Zigzag edge modes in a $Z_2$ topological insulator: Reentrance and completely flat spectrum

The spectrum and wave function of helical edge modes in $Z_2$ topological insulator are derived on a square lattice using Bernevig-Hughes-Zhang (BHZ) model. The BHZ model is characterized by a “mass” term $M\left(\mathit{k}\right)=\Delta -B{\mathit{k}}^2$. A topological insulator realizes when the parameters $\Delta$ and $B$ fall on the regime, either $0<\Delta /B<4$ or $4<\Delta /B<8$. At $\Delta /B=4$, which separates the cases of positive and negative (quantized) spin Hall conductivities, the edge modes show a corresponding change that depends on the edge geometry. In the (1,0) edge, the spectrum of edge mode remains the same against change in $\Delta /B$, although the main location of the mode moves from the zone center for $\Delta /B<4$, to the zone boundary for $\Delta /B>4$ of the one-dimensional (1D) Brillouin zone. In the (1,1)-edge geometry, the group velocity at the zone center changes sign at $\Delta /B=4$ where the spectrum becomes independent of the momentum, i.e., flat, over the whole 1D Brillouin zone. Furthermore, for $\Delta /B<1.354$, the edge mode starting from the zone center vanishes in an intermediate region of the 1D Brillouin zone, but reenters near the zone boundary, where the energy of the edge mode is marginally below the lowest bulk excitations. On the other hand, the behavior of reentrant mode in real space is indistinguishable from an ordinary edge mode.

Figure 1

(Color online) Recipe (conceptual) for constructing the exact edge wave function in the zigzag edge geometry—a half-empirical way.

Figure 2

(Color online) Bulk energy spectrum: $E=E\left(k_x,k_y\right)$ (vertical axis) of a BHZ lattice model, here, implemented as a NN tight-binding model on a square lattice: cf. Eq. (9). The spectrum is shown over the entire Brillouin zone: $-\pi /a<k_x$, $k_y<\pi /a$ (horizontal plane). Parameters are chosen such that $A=B=1$ and $\Delta =2$, i.e., the system is in the topologically nontrivial phase: $0<\Delta /B<8$. The spectrum shows a typical wine-bottle structure around the $\Gamma$ point.

Figure 3

(Color online) ${\sigma }_{xy}^{\left(s\right)}$ in units of $e/2\pi$, obtained by numerically evaluating the $\mathit{k}$ integral in Eq. (11), plotted as a function of $\Delta /B$.

Figure 4

(Color online) Straight edge geometry. The two boundaries of the strip $\left(=\text{edges}\right)$ are, here, chosen to be perpendicular to the (0,1) direction. In this figure, the number of rows in the strip is $N_r=5$.

Figure 5

(Color online) Energy spectrum (numerical) in the straight edge geometry for different values of $\Delta$ $\left(A=B=1\right)$. The number of rows $N_r$ is, here, chosen to be $N_r=100$. The dotted curve is a reference, showing the exact edge spectrum given in Eq. (33). Starting with the left panel $\Delta =B$ (spectrum shown in red), $\Delta =4B$ (center panel, spectrum in green) and $\Delta =5B$ (right, blue).

Figure 6

(Color online) $\left|{\rho }_1\right|$ and $\left|{\rho }_2\right|$ plotted as a function $\Delta /B$ in the limit $k_x\to 0$.

Figure 7

(Color online) Upper panel: $\left|{\rho }_1\right|$ and $\left|{\rho }_2\right|$ plotted as a function of $k_x$ at $\Delta =2$ $\left(A=B=1\right)$. Reference lines are at $k_x/\pi =\pm 0.5$, and at $\left|\rho \right|=1$ and $\left|\rho \right|=1/\sqrt{3}$—as for the latter, cf. Eq. (40). $\left|\rho \right|=1$ corresponds indeed to the point at which the edge modes merge with the bulk spectrum (central panel, $N_r=50$). Lower panel: an enlarged image of the central panel, plotted also for a system of larger size: $N_r=100$.

