Polarization Jumps across Topological Phase Transitions in Two-dimensional Systems

In topological phase transitions involving a change in topological invariants such as the Chern number and the $\mathbb{Z}_2$ topological invariant, the gap closes, and the electric polarization becomes undefined at the transition. In this paper, we show that the jump of polarization across such topological phase transitions in two dimensions is described in terms of positions and monopole charges of Weyl points in the intermediate Weyl semimetal phase. We find that the jump of polarization is described by the Weyl dipole at $\mathbb{Z}_2$ topological phase transitions and at phase transitions without any change in the value of the Chern number. Meanwhile, when the Chern number changes at the phase transition, the jump is expressed in terms of the relative positions of Weyl points measured from a reference point in the reciprocal space.