Creating topological interfaces and detecting chiral edge modes in a two-dimensional optical lattice

We propose a general scheme to create chiral topological edge modes within the bulk of two-dimensional engineered quantum systems. Our method is based on the implementation of topological interfaces, designed within the bulk of the system, where topologically protected edge modes localize and freely propagate in a unidirectional manner. This scheme is illustrated through an optical-lattice realization of the Haldane model for cold atoms [G. Jotzu et al., Nature (London) 515, 237 (2014)], where an additional spatially varying lattice potential induces distinct topological phases in separated regions of space. We present two realistic experimental configurations, which lead to linear and radial-symmetric topological interfaces, which both allow one to significantly reduce the effects of external confinement on topological edge properties. Furthermore, the versatility of our method opens the possibility of tuning the position, the localization length, and the chirality of the edge modes, through simple adjustments of the lattice potentials. In order to demonstrate the unique detectability offered by engineered interfaces, we numerically investigate the time evolution of wave packets, indicating how topological transport unambiguously manifests itself within the lattice. Finally, we analyze the effects of disorder on the dynamics of chiral and nonchiral states present in the system. Interestingly, engineered disorder is shown to provide a powerful tool for the detection of topological edge modes in cold-atom setups.

Figure 1

Illustration of topological interfaces in a two-band model. (a) Abstract two-band model realizing different topological phases. Depending on the value of a controllable parameter $\alpha$, the two bulk bands are either topologically trivial (Chern number $C=0$) or nontrivial ($C=\pm 1$), where ${\alpha }_1^{*}$ and ${\alpha }_2^{*}$ indicate the transition points (i.e., gap-closing points). (b) Spatially varying the parameter $\alpha (x,y)$ in the 2D plane generates different regions, which are associated with distinct (spatially resolved) topological phases. In this figure, each region is labeled by the Chern number $C$ of the lowest bulk band [see panel (a)], which is evaluated locally in space. The singular spatial regions where $\alpha (x,y)={\alpha }_{1,2}^{*}$ define the topological interfaces within the 2D plane, where topologically protected “edge” modes are located and propagate.

Figure 2

Edge-state structures in various external potentials and interfaces, for an abstract two-band model. Left of each panel: schematics of the bulk energy bands $E_{\pm }\left(\mathit{x}\right)$, as evaluated locally in space along the $x$ direction. The Chern numbers $C$ associated with the bulk bands are indicated; the Fermi energy $E_{\text{F}}$ is set within the bulk gap, where edge modes (not shown) are expected whenever the bands reach the topological regimes $C=\pm 1$. The location and localization length ${\lambda }_{\text{loc}}$ of the edge modes is represented on the $x$ axis. The external potential or confinement is indicated by thick black lines. Right of each panel: illustration of the corresponding topological edge modes in the 2D plane. The chirality and the localization length of the edge modes are indicated, as well as the Chern number of the lowest band (note that vacuum is associated with $C=0$). (a) In an ideal boxlike potential $V_{\mathrm{box}}$, the bulk bands are shifted to higher energies. This naturally generates a topological interface between the inner topological system ($C=1$) and vacuum ($C=0$). In this standard case, chiral edge modes appear at the topological interfaces defined by the edges of the box. (b) In the case of a smoothly varying potential (e.g., a harmonic trap), the bulk bands are locally deformed, and edge modes potentially survive within the bulk gaps. However, smooth traps significantly increase the localization length and reduce the group velocity of the edge modes. (c) Varying a system parameter in space allows for the creation of tunable topological interfaces within the 2D plane (Fig. 1): topological edge modes are located in the vicinity of the transition point, which can be designed at the center of the (smooth) trap. In the latter case, robust edge modes appear at a genuine topological phase transition (here, $C=0\leftrightarrow 1$), without the simultaneous action of a potential step [panel (a)].

Figure 3

The Haldane model defined on the “brick-wall” lattice used in this work. The lattice spacing between NN sites is denoted $a$, the NN (resp. NNN) tunneling amplitude is denoted $t_{\text{NN}}$ (resp. $t_{\text{NNN}}$). The NNN tunneling matrix elements $\pm i{t}_{\text{NNN}}$ are complex; their sign is positive (resp. negative) for clockwise (resp. anticlockwise) paths. Nonequivalent lattice sites are denoted $\text{A}$ and $\text{B}$. Note that we neglect purely vertical NNN hoppings, which are small in experimental realizations of the model; this slightly modifies the original Haldane model, but does not alter its topological properties.

