Edge states and the integer quantum Hall conductance in spin-chiral ferromagnetic kagome lattice

TL;DR

This paper studies chiral edge states in a ferromagnetic kagome lattice with spin anisotropies, linking edge state properties to the quantized Hall conductance through topological invariants.

Contribution

It derives the edge state energies using the Harper equation and relates the Hall conductance to the winding number of edge states, confirming topological bulk-edge correspondence.

Findings

Two edge states per bulk energy gap identified.

Edge state energy loops cross the Riemann surface holes.

Quantized Hall conductance depends on the spin chiral parameter.

Abstract

We investigate the chiral edge states in the two-dimensional ferromagntic kagom\'{e} lattice with spin anisotropies included. The system is periodic in the $x$ direction but has two edges in the $y$ direction. The Harper equation for solving the energies of edge states is derived. We find that there are two edge states in each bulk energy gap, corresponding to two zero points of the Bloch function on the complex-energy Riemann surface (RS). The edge-state energy loops parametrized by the momentum $k_{x}$ cross the holes of the RS. When the Fermi energy lies in the bulk energy gap, the quantized Hall conductance is given by the winding number of the edge states across the holes, which reads as $\sigma_{xy}^{\text{edge}}$=$-\frac{e^{2}}{h}% $sgn$(\sin\phi) $, where $\phi$ is the spin chiral parameter (see text). This result keeps consistent with that based on the topological bulk theory.