Hinge modes and surface states in second-order topological three-dimensional quantum Hall systems induced by charge density modulation

We consider a system of weakly coupled one-dimensional wires forming a three-dimensional stack in the presence of a spatially periodic modulation of the chemical potential along the wires, equivalent to a charge density wave (CDW). An external static magnetic field is applied parallel to the wire axes. We show that, for a certain parameter regime, due to interplay between the CDW and magnetic field, the system can support a second-order topological phase characterized by the presence of chiral quasi-1D quantum Hall effect (QHE) hinge modes. Interestingly, we demonstrate that direction of propagation of the hinge modes depends on the phase of the CDW and can be reversed only by electrical means without the need of changing the orientation of the magnetic field. Furthermore, we show that the system can also support 2D chiral surface QHE states, which can coexist with one-dimensional hinge modes, realizing a scenario of a hybrid high-order topology. Performing two-terminal transport simulations in the linear response regime, we confirm quantized QHE resistance plateaus, which are highly robust to disorder giving a clear signature of hinge and surface states.

Figure 1

(a) A 3D stack of weakly coupled wires in a magnetic field $B$ applied along their axis. (b) The wires (thick black lines) aligned along the $x$ axis and labeled by the indexes $m$ ($l$) are weakly coupled with tunnel amplitude $t_y, \left(t_z\right)$ in the $y$ ($z$) direction. The vector potential $\mathbf{A}=By{\mathit{e}}_z$ is chosen to be in the $z$ direction such that the corresponding tunneling phase $\varphi \left(y\right)=eB{a}_{z}y/\hslash c$ is position dependent, where $a_z$ is the distance between neighboring wires in the $z$ direction. In addition, there is a CDW modulation $V\left(x\right)$ [orange wavy line (b)] of the chemical potential of the wires with period ${\lambda }_w$. The sketch of (c) the hinge modes localized on the left (red) and right (blue) surfaces and of (d) surface states localized on all surfaces except left and right. Green areas indicated leads attached to the system for the purpose of transport studies.

Figure 2

(a) Energy spectrum of a single wire with CDW modulation as function of phase $\phi$ around the lowest gap at filling factor 1/4. For $\phi =3\pi /4$ the gap contains doubly degenerate end states at energy $E_0$ and for $\phi ={\phi }_0$ two nondegenerate end states at energies $E_{0L}$ and $E_{0R}$. (b),(c) The corresponding probability densities $|{\mathrm{\Psi }}_{\text{CDW}}{(x;E)|}^2$ for $E=E_0,E_{0L},E_{0R}$ indicate that double-degenerate states at $E_0$ are localized at both ends of the wire, while states at energies $E_{0L}$ and $E_{0R}$ are localized at the left ($x_L$) and the right ($x_R$) end of the wire. Here, we choose $U_0=0.9t_x, N_x=40$, and $2k_w{a}_x=\pi /2$.

Figure 3

(a) The spectrum of the system from Fig. 1 with periodic boundary conditions in the $z$ direction around the lowest CDW-induced gap for $\phi =3\pi /4$. Two degenerate copies of a 2D QHE spectrum are centered around $E_0$ within two QHE gaps. The 2D QHE bulk bands (green) and 1D hinge modes $E_{LR}^{\sigma }$ (dashed red-blue) are localized at the left and right surfaces with $\sigma =⥁\left(1\right)$ and $\sigma =⥀(-1)$ denoting the propagation direction of the hinge modes. The two-terminal resistance $\rho \left(E_F\right)$ for clean (orange) and disordered (purple) systems from Figs. 1 and 1 as a function of Fermi energy $E_F$ with QHE plateaus at $\rho ={\rho }^{\mathrm{dis}}=0.5[h/e^2$] for $E_F$ lying in the gaps hosting a pair of degenerate QHE hinge modes. (b),(c) The cross sections of the QHE hinge modes probability density $|{\mathrm{\Psi }}_{\text{hinge}}^{\sigma }(x=x_L,y,k_z){|}^2={\left|{\mathrm{\Psi }}_{\text{hinge}}^{\sigma }(x=x_R,y,k_z)\right|}^2$. (d) The probability density $|{\mathrm{\Psi }}_1{(x,y)|}^2, |{\mathrm{\Psi }}_{\overline{1}}{(x,y)|}^2$ for the QHE hinge modes at selected $k_z$ points, 1 and $\overline{1}$, indicated in (a). Here, we set $t_x=1, t_y=0.11t_x, t_z=0.09t_x, U_0=0.9t_x, 2k_w{a}_x=\pi /2$ with $N_x\times N_y(\times L_z)=40\times 39(\times 40$) lattice points (for transport) and take a resonant magnetic field $\varphi =2k_F{a}_y=\pi /2$, corresponding to filling factor $\nu =1$.

