Microscopic theory of Chern polarization via crystalline defect charge

TL;DR

The paper resolves how to define absolute polarization in Chern insulators by linking bulk polarization to fractional charges bound to lattice dislocations, circumventing edge-state ambiguities that plague surface-charge approaches. It derives a gauge-invariant bulk polarization expression, $\mathbf{P}_n = -\frac{e}{V_c}\overline{\mathbf{r}}_n + \frac{eC}{2\pi}\hat{\mathbf{z}}\times\mathbf{k}_{vn}$, from dislocation bound charges and Zak-phase structure, and shows consistency with prior results while extending to systems lacking crystalline symmetries beyond translations. The authors validate the theory with a Qi-Wu-Zhang Chern-insulator model, demonstrating quantitative agreement between dislocation-charge calculations and the bulk polarization. Overall, the work provides a general, symmetry-agnostic route to polarization in topological insulators and suggests broader implications for bulk electrodynamic responses in topological phases.

Abstract

The modern theory of polarization does not apply in its original form to systems with non-trivial band topology. Chern insulators are one such example. Defining polarization for them is complicated because they are insulating in the bulk but exhibit metallic edge states. Wannier functions formed a key ingredient of the original modern theory of polarization, but it has been considered that these cannot be applied to Chern insulators since they are no longer exponentially localized and the Wannier center, obtained from the Zak phase, is no longer gauge invariant. In this article, we provide an unambiguous definition of absolute polarization for a Chern insulator in terms of the Zak phase. We obtain our expression by studying the non-quantized fractional charge bound to lattice dislocations. Our expression can be computed directly from bulk quantities and makes no assumption on the edge state filling. It is fully consistent with previous results on the quantized charge bound to dislocations in the presence of crystalline symmetry. At the same time, our result is more general since it also applies to Chern insulators which do not have crystalline symmetries other than translations.

Figures (5)

• Figure 1: (a) Partitioning a 1D system deep in the bulk for the edge charge calculation. The ions are shown in red while the electrons (in the form of their Wannier centers) are shown in blue. The vertical dashed line denotes the partition. The regions into which the left and right spatial projectors, as well as individual cell projectors, project are shown. (b) Schematic of the flow of the HWF center of the effective 1D system as $k_y$ is taken from $0$ to $2\pi/a$. The solid (brown) arrows depict the non-trivial winding of the Wannier center for a Chern insulator, while the dotted (black) arrows depict the flow in the trivial case.
• Figure 2: Dislocation in a square lattice with a cylindrical geometry. Note that the right hand edge has one fewer lattice site compared to the left hand edge. The core of the dislocation is the lattice site colored yellow, which has only three nearest neighbors.
• Figure 3: Plot comparing the theoretical absolute polarization for the QWZ model (blue line) with the numerical value obtained from a real space dislocation charge computation with fully open boundary conditions (red dots). Note that $P_x$ and $\tilde{P}_x$ have units of $-e/a$. The theoretical values were calculated by constructing Wannier functions and using (\ref{['abspol']}). The real space dislocation charge calculation was done using clean system sizes ranging from approximately $70a\times 140a$ up to $100a\times 200a$, with charge counted in $R\times R$ regions around the defect for $R = 17a$ and $R = 25a$ respectively.
• Figure 4: Figure showing the integration region to demonstrate the jump in the Zak phase.
• Figure 5: Figure showing values of $\varphi_x(k_y)$ at different values of $k_y$ in the Brillouin zone for a gauge transformation with a vortex-antivortex pair. An anti-clockwise arrow represents a circulation of $2\pi C$.