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

Polarization Jumps across Topological Phase Transitions in Two-dimensional Systems