Figure 8

(Color online) Zigzag or (1,1)-edge geometry (here, chosen to be normal to the $\left(1,-1\right)$ axis), defined in terms of the original square lattice, on which the new $\tilde{x}$ and $\tilde{y}$ axes are superposed in blue (online). Numbers on these axes are redefined indices $I$ and $J$, i.e., $\left(\tilde{x},\tilde{y}\right)=\left(Ia/\sqrt{2},Ja/\sqrt{2}\right)$. The number of rows $N_r$ is here chosen to be, $N_r=6$. When $N_r$ is even (odd), the two edges are inversion asymmetric (symmetric) with respect to the center of strip.

Figure 9

(Color online) Energy spectrum in the zigzag edge geometry for different values of $\Delta \left(A=B=1\right)$. Upper-left panel: $\Delta =0.2B$ (spectrum shown in red), upper-central: $\Delta =0.8B$ (spectrum in green), upper-right: $\Delta =2B$ (spectrum in blue), lower-left: $\Delta =3.2B$ (spectrum in cyan), and lower central: $\Delta =4B$ (spectrum in magenta). At $\Delta =4B$, the edge modes become completely flat and covers the entire Brillouin zone [cf. case of graphene in the zigzag edge geometry (Refs. 7, 26)]. Notice that here the horizontal axe is suppressed to make the edge modes legible. Note that at $\Delta =4B$ the bulk spectrum is also gapless; the completely flat edge modes indeed touch the bulk continuum at the zone boundary (projection of 2D Dirac cones). Compare this panel with the corresponding panel of straight edge case: Fig. 5. The number of rows $N_r$ is here chosen to be $N_r=100$. These five plots are superposed in the lower-right panel to show that the edge spectra at different values of $\Delta$ are, in contrast to the straight edge case, not on the same curve. Even in the long-wavelength limit: $k\to 0$, their slopes are different.

Figure 10

(Color online) ${\lambda }_{-}$ plotted as a function of $k$ at $A=1$ but for different values of $B$, i.e., $B=1,0.6,0.4,0.35,0.2$ and $B=0$, corresponding, respectively, the colors: red, green, blue, cyan, magenta, and orange (upper panel). Edge spectrum at $\Delta =4B$ and $B=0.2$ ($A=1$, $N_r=100$, lower panel). Two reference lines are at $k=\pm 0.438977\dots$, the value of $k$ at which ${\lambda }_{-}$ vanishes at $B=0.2$. The axes are suppressed in the lower panel so as to highlight the completely flat edge modes.

Figure 11

(Color online) The ratio, ${\rho }_j=c_{2j+2}/c_{2j}$ $\left(j=1,2,3,\dots \right)$ plotted (red points with filling to the horizontal axis) at $k=0.05$ for $\Delta /B=0.25$ (upper panel) and $\Delta /B=0.30$ (lower panel). $A=B=1$, $N_r=100$. In the upper panel, the blue line corresponds to the “theoretical” value, ${\rho }_j=0.759908\dots$ whereas in lower panel, the plots are fitted by a curve, ${\rho }_j=r\text{sin}\left(j+1\right)\theta /\text{sin}j\theta$, with the choice of parameters, $r=0.691189\dots$ and $\theta =0.133449\dots$

Figure 12

(Color online) Theoretical value (derived later) of $\rho$ (its magnitude, $\left|\rho \right|$) plotted as a function $k$ for $\Delta /B=0.25$ $\left(A=B=1\right)$. At $k/\pi =0.05$ there are two possible solutions for $\rho$: ${\rho }_{-1}=0.759908\dots$ and ${\rho }_{-2}=0.628683\dots$, the larger value of which determines large-$j$ behavior of ${\rho }_j=c_{2j+2}/c_{2j}$. Merger with bulk occurs when ${\rho }_{-1}=1$, i.e., at $k/\pi =0.263808\dots$ Close to the zone boundary $\left(k/\pi >0.900237\dots \right)$, reentrance of edge solution occurs (see Fig. 18 for details).

Figure 13

(Color online) Same as Fig. 12 for $\Delta /B=0.3$ $\left(A=B=1\right)$. On the reference line at $k=0.05$, $\rho \simeq 0.685038\pm 0.0919993i$ $\left(\left|\rho \right|=0.691189\dots ,\text{Arg}\left[\rho \right]=\pm 0.133499\dots \right)$.

