Chiral and nonchiral edge states in quantum Hall systems with charge density modulation

We consider a system of weakly coupled wires with quantum Hall effect (QHE) and in the presence of a spatially periodic modulation of the chemical potential along the wire, equivalent to a charge density wave (CDW). We investigate the competition between the two effects which both open a gap. We show that by changing the ratio between the amplitudes of the CDW modulation and the tunneling between wires, one can switch between nontopological CDW-dominated phase to topological QHE-dominated phase. Both phases host edge states of chiral and nonchiral nature robust to on-site disorder. However, only in the topological phase, the edge states are immune to disorder in the phase shifts of the CDWs. We provide analytical solutions for filling factor $\nu =1$ and study numerically effects of disorder as well as present numerical results for higher filling factors.

Figure 1

2D strip of weakly coupled wires in a perpendicular magnetic field $B_{\perp }$ applied along the $z$ axis. The wires (thick black lines) aligned along the $x$ axis and labeled by the index $m$ are weakly coupled with tunnel amplitude $t_y$ in the $y$ direction. The vector potential $\mathbf{A}=Bx{\mathit{e}}_y$ is chosen to be in the $y$ direction such that the phase $\varphi \left(x\right)=eB{a}_{y}x/\hslash c$ acquired in the tunneling is position dependent, where $a_y$ is the distance between neighboring wires. There is a CDW modulation $U\left(x\right)$ (orange wavy line) of the chemical potential inside the wires, with period ${\lambda }_w$ tuned to the Fermi wavelength ${\lambda }_F$.

Figure 2

Energy spectrum $E\left(k_y{a}_y\right)$ near the band gap around the Fermi level defined at $\mu =-\sqrt{2}t$: (a)–(d) in the topological phase $\left(t_y>U_0=0.05t\right),({\mathrm{a}}^{\prime })$–$({\mathrm{d}}^{\prime })$ at the phase transition point $\left(t_y=U_0=0.1t\right)$, and $({\mathrm{a}}^{\prime \prime })$–$({\mathrm{d}}^{\prime \prime })$ in the nontopological phase $\left(t_y<U_0=0.15t\right)$ for $\nu =1$. The panel columns (a), (b), (c), and (d) correspond to the phase shift of the CDW modulation $\phi =-\pi /2,-0.1\pi ,0,\text{and}\pi /2$, respectively. The energy of the edge state and the bulk spectral edge is found both numerically (solid line) and analytically (dashed line), while the color map represents $|\mathrm{\Psi }(n,k_y){|}^2$ for the edge state wave function probability. The position at site $n$ (energy) is marked on the left (right) axis. We note that edge states can be found both in the topological and nontopological phase. By changing $\mu$, the edge states can be tuned between being chiral and nonchiral independent in both phases; see panels (a) and $({\mathrm{b}}^{\prime \prime })$.

Figure 3

Wave functions ${\left|\mathrm{\Psi }(n,m,E\approx 0)\right|}^2$ of edge states for a 2D finite size lattice inside the bulk gap (near $E=0)$ in the topological phase, $t_y>U_0=0.05t$ for (a) $\phi =0$, (b) $\phi =3/2\pi$, and (c) in the nontopological phase, $t_y<U_0=0.15t$ and $\phi =3/2\pi$. The lower panels $\left({\text{a}}^{\prime }\right)–\left({\text{c}}^{\prime }\right)$ and $\left({\text{a}}^{\prime \prime }\right)–\left({\text{c}}^{\prime \prime }\right)$ correspond to the case of on-site disorder $\left({\sigma }_V=0.05t\right)$ and disorder in the CDW phase $\left({\sigma }_{\phi }=\pi /8\right)$. In the topological phase, the presence of edge states is not affected by weak disorder (with effective amplitude not exceeding the size of the gap), while edge states in the nontopological phase are affected by weak disorder leading to Anderson localization (around some random edge site). However, even in the latter case edge states can survive some small amount of disorder of both types when ${\sigma }_V\le 0.1t$ and ${\sigma }_V\lesssim \pi /16$.

