Propagating edge states in strained honeycomb lattices

We investigate the helically propagating edge states associated with pseudo-Landau levels in strained honeycomb lattices. We exploit chiral symmetry to derive a general criterion for the existence of these propagating edge states in the presence of only nearest-neighbor hoppings and we verify our criterion using numerical simulations of both uniaxially and trigonally strained honeycomb lattices. We show that the propagation of the helical edge state can be controlled by engineering the shape of the edges. Sensitivity to chiral-symmetry-breaking next-nearest-neighbor hoppings is assessed. Our result opens up an avenue toward the precise control of edge modes through manipulation of the edge shape.

Figure 1

A honeycomb lattice. The different sites $A$ and $B$, which form two different sublattices, are colored in blue and red, respectively. The vectors ${\mathbf{R}}_j$ connect nearest-neighbor sites. In the main text, we always refer to this orientation of the lattice. On the left edge, we show the bearded termination, while on the right edge, we show the zigzag termination. The bottom and top edges have armchair terminations. This ribbon has $N_x=7$ unit cells counted along the armchair direction (although the last unit cell is not complete, as it is missing a $B$ site), and $N_y=5$ unit cells counted along the zigzag/bearded direction.

Figure 2

Energy dispersion of a uniformly strained ribbon, i.e., when hoppings are spatially constant but not equal. Dirac cones are shifted in momentum space because of the synthetic pseudomagnetic vector potential, and these Dirac cones are connected by flat lines corresponding to the nonpropagating edge states. (a)–(d) Different configurations with $t_1\ne t$ and $t_{2,3}=t$ and periodic boundary conditions along $y$. Specifically, (a) is for bearded terminations on both ends and $t_1<t$, (b) is for bearded terminations on both ends and $t_1>t$, (c) is for zigzag terminations on both ends and $t_1<t$, and (d) is for zigzag terminations on both ends and $t_1>t$. (e) Armchair terminations and $t_2<t_1=t<t_3$ with periodic boundary conditions along $x$. The vertical dashed lines indicate the position of the Dirac points $K$ and $K^{\prime }$ of the pristine lattice. In all cases, the nonpropagating edge states are doubly degenerate and live on both open edges.

Figure 3

Energy dispersion for a uniaxially strained ribbon along $x$, oriented as in Fig. 1, with $N_x=99$ along the armchair direction and periodic boundary conditions along $y$, for different terminations. The strain strength is $\tau =0.015$, corresponding for parameters of solid-state graphene, to a pseudomagnetic field of $\mathcal{B}=30$ T. (a) Bearded terminations on both ends, (b) bearded termination on the left and a zigzag termination on the right, (c) zigzag terminations on both ends, and (d) a zigzag termination on the left and a bearded termination on the right. Each state is colored according to the mean $\langle x\rangle$ position of its spatial wave function, where cyan stands for states localized on the left edge, magenta stands for states localized on the right edge, and black is for bulk states. The spatial wave function of the states associated with the cyan, magenta, violet, and black dots are plotted in Fig. 4. Labels L and R indicate whether the eigenstate is localized on the left or on the right edge. The label E indicates the nonpropagating edge state, while the label LL0 is used to indicate an example of a $n=0$ Landau level state.

Figure 4

Modulus of the numerical eigenfunctions for parameters in Fig. 3. L corresponds to the cyan dots in Fig. 3, R corresponds to the magenta dots in Fig. 3. LL0 corresponds to the black dot in Fig. 3, where we see the wave function of the pseudo-Landau level with $n=0$ together with the nonpropagating edge state of the bearded left end. E corresponds to the violet dot in Fig. 3, showing the superposition of the doubly degenerate nonpropagating edge states.

Figure 5

Energy dispersion and modulus of the numerical wave functions for a uniaxially strained ribbon along $y$ with periodic boundary conditions along $x$. The strain strength is $\tau =0.02$, corresponding to a pseudomagnetic field of $\mathcal{B}=120$ T, for parameters of solid-state graphene. In (a) $N_y=99$ along the vertical direction and $t_{2,3}=t$ is in the center of the ribbon, thus both the top and the bottom armchair edge can sustain propagating edge states. In (b) $N_y=49$ along the vertical direction and the hoppings are equal at the bottom edge, where no propagating edge state is supported. Each state is colored according to the mean $\langle y\rangle$ position of its spatial wave function, where cyan stands for states localized on the bottom edge, magenta stands for states localized on the top edge, and black is for bulk states. The spatial wave function of the states associated with the cyan and magenta dots are plotted in (B) and (T), corresponding to the eigenstate localized on the bottom or on the top edge, for parameters in (a).

Figure 6

Energy dispersion in the presence of a real magnetic field for a ribbon of $N_x=99$ along the armchair direction and periodic boundary conditions along $y$, with different terminations on the left and right edges. The magnetic flux per plaquette has strength $2\pi \phi =0.0087$. The intensity of the associated real magnetic field ${\mathcal{B}}_r=30$ T for parameters of solid-state graphene, is the same as the pseudomagnetic field generated by the strain in Fig. 3. (a) Bearded terminations on both ends, (b) a bearded termination on the left and a zigzag termination on the right, (c) zigzag terminations on both ends, and (d) a zigzag termination on the left and a bearded termination on the right. The states are colored according to the mean position of their wave function.

