Equilibrium currents in chiral systems with nonzero Chern number

We describe a simple quantum-mechanical approach to calculating equilibrium particle current along the edge of a system with nontrivial band spectrum topology. The approach does not require any a priori knowledge of the band topology and, as a matter of fact, treats topological and nontopological contributions to the edge currents on the same footing. We illustrate its usefulness by demonstrating the existence of “topologically nontrivial” particle currents along the edges of three different physical systems: two-dimensional electron gas with spin-orbit coupling and Zeeman magnetic field, surface state of a topological insulator, and kagomé antiferromagnet with Dzyaloshinskii-Moriya interaction. We describe the relationship of our results to the notion of orbital magnetization.

Figure 1

Spectrum of a two-dimensional electron gas with spin-orbit interaction and Zeeman splitting, Eq. (8). The left panel shows the spectrum in the case of spin-orbit coupling exceeding the Zeeman field: A ring of minima is formed at a finite value of momentum and a local maximum appears at $p=0$. The right panel corresponds to the case of strong Zeeman coupling: Both electron subbands are monotonic functions of momentum. Different possibilities for the position of the chemical potential $\mu \left(x\right)$ are indicated by the dashed lines.

Figure 2

Smooth boundary of a two-dimensional electron gas. The upper panel illustrates the depletion of the electron density near the edge. The dashed line indicates the position of the electrochemical potential $\zeta$ as counted from the bottom of the band deep inside 2DEG. The chemical potential $\mu \left(x\right)$ is a function of the coordinate [$\mu \left(\infty \right)=\zeta$]. The lower panel indicates the direction of the equilibrium electric edge currents $\vec{J}$.

Figure 3

Two-dimensional electron gas formed by the surface states of a topological insulator. Smooth gate potential creates a $p$-$n$ junction. The net current $eh/2\pi$ is flowing along the junction.

Figure 4

Kagomé lattice antiferromagnet. Bond arrows point from site $i$ to site $j$ in DM interaction term $D\widehat{z}·{\mathbf{S}}_i\times {\mathbf{S}}_j$. Also indicated are sublattices $a$, $b$, and $c$.