Figure 4

Phase diagram for filling factor $\nu =1$. If $t_y/U_0>1$ $\left(t_y/U_0<1\right)$, the system is in the topological (nontopological) phase, indicated as QHE (CDW) phase. The phase transition between phases occurs at $t_y/U_0=1$, where the gap closes. The nontopological phase is subdivided into the following subphases: (I) (orange area) edge states are totally inside the bulk gap, (II) (blue area) no edge states, and edge states spread between two wave vectors $k_{-}$ and $k_{+}$ belonging either both to the electron (III) (light green area) or both to the hole $({\mathrm{III}}^{\prime })$ (dark green areas) band. At the phase boundary between the subphases (dashed lines), we have $k_{-}=k_{+}$. Generally, depending on the phase $\phi$ the edge states can be either chiral for all values of $\mu$ or be piecewise chiral, such that by shifting $\mu$, opposite chiralities can be observed.

Figure 5

Energy spectrum $E\left(k_y{a}_y\right)$ near the band gap at the Fermi level defined by $\mu =-\sqrt{2}t$ for the isotropic system $\left(t_y=t\right)$ (a) in the topological phase $\left(U_0=0.5t\right)$, (b) at the phase transition point $\left(U_0=t\right)$, and (c) in the nontopological phase $\left(U_0=1.5t\right)$ found numerically. The phase of the CDW modulation is set to $\phi =\pi /2$ $\left[\right({\mathrm{a}}^{\prime }),({\mathrm{b}}^{\prime }),({\mathrm{c}}^{\prime })$ for $\phi =-0.1\pi ]$. The color map represents $|\mathrm{\Psi }(n,k_y){|}^2$ for the edge state wave function probability. The position at site $n$ (energy) is marked on the left (right) axis. We note that edge states can be found both in the topological and nontopological phase. By changing $\mu$, the edge states can be tuned between being chiral and nonchiral; see panels (a) and $({\mathrm{c}}^{\prime })$.

Figure 6

Wave functions $|\mathrm{\Psi }(n,m,E_0){|}^2$ (color maps) of edge states for an isotropic $\left(t_y=t\right)2\text{D}$ finite size lattice inside the bulk gap (near the center of the gap $E_0)$ (a) in the topological phase, $U_0=0.5t$, and (b) in the nontopological phase $U_0=1.5t$ for $\phi =\pi /2$. The discrete energy spectrum in the gap is plotted in the insets with the corresponding state marked by the red dot. In the nontopological regime localized edge states are present only on one side of the 2D sample.

Figure 7

Energy spectrum in the anisotropic regime $\left(t_y=0.1t\right)$ $E\left(k_y\right)$ near the band gap for the system (a)–(d) in the topological (QHE dominated) phase $\left(t_y>U_0=0.05t\right),({\mathrm{a}}^{\prime })$–$({\mathrm{d}}^{\prime })$ at the phase transition point $\left(t_y=U_0=0.1t\right)$, and $({\mathrm{a}}^{\prime \prime })$–$({\mathrm{d}}^{\prime \prime })$ in the nontopological (CDW dominated) phase $\left(t_y<U_0=0.15t\right)$. The panels (a), (b), (c), and (d) refer to the phase of CDW $\phi =\pi ,5\pi /4,3\pi /2,7\pi /4$, respectively. The spectrum of localized edge state is found numerically (dashed line) and analytically (red line). The color map represents $|\mathrm{\Psi }(n,k_y){|}^2$ for the edge state. The position, site $n$ (energy) is marked on the left (right) axis. The found behavior of edge states is, generally, the same as described in the main text, confirming that proposed phases do not depend qualitatively on the system width.

Figure 8

Wave function ${\left|\mathrm{\Psi }(n,m,E\approx 0)\right|}^2$ for the zero edge states inside the gap in the nontopological phase $|t_y|<|U_0|=0.15t$ and $\phi =3/2\pi$ where both edge states are present; see Fig. 7.

Figure 9

Same as in Fig. 2 of the main text $(t_y=0.1t$ and $\mu =-\sqrt{2}t)$ but for the filling factor $\nu =2$ (resulting in $\varphi =k_F{a}_x)$. (a),(b) In the topological phase $\left(U_0=0.02t,\phi =0,-\pi /2\right)$, there are two chiral edge states present at each edge. $({\mathrm{a}}^{\prime }),({\mathrm{b}}^{\prime })$ In the nontopological phase $\left(U_0=0.1t,\phi =0,-\pi /2\right)$, depending on the chemical potential, there could be two or four nonchiral edge states.