Figure 4

(a) Spectrum of the 3D system from Fig. 1 around the CDW-induced gap as a function of $\phi$ for $k_z=0$. The interesting part, where the QHE spectra of states localized at the left and right surfaces cross and overlap, is marked by a gray rectangle and zoomed in on panel (b) for $k_z{a}_z=\pi /2$. The pale orange (green) color denotes QHE gaps with a single hinge mode characterized by counterclockwise $\sigma =1=⥀$ (clockwise $\sigma =-1=⥁$) direction of propagation. The dark orange (green) areas denotes QHE gaps with a pair of chiral unidirectional QHE hinge modes with $\sigma =1$ ($\sigma =-1$) localized at the left and right surfaces. The yellow area denotes a scenario with a pair of in-gap hinge modes propagating in opposite directions on the left and right surface. The solid red (blue) lines correspond to hinge modes localized at hinges adjacent to the left (right) surface.

Figure 5

Same as in Fig. 3 but for $\phi ={\phi }_0$. The two degenerate copies of the bulk QHE spectrum from Fig. 3 are split in energies such that each one is centered around $E_{0R}$ and $E_{0L}$, respectively. The part of the QHE spectrum around $E_{0R}$ ($E_{0L}$) corresponds to states localized on left (right) surface. Interestingly, for the chosen CDW phase $\phi ={\phi }_0$, in the gap around $E_0$ we find a pair of chiral hinge modes that propagate in the opposite directions on the left and the right surface. We can see stable quantized resistance plateaus at $\rho =0.5[h/e^2$] ($\rho =1[h/e^2$]) for the energies lying in the gap which hosts two (one) hinge modes for clean (orange) and disordered (purple) system. (c) The corresponding wave function probability ${\mathrm{\Psi }}_{1,\overline{1},2,\overline{2}}$ for $k_z$ momenta marked by 1, $\overline{1}$, 2, $\overline{2}$ points on panel (a).

Figure 6

(a) Spectrum of the 3D system from Fig. 1 with the periodic boundary conditions in the $z$ direction around the lowest in energy 3D QHE gaps (yellow area) as a function of $k_z$ for $\phi =3\pi /4$. The parameters have been chosen so that the lower in energy gap hosts exclusively chiral QHE surface states $E_{\mathrm{surf}}^{⥁}\left(k_z\right)$ propagating in clockwise direction, while the upper gap hosts chiral QHE surface states $E_{\mathrm{surf}}^{⥀}\left(k_z\right)$ propagating in counterclockwise direction and pair of degenerate hinge modes $E_{LK}^{⥁}\left(k_z\right)$ propagating in clockwise direction realizing hybrid high-order topology. The spectrum of the 3D system E(k) along high symmetry k points in 3D Brillouin zone (upper horizontal axis) with periodic boundary conditions in all the directions (brown) demonstrates the bulk character of the 3D QHE gaps. On panel (b), we plot two terminal resistance $\rho \left(E_F\right)$ for clean (orange line) and disordered system ${\rho }^{\mathrm{dis}}\left(E_F\right)$ and ${\rho }^{{\mathrm{dis}}_2}\left(E_F\right)$ (purple and blue). One can clearly see quantized QHE resistance plateaus at $\rho =1/11[h/{\mathrm{e}}^2]$ ($\rho =1/9[h/{\mathrm{e}}^2]$) for Fermi energies $E_F$ lying in the upper (lower) gap. (c) Interestingly, in the presence of disorder for in-gap energies, conductance is reduced and not perfectly quantized for hybrid scenario with coexisting counterpropagating surface and hinge modes. (${\mathrm{c}}^{\prime }$) This is not the case for $E_F$ lying in the lower gap which hosts only surface modes and results in perfectly quantized conductance at $G\left(E_F\right)=G^{{\mathrm{dis}}_1}=G^{{\mathrm{dis}}_2}=9[{\mathrm{e}}^2/h]$ even in the presence of strong disorder. Here, we set $t_y=0.31, t_z=0.21, U_0=0.9t_x, 2k_w{a}_x=\pi /2$ with $N_x\times N_y(\times N_z)=40\times 39(\times 40)$ lattice points (for transport simulations) and take the resonant value for the magnetic field $\varphi =\pi /2$.

