion and band topology. Constructed from the oscillatory interband displacement and velocity, this operator distinguishes clockwise from anticlockwise motion. For a generic gapped two-band Hamiltonian in two dimensions, we show that the Zitterbewegung areal rate is proportional to the Berry curvature. The derivation uses only spectral projectors and is therefore gauge invariant. We also demonstrate that the Zitterbewegung areal rate operator is time-independent and a constant of motion at each momentum, although the displacement and velocity from which it is constructed oscillate in time. Its sign distinguishes anticlockwise from clockwise Zitterbewegung and coincides with the sign of the Berry curvature. For wave packets narrowly localized around the isolated massive Dirac points, the Zitterbewegung chirality equals the sign of the local Chern contribution. The Chern number can therefore be reconstructed from the orientations of the Zitterbewegung motion around the individual Dirac points.