Unified Formalism for calculating Polarization, Magnetization, and more in a Periodic Insulator

TL;DR

This paper introduces a Green's function-based formalism to compute polarization, magnetization, and related properties in periodic insulators under uniform electromagnetic fields, generalizing previous results to systems without time reversal symmetry.

Contribution

It develops a unified perturbative formalism using Green's functions to calculate electromagnetic responses in insulators, extending previous theories to non-time-reversal-invariant systems.

Findings

Derived a perturbative Green's function formula for uniform fields.

Expressed linear electric field terms via Green's functions in extended dimensions.

Connected the $ heta$ term coefficient to a Wess-Zumino-Witten-type integral.

Abstract

In this paper, we propose a unified formalism, using Green's functions, to integrate out the electrons in an insulator under uniform electromagnetic fields. We derive a perturbative formula for the Green's function in the presence of uniform magnetic or electric fields. Applying the formula, we derive the formula for the polarization, the orbital magnetization, and the orbital magneto-polarizability, without assuming time reversal symmetry. Specifically, we realize that the terms linear in the electric field can only be expressed in terms of the Green's functions in one extra dimension. This observation directly leads to the result that the coefficient of the $\theta$ term in any dimensions is given by a Wess-Zumino-Witten-type term, integrated in the extended space, interpolating between the original physical Brillouin zone and a trivial system, with the group element replaced by the Green's function. This generalizes an earlier result for the case of time reversal invariance [see Z. Wang, X.-L. Qi, and S.-C. Zhang, Phys. Rev. Lett. {\bf 105}, 256803 (2010)].