Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering

We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. In electron gases with stripes, these gaps result in plateaus for the Hall conductivity exactly at the known fillings $n/m$ (both positive integers and $m$ odd) for the integer and fractional quantum Hall effect. For double strips, we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.

Figure 1

(a) Strip: two-dimensional lattice of width $W$ in the $x$ direction with the unit cell defined by the lattice constants $a_x$ and $a_y$. The hopping amplitudes in the $y$ direction, $t_{y1}$ and $t_{y2}$, carry the phase $\varphi$, arising from a perpendicular magnetic field $\mathbf{B}$, and we assume $t_x\gg t_{y1}$, $t_{y2}$. (b) Doubly degenerate spectrum of $H_x$ [see Eq. (1)] for the rows with $\sigma =1$ (upper blue line) and with $\sigma =\overline{1}$ (lower green line). The hoppings $t_{y1}$ (dashed line) and $t_{y2}$ (dotted line) induce resonant scattering between right ($R_{\sigma }$) and left ($L_{\overline{\sigma }}$) movers, which open gaps (not shown) at the Fermi wave vectors $\pm k_F$ defined by the chemical potential $\mu$.

Figure 2

Spectrum $E(k_y)$ of the left edge state (red line or dots) propagating along $y$ for a strip ($t_{y1}/t_x=0.02$, $t_{y2}/t_x=0.1$) of width $W/a_x=801$ and with phase $\varphi =\pi /2$, obtained (a) analytically [see Eq. (4)] and (b) numerically [see Eqs. (1, 2)]. For ${\overline{k}}_{-}<k_y<{\overline{k}}_{+}$, there exists one edge state at each edge. The left (red dots) and the right (blue dots) edge states are chiral and propagate in opposite $y$ directions. For each $k_y$ [(c) $k_y{a}_y=\pi$, (d) $k_y{a}_y=13\pi /12$] there is one left (red dots) and one right (blue dots) edge state if Eq. (5) is satisfied. Here, $ϵ(N)$ corresponds to the $N$th energy level. The probability density $|{\psi }_{\sigma }{|}^2$ (e) [(f)] of the left [right] localized state decays exponentially in agreement with the analytical result, Eq. (4) in Ref. 45.

Figure 3

Double strip for spinless fermions. The intrastrip couplings are the same as in Fig. 1. The interstrip coupling along $z$ is described by the hopping amplitude $t_z$.

Figure 4

Spectrum of a double strip obtained by numerical diagonalization of the tight-binding Hamiltonian $H_2=H_x+H_y+H_z$ for the same parameters as in Fig. 2. (a) If $|t_z|<t_{y1}$ [$t_z/t_x=0.01$], there are four edge states at any energy within the gap; two of them localized at the left and two at the right edge. (b) If $t_{y1}<|t_z|<t_{y2}$ [$t_z/t_x=0.05$], there is one zero-energy edge state (i.e., with flat dispersion) at each edge. (c) If $|t_z|>t_{y2}$ [$t_z/t_x=1.5$], there are no edge states in the gap.