Integer and fractional quantum anomalous Hall effect in a strip of stripes model

We study the quantum anomalous Hall effect in a strip of stripes model coupled to a magnetic skyrmion texture with zero total magnetization and in the presence of strong electron-electron interactions. A helical magnetization along the stripes and a spin-selective coupling between the stripes gives rise to a bulk gap and chiral edge modes. Depending on the ratio between the period of the magnetic structure and the Fermi wavelength, the system can exhibit the integer or fractional quantum anomalous Hall effect. In the fractional regime, the quasiparticles have fractional charges and nontrivial Abelian braid statistics.

Figure 1

A strip of stripes aligned in the $y$ direction with the stripes (green rectangles) lying along the $x$ direction hosting itinerant electrons. Two neighboring stripes labeled $m=\pm 1$ form a unit cell labeled $n$. The localized magnetic moments inside each stripe produce a helical magnetization (blue arrows) with period $\pi l/k_F$ and of opposite chiralities for $m=\pm 1$. Here, $k_F$ is the Fermi wave vector inside the stripes. The tunneling between stripes is spin conserving and spin selective, i.e., only spin-up (spin-down) electrons can tunnel between stripes belonging to the same unit cell (to different unit cells). The tunneling amplitude is spatially nonuniform and has a substantial Fourier component at $2k_F/l$. The total magnetization of the texture is zero.

Figure 2

The energy spectrum of a strip of stripes in the momentum representation for $l=1$. Each parabola, being initially twofold degenerate in spin, represents the spectrum of a particle freely propagating in the $x$ direction. The magnetic texture opens partial gaps of the size $2{\Delta }_M$ in the bulk spectrum but leaves gapless modes $L_{n11}\left(x\right),R_{n1\overline{1}}\left(x\right),L_{n\overline{1}\overline{1}}\left(x\right)$, and $R_{n\overline{1}1}\left(x\right)$, with $n$ being the unit-cell index. The spin-selective hopping $t_1$ between stripes gaps out these gapless modes in the bulk. However, there remain still gapless propagating modes at the edges: one chiral mode $R_{N1\overline{1}}\left(x\right)$ at the upper edge and one chiral mode $L_{1\overline{1}\overline{1}}\left(x\right)$ at the lower edge.

Figure 3

The scattering processes for a strip of stripe model at the filling factor $l=1/3$ which include strong interactions. The spectrum of spin-up (right parabola, green) and spin-down (left parabola, blue) electrons are shown separately for each stripe. The helical magnetic field of period $\pi /3k_F$ can open a gap in the spectrum only if spin-conserving backscattering processes due to electron-electron interactions are taken into account (red dotted lines). The same applies for the tunneling terms (brown dotted lines).

Figure 4

The topological phase diagram as a function of the system parameters $t_0/{\Delta }_M$ and $t_1/{\Delta }_M$, see Eq. (A3). The topological phase (green area) hosts edge modes. The topologically trivial phase (yellow area) corresponds to the fully gapped system without any state inside the bulk gap.

Figure 5

The energy spectrum of a strip of stripes in the momentum representation for the IQAHE regime with $l=2$. The notations are the same as in Fig. 2 of the main text. The magnetic skyrmion texture combined with spin-selective tunneling terms (both have a period $\pi /2k_F)$ opens partial gaps of the size ${\Delta }_M{t}_2/E_F$ in the bulk spectrum but leaves two gapless modes at each edge of the stripe.