Electric polarization in Chern insulators: Unifying many-body and single-particle approaches

TL;DR

The paper tackles the challenge of defining electric polarization in Chern insulators, where traditional Berry-phase methods face issues from nontrivial topology and nonlocal Wannier functions. It unifies the many-body origin-based polarization $\vec{P}_{\text{o}}$ with the Berry-phase polarization $\vec{\mathcal{P}}_{\vec{r}_0,\vec{k}_0}$ by deriving a bulk constraint that relates real-space origin $\text{o}$, momentum-space reference $\vec{k}_0$, the real-space shift $\vec{r}_0$, the gauge origin $\bar{\text{o}}$, and system size $L_x$. The central result, Eq. id1 with $c_1=\tfrac{1}{2}$ and $c_2=0$, prescribes how to choose $\vec{k}_0$ and other parameters so that $\vec{\mathcal{P}}_{\vec{r}_0,\vec{k}_0}$ matches $\vec{P}_{\text{o}}$ in interacting and noninteracting Chern insulators. The authors verify the relation numerically in the Harper-Hofstadter and Haldane models, demonstrating that the two polarization notions coincide under the stated constraints and clarifying the role of origin-dependence in magnetic fields. This provides a robust, bulk-based criterion for defining Berry-phase polarization in topological bands and links microscopic response quantities to geometric phases.

Abstract

Recently, it has been established that Chern insulators possess an intrinsic two-dimensional electric polarization, despite having gapless edge states and non-localizable Wannier orbitals. This polarization, $\vec{P}_{\text{o}}$, can be defined in a many-body setting from various physical quantities, including dislocation charges, boundary charge distributions, and linear momentum. Importantly, there is a dependence on a choice of real-space origin $\text{o}$ within the unit cell. In contrast, Coh and Vanderbilt extended the single-particle Berry phase definition of polarization to Chern insulators by choosing an arbitrary point in momentum space, $\vec{k}_0$. In this paper, we unify these two approaches and show that when the real-space origin $\text{o}$ and momentum-space point $\vec{k}_0$ are appropriately chosen in relation to each other, the Berry phase and many-body definitions of polarization are equal.

Figures (7)

• Figure 1: (a-c)Three choices of magnetic unit cell for $\phi=\frac{1}{4}2\pi$. $m_x$ and $m_y$ are the integer linear sizes of the magnetic unit cell in the $x$ and $y$ direction. The $(0,0)$ position is always set to the site at the bottom left corner of the magnetic unit cell. (d) Square lattice unit cell with high symmetry points $\alpha,\beta,\gamma_1,\gamma_2$.
• Figure 2: Two gauge choices for $\phi=2\pi/4$ and corresponding magnetic unit cell. Note the $x$ and $y$ axes are not orthogonal. Each blue arrow represents a hopping phase of $e^{i2\pi/8}$. The gauge origins are marked with '$\times$' (a)$\bar{\text{o}}=(0,0)$, (b) shifting the gauge origin to $\bar{\text{o}}=(3/2,0)$. The flux in the red shaded region determines the enclosed flux: $\Phi_{\text{enc}}/L_y=3\phi/2$.
• Figure 3: (a). Left panel: Part of the Hofstadter butterfly, colored in with the values of $P_{\alpha,x} = 0,1/2 \mod 1$, originally calculated in zhang2022pol. Each white / blue colored space in the butterfly represents the value of ${P}_{\alpha,x}$ where every state with energy below $\mu$ is filled. In each of the red boxes at $\phi = \pi/2$, there is a continuous band of states; fully filling these states gives an insulator with polarization $P_{\alpha,x}$ shown in red. Middle and right panels: $\mathcal{P}^{\text{Bloch}}_{\vec{r}_0,x}$ and $\mathcal{P}_{\vec{r}_0,\vec{k}_0,x}$ are also calculated for these bands with the parameters $\{k_{0y}, \bar{\text{o}}_x, r_{0x},L_x\} =\{0,0,0,\text{Even}\}$. Note the agreement between $P_{\alpha,x}$ and $\mathcal{P}_{\vec{r}_0, \vec{k}_0, x} \mod 1$. (b). Same as (a), but comparing $P_{\beta,x}$ with $\mathcal{P}_{\vec{r}_0,\vec{k}_0,x}$, with the parameters $\{k_{0y}, \bar{\text{o}}_{x}, r_{0x},L_x\}=\{0,1/2,1/2,\text{Even}\}$.
• Figure 4: Left. The choice of unit cell in the Haldane model, black lines represent hoppings with coefficient $t_1$ and red lines represent hoppings with coefficient $it_2$. Middle. Maximal Wyckoff positions of the $M=6$ unit cell. The choice of $(0,0)$ position of the unit cell is marked. The high symmetry points $\beta_1$, $\beta_2$ have the same point group symmetry, but are inequivalent under lattice translations; same with $\gamma_i$ points. Right. Band structure of the Haldane model in the first Brillouin zone with parameters $\{t_1,t_2,m\}=\{1,0.1,0.2\}$. The linear size of the reciprocal lattice is $b_x=b_y=\frac{4\pi}{\sqrt{3}}$.
• Figure 5: Comparison between $\mathcal{P}_{\vec{r}_0,\vec{k}_0,x}$ and $P_{\text{o},x}$ for the Haldane model.