Electric Polarization in Inhomogeneous Crystals

TL;DR

This paper develops a comprehensive semiclassical framework to calculate electric polarization in inhomogeneous crystals, revealing new topological and quadrupole-like contributions that ensure gauge invariance and relate to fractional charges.

Contribution

It introduces a second-order gradient expansion of charge density, identifying a novel quadrupole-like polarization term and extending the theory to multi-band systems.

Findings

Derived second-order polarization contributions including a new quadrupole-like term.

Linked topological polarization to fractional charges in 2D vortex systems.

Validated the theoretical framework with model system simulations.

Abstract

We derive the charge density up to second order in spatial gradient in inhomogeneous crystals using the semiclassical coarse graining procedure based on the wave packet method. It can be recast as divergence of polarization, whose first-order contribution consists of three parts, a perturbative correction to the original Berry connection expression, a topological part that can be written as an integral of the Chern-Simons 3-form, and a previously-unknown, quadrupole-like contribution. The topological part can be related to the quantized fractional charge carried by a vortex in two dimensional systems. We then generalize our results to the multi-band case and show that the quadrupole-like contribution plays an important role, as it makes the total polarization gauge-independent. Finally, we verify our theory in several model systems.