Figure 14

(Color online) ${\alpha }_j=c_{2j-1}/c_{2j}$ $\left(j=1,2,3,\dots \right)$ calculated in a strip geometry (here, $N_r=100$) is plotted against the theoretical curve for ${\alpha }_{-}\left(k\right)$ (shown in black) given in Eq. (56) up to $j=N_r/4$ for the lowest-energy eigenmode at different values of $\Delta /B=0.1,0.25,0.5,1,1.35$ ($A=B=1$ fixed), corresponding, respectively, to the colors: red, green, blue, cyan, and magenta. Reference lines indicate the regime of $k$ in which the edge modes disappear at the given value of $\Delta /B$.

Figure 15

(Color online) Four solutions for $\rho$ (each curve corresponds to one solution). Only the magnitude of such solution, which is generally a complex number, i.e., $\left|\rho \right|$ is plotted as a function $k$ at different values of $\Delta /B=0.25,$, 0.3, 1.35, and 1.45 $\left(A=B=1\right)$.

Figure 16

(Color online) Merger of the edge mode with bulk continuum and “absence” of reentrance in the spectrum. As for the latter, it turns out later that binding energy of the edge mode is too small to be seen at this scale (see Fig. 19). This is an enlarged image of the spectrum in the zigzag edge geometry as shown in Fig. 9. Here, the parameters are chosen such that $\Delta /B=1.2$, $A=B=1$, and $N_r=100$. Two reference lines at $k/\pi =0.643658\dots \equiv k_{c1}/\pi$ and at $k/\pi =0.826568\dots \equiv k_{c2}/\pi$ introduce three different momentum regions: (i) $k/\pi ∊\left[0,k_{c1}/\pi \right]$, (ii) $k/\pi ∊\left[k_{c1}/\pi ,k_{c2}/\pi \right]$, and (iii) $k/\pi ∊\left[k_{c2}/\pi ,1\right]$, corresponding, respectively, to (i) the ordinary edge, (ii) the bulk, and (iii) the reentrant regimes.

Figure 17

(Color online) Reentrant edge modes I: ${\rho }_j$ plotted as a function of $j=1,2,3,\dots$ at $A=B=1$ and $\Delta =0.25$ (as in Fig. 11, upper panel) but for $k/\pi =0.901$ (upper panel) and $k/\pi =0.903$ (lower panel). In the upper panel, the blue line corresponds to the theoretical value, $\rho =0.956432$ whereas, in the lower panel the plots (in red) are fitted by the curve, ${\rho }_j=r\text{sin}\left(j+1\right)\theta /\text{sin}j\theta$, with the choice of parameters, $r=0.867804$ and $\theta =0.140687$.

Figure 18

(Color online) (Theoretical value of) $\left|\rho \right|$ plotted as a function $k$ at $\Delta =0.25$ (again, as in Fig. 11, upper panel). Enlarged picture for $k$ close to the zone boundary: $k/\pi \sim 0.9$ The reentrance of edge solution occurs at $k/\pi =0.900237\dots$ whereas two real solutions for $\rho$ is possible when $k/\pi <0.901675\dots$ Two reference lines are at $k/\pi =0.901$ and at $k/\pi =0.903$, on which ${\rho }_j$ was plotted in Fig. 17. The value of $\rho$ on these lines are, $\rho =0.783429\dots$ and $\rho =0.956432\dots$ on $k/\pi =0.901$ (case of real solutions) whereas $\rho \simeq 0.859242\pm 0.121601i$ on $k/\pi =0.903$.

Figure 19

(Color online) Binding energy of the reentrant edge mode: $E_1-E_0$ is plotted as a function of $k$ at $\Delta =0.3B$, $A=B=1$. Different curves correspond to different size (width) of the system: $N_r=100$ (blue), 200 (green), and 300 (red). The two reference lines are placed at $k/\pi =0.289936\equiv k_{c1}/\pi$ and $k/\pi =0.8983\equiv k_{c2}/\pi$ ($k∊\left[k_{c1},k_{c2}\right]$ corresponds to the bulk regime). The plots reveal an extremely small but still a finite binding energy of the reentrant edge modes.