Figure 4

Topological phase diagram and two representative spectra. (Top) The boundary lines in the phase diagram indicate gap-closing events in the two-band spectrum, through which the Chern number of the bulk bands is allowed to change. (Bottom) Spectrum $E=E\left(k_y\right)$ of the Haldane model (1), set on a cylinder geometry aligned along $x$: $① {\mathrm{\Delta }}_{\text{AB}}=0$ and $② {\mathrm{\Delta }}_{\text{AB}}=2{\mathrm{\Delta }}_{\text{trans}}$; here $t_{\text{NNN}}=0.15t_{\text{NN}}$ and ${\mathrm{\Delta }}_{\text{trans}}\equiv 4\sqrt{3}{t}_{\text{NNN}}$. The situation in $①$ corresponds to a topological band configuration, where the bulk bands are associated with Chern numbers $C=\pm 1$, and where the bulk gap hosts a single edge-state mode (per edge); note that the two dispersion branches, with opposite group velocity (chirality), correspond to edge modes located on opposite edges of the cylinder. The case in $②$ is a trivial band insulator with $C=0$. These two situations are indicated in the topological phase diagram.

Figure 5

(a) Space-dependent offset ${\mathrm{\Delta }}_{\text{AB}}\left(x\right)$, as defined in Eq. (3), for ${\delta }_{\text{max}}=20t_{\text{NN}}, t_{\text{NNN}}=0.4t_{\text{NN}}$, and $L_x=100a$. The offset ${\mathrm{\Delta }}_{\text{AB}}\left(x\right)$ is expressed in units of the critical value ${\mathrm{\Delta }}_{\text{trans}}\equiv 4\sqrt{3}{t}_{\text{NNN}}$, at which topological transitions occur (see horizontal dotted lines). (b) The corresponding separation between topological and trivial regions, as dictated by the two interfaces located at $x=x_R=L_x/2$ and at $x=x_L$; see Eq. (5). The inner region of the lattice corresponds to the topological region $[-{\mathrm{\Delta }}_{\text{trans}}<{\mathrm{\Delta }}_{\text{AB}}\left(x\right)<{\mathrm{\Delta }}_{\text{trans}}]$, while the regions outside these frontiers are topologically trivial. Chiral topological edge modes are localized and propagate in the vicinity of these two interfaces, as indicated by the colored arrows.

Figure 6

Energy spectrum $E=E\left(k_y\right)$, as a function of quasimomentum $k_y$, and edge-state amplitudes for two values of the NNN hopping amplitude: (a) $t_{\text{NNN}}=0.4t_{\text{NN}}$, (b) $t_{\text{NNN}}=0.2t_{\text{NN}}$. The system is solved on a cylinder of length $L_x=100a$, and with a space-dependent offset ${\mathrm{\Delta }}_{\text{AB}}\left(x\right)$ [Eq. (3)], characterized by ${\delta }_{\text{max}}=20t_{\text{NN}}$. Each eigenvalue $E\left(k_y\right)$ is colored in terms of the mean position $\langle x\rangle$ of its corresponding eigenstate (see color bars). The eigenstates represented on the right correspond to the eigenenergies $E\left(k_y\right)$ indicated by blue and green circles in the spectrum, respectively. Vertical dotted lines in the right panels correspond to the interface positions $x_{L,R}$, as predicted by Eqs. (3, 4, 5). We note that the group velocity $v_g^y$ associated with the edge modes remains approximately constant as $t_{\text{NNN}}$ is increased. Moreover, the bulk bands are only slightly distorted in the vicinity of the bulk gap, whose size is approximatively ${\mathrm{\Delta }}_{\text{gap}}\approx 0.9t_{\text{NN}}$. We point out that only the states located within the bulk gap, with an approximately linear dispersion relation, are well (spatially) localized.

Figure 7

(a) Amplitude of an edge state localized around $x_R$ for $t_{\text{NNN}}=0.4t_{\text{NN}}$, and four different values of the slope parameter ${\delta }_{\text{max}}$, as indicated on the figure (the parameter ${\delta }_{\text{max}}$ is measured in units of $t_{\text{NN}}$). The edge mode propagating along the central interface is more localized as ${\delta }_{\text{max}}$ is increased. (b) Dispersion relation $E=E\left(k_y\right)$ of the edge mode at $x_R$, for the same values of the system parameters. The group velocity of this localized mode is found to be stable around the value $v_g^y\approx 1.7a{t}_{\text{NN}}/\hslash$. Here, the system size along the $x$ direction is $L_x=100a$.

Figure 8

(a) Energy spectrum for a finite-size open-boundary honeycomb lattice [Eq. (1)], with spatially dependent offset given by Eq. (3). Here, $\lambda$ is an integer labeling the eigenvalues by increasing order. The bulk gap has a size ${\mathrm{\Delta }}_{\text{gap}}\approx t