Figure 7

(a) Same as in Fig. 3 but for smaller $U_0=0.7t$. (b) The zoomed-in part of the spectrum [marked by orange rectangle on (a)] contains the gap in which doubly degenerate chiral QHE hinge modes ${\text{E}}_{\text{h,}}\text{LR}\left(k_z\right)$ and nondegenerate QHE chiral surface states ${\text{E}}_{\text{surf}}\left(k_z\right)$ coexist. The resistance with quantized plateaus at $\rho ={\rho }^{\mathrm{dis}}=1/3[h/e^2]$ for clean (orange) and disordered (purple) systems and for the system with ‘truncated’ leads [green lines on (c)] so that only surface modes contribute to transport with resistance plateau at $\tilde{\rho }=1[h/e^2]$ (blue). (c) The corresponding probability density $|{\mathrm{\Psi }}_{\text{j}}(x,y,k_z^{\text{fixed}}){|}^2$ for fixed $k_z$ denoted by $j=1,\overline{1},2,\overline{2}$ which shows chiral character of the states. (d) Zoom of the plateau region from (b) with conductance which shows how the presence of disorder affects conductance $G^{\mathrm{dis}}\approx 3[e^2/h]$ in a hybrid regime.

Figure 8

(a) Spectrum of the 3D system from Fig. 1 from the main text with the periodic boundary conditions in the $z$ direction around the lowest in energy CDW induced gap as a function of $k_z$ for $\phi =-\pi /4$. The zoom of upper edge of CDW gap is plotted in panel (b) and further zoom of one of the gaps in (${\mathrm{b}}^{\prime }$). The color maps (c,d) shows probability density $|{\mathrm{\Psi }}_{\text{surface}}(x=x_M,y,k_z){|}^2$ for the first and second QHE surface mode hosted in the gap marked by the brown rectangle and further zoomed in (${\mathrm{b}}^{\prime }$). The probability density $|{\mathrm{\Psi }}_{\text{surface}}(x,y,k_z^{\text{fixed}}){|}^2$ for the first ($\eta =1$) and second mode ($\eta =2$) of the QHE chiral surface states [selected $k_z^{\text{fixed}}$ points: 1, $\overline{1}$, 2, $\overline{2}$ from panels (b), (${\mathrm{b}}^{\prime }$) is represented on panel (e)]. Here, we set $t_y=0.11, t_z=0.09, U_0=0.9t_x, 2k_w{a}_x=\pi /2$ with $N_x\times N_y=40\times 39$ lattice points and take the resonant value for the magnetic field $\varphi =\pi /2$.

Figure 9

Spectrum of the 3D system from Fig. 1 of the main text with periodic boundary conditions in the $z$ direction around the energetically lowest CDW-induced gap as a function of $k_z$ for $\phi =3\pi /4$: (a) without disorder and (b) in the presence of onsite static disorder corresponding to a uniformly fluctuating chemical potential in the range of the QHE gap $|\delta {\mu }_{n,m}|\le 0.09=|t_z|$. Despite the fact that the bulk gap gets smaller, the hinge modes are clearly present in the spectrum.

Figure 10

Spectrum of the 3D system from Fig. 1 of the main text with periodic boundary conditions in the $z$ direction around the energetically lowest CDW-induced gap as a function of $k_z$ for (a) $\phi =3\pi /4$ and (b) $\phi =3\pi /4+\delta \phi$. One can clearly see in panel (b) that by moving away from $\phi =3\pi /4$, e.g., for $\phi =3\pi /4+\delta \phi$, the degeneracy of the two QHE spectrum copies is lifted, which results in splitting in energy of the hinge modes and bulk bands belonging to two different surfaces. Now, the QHE gaps are reduced and host two hinge modes localized on the left (red curve) and right (blue curve) surfaces which corresponds to the regimes marked by dark green and orange on Fig. 4 of the main text.