Figure 7

Energy dispersion in the presence of a real magnetic field for a ribbon of $N_y=99$ along the vertical direction and periodic boundary conditions along the $x$ direction. The strength of the magnetic flux per plaquette is $2\pi \phi =0.0346$. The intensity of the associated real magnetic field ${\mathcal{B}}_r=120$ T for parameters of solid-state graphene, is the same as the pseudomagnetic field generated by the strain in Fig. 5. The states are colored according to the mean position of their wave function.

Figure 8

(a)–(c) Spatial amplitudes in the steady state showing the propagation of the edge state in the uniaxially strained system of $N_x=49$ and $N_y=199$ unit cells in the horizontal and vertical direction, respectively. Parameters are $\tau =0.02$, $\hslash \gamma /t=0.0025$, $\hslash {\omega }_0/t=(E_{n=0}+E_{n=1})/2t=0.0707$. The pumped site is indicated by the arrow. In (a), the left edge is bearded, and so supports a 0th Landau level propagating edge states. In (b), the left edge is changed to zigzag, and the propagation of the edge state is clearly suppressed. In (c), the left bearded edge has six defect sites around $y=0$, and the propagation of the edge state is strongly suppressed.

Figure 9

(a) and (b) Spatial amplitudes in the steady state showing the propagation of the edge state in the armchair-strained system. The maximum numbers of unit cells along the horizontal and vertical directions are $N_x=79$ and $N_y=99$. The pumped site is indicated by the arrow. In (a), all edges support helically propagating edge states. Parameters are $\tau =0.02$, $\hslash \gamma /t=0.0025$, $\hslash {\omega }_0/t=(E_{n=0}+E_{n=1})/4t=0.1225$. In (b) the strain is such that the propagation of the edge state is forbidden on the bottom edge. Parameters are $\tau =0.01$, $\hslash \gamma /t=0.0025$, $\hslash {\omega }_0/t=(E_{n=0}+E_{n=1})/4t=0.0866$.

Figure 10

(a) and (b) Spatial amplitudes in the steady state showing the propagation of the edge state in the trigonally strained hexagonally shaped system. Edges are labeled by Z and B to indicate if they are, respectively, zigzag or bearded terminations. The maximum numbers of unit cells along the horizontal and vertical directions are $N_x=99$ and $N_y=199$, respectively. The pumped site is indicated by the arrow. In (a), all edges support helically propagating edge states. In (b), the bottom right edge is changed from a zigzag to bearded, and the propagation of the edge state is clearly blocked on that edge. Parameters are $\tau =-0.02$, $\hslash \gamma /t=0.0025$, $\hslash {\omega }_0/t=(E_{n=0}+E_{n=1})/4t=0.061$. (c) and (d) The Fourier transform of (a) and (b), respectively. We see that all valleys are excited in (c) that corresponds to the case of both helical edge states going around the system in (a). In (d) one valley is predominantly excited, corresponding to the case of mostly the clockwise edge state in (b).

Figure 11

Spatial amplitudes in the steady state showing the propagation of the edge state in the trigonally strained hexagonally shaped system. All edges are armchair. The maximum numbers of unit cells along the horizontal and vertical directions are $N_x=79$ and $N_y=99$, respectively. The pumped site is indicated by the arrow. Parameters are the same as in Fig. 10.

Figure 12

(a) Energy dispersion for a unstrained system $\tau =0$, with $N_y=99$ along the vertical direction and $t^{\prime }=0.1t$, with a bearded termination on the left and a zigzag termination on the right. (b) Energy dispersion for a uniaxially strained ribbon along $y$ with periodic boundary conditions along $x$ and armchair terminations at the top and bottom, with a next-nearest-neighbor hopping strength of $t^{\prime }=0.1t$. The strain strength is $\tau =0.02$, with $N_y=99$ along the vertical direction and $t_{2,3}=t$ is in the center of the ribbon. (c)–(f) Energy dispersions for a uniaxially strained ribbon along $x$, with $N_x=99$ along the armchair direction and periodic boundary conditions along $y$, for different terminations with a constant next-nearest-neighbor hopping strength of $t^{\prime }=0.1t$, and a strain strength of $\tau =0.015$. (c) Bearded terminations on both ends, (d) is for a bearded termination on the left and a zigzag termination on the right, (e) is for zigzag terminations on both ends, and (f) is for a zigzag termination on the left and a bearded termination on the right. In (b)–(f), the states are colored according to the mean position of their wave function, as indicated by the color bar.

Figure 13

(a)–(d) Spatial amplitudes in the steady-state showing the propagation of the edge state in the uniaxially strained system of $N_x=49$ and $N_y=199$ unit cells in the horizontal and vertical direction, respectively. Parameters are $t^{\prime }=0.1t$, $\tau =0.02$, $\hslash \gamma /t=0.0025$. (a) and (b) A pump frequency in the lower LL0 gap $\hslash {\omega }_0/t=-(E_{n=0}+E_{n=1})/2t=-0.36$, while (c) and (d) are for pump frequency in the upper LL0 gap $\hslash {\omega }_0/t=(E_{n=0}+E_{n=1})/2t=-0.24$. The pumped site is indicated by the arrow. In (a), the left edge is bearded, and so supports a 0th Landau level propagating edge state. In (b), the left edge is changed to zigzag, and the propagation of the edge state is clearly suppressed. (c) and (d) The same configuration as in (a) and (b), respectively. We see the mixing of the propagating edge state associated with pseudo-Landau levels with previously nonpropagating edge states, which are localized on a shorter length. In particular, now we see that the zigzag termination in the upper LL0 gap has an edge state that propagates along the edge.