Polarization jumps by breaking symmetries of two-dimensional Weyl semimetals

The electric polarization as a bulk quantity is described by the modern theory of polarization in insulating systems and cannot be defined in conducting systems. Upon a gradual change of a parameter in the system, the polarization always varies smoothly as long as the gap remains open. In this paper, we focus on the two-dimensional Weyl semimetal, which hosts Weyl nodes protected by symmetries, and study the behavior of the polarization when a symmetry-breaking term $M$ is introduced and a gap opens. We show that there can be a jump between $M\ensuremath{\rightarrow}{0}^{+}$ and ${0}^{\ensuremath{-}}$ limits. We find that the jump is universally described by the ``Weyl dipole'' representing how the Weyl nodes with monopole charges are displaced in the reciprocal space. Our result is applicable to general two-dimensional Weyl semimetals.