Figure 11

Cross section of the spectrum of the 3D system from Fig. 1 of the main text with periodic boundary conditions in the $z$ direction as a function of (a) $U_0$ and (b) $t_y$ for fixed $\phi =3\pi /4, k_z=\pi /2, t_z=0.09$, and (a) $t_y=0.11$, (b) $U_0=0.9$. The interesting regimes are marked schematically by ovals. For $U_0\gtrsim 0.8$, the part of the spectrum with hinge modes is well separated from one with QHE surface states. If $U_0$ is decreased, the 2D QHE spectrum with the QHE gap hosting quasi-1D hinge modes is pushed into the 3D bulk states and starts to overlap with gaps hosting 2D QHE chiral surface states. In such a regime, the scenario with a hybrid topology is realized. Further decrease of $U_0$ below $U_0\lesssim 0.4$ leads to an emergence of another region with mixed higher-order topology. On the other hand, when $|t_y|$ is increased, the QHE hinge modes start to hybridize with the QHE surface states. However, in this case, the bulk states mask them, making them undetectable in transport experiments.

Figure 12

Spectrum of the 3D system from Fig. 1 of the main text with periodic boundary conditions in the $z$ direction around the energetically lowest CDW-induced gap as a function of $k_z$ for $\phi =3\pi /4$ (a) for isotropic hopping in the $yz$ plane with $t_y=t_z=0.11$, (b) for a slightly anisotropic case like in the main text with $t_y=0.11, t_z=0.09$, and (c) for a highly anisotropic case with $t_y=t_z/2=0.055$. The lower panels (${\mathrm{a}}^{\prime }$)–(${\mathrm{c}}^{\prime }$) show the zoomed-in part of the spectrum with the 3D QHE gaps hosting nondegenerate 2D QHE chiral surface states. One can clearly see that decreasing $t_z$ results in flattening of the bulk bands and in reducing the size of the gaps. Moreover, for $|t_y|\gg |t_z|$, (c) the chiral QHE hinge modes and (${\mathrm{c}}^{\prime }$) chiral QHE surface states have a cosinelike shape.

Figure 13

(a)–(d) The same as in Figs. 2, 2 and 2 of the main text but for a different periodicity of the CDW potential, i.e., $k_w=\pi /5a_x$. In this case, the system has an inversion symmetry for $\phi =4\pi /5$ and supports doubly degenerate hinge modes in the same way as we presented in Figs. 3 and 3 of the main text.

Figure 14

Spectrum of the 3D system shown in Fig. 1 of the main text with periodic boundary conditions in the $z$ direction around the energetically lowest CDW-induced gap as a function of $k_z$ for $\phi =3\pi /4$—same as Fig. 3 of the main text but for the filling factor $\nu =2$ with $\varphi =\pi /4$. The corresponding doubly degenerate hinge modes $E_{LR}^{\sigma }\left(k_z\right)$ are marked by the red/blue dashed lines.

Figure 15

Spectrum of the 1D CDW modulated wire with periodic boundary conditions described by Hamiltonian $H_{\text{CDW}}^{1D}$. (a) The spectrum $E(k_x,\phi$) as a function of CDW phase $\phi$ for fixed $k_x=0$ (brown line) and $k_x=\pi$ (green line). (b) The spectrum $E(k_x,\phi )$ as a function of CDW phase $\phi$ for fixed $\phi =0$ (brown line) and $\phi =\pi /4$ (green line). For comparison, we also plot the spectrum of the finite-size system (see Fig. 2 main text) in panel (a) and marked it by black lines with end states denoted by thick ones. We can see perfect agreement between the gaps and position of the bulk states in the open and closed systems. The parameters are the same as in the main text, $t_x=1, U_0=0.9, k_w=\pi /4$, and $N_x=40$. We checked numerically for the finite size system that the increase of $N_x$ leads to the increase of the number of bulk bands which lay in energy between $E(k_x=\pi ,\phi )$ and $E(k_x=\pi ,\phi )$ bands obtained for the periodic system. However, the size of the CDW induced gap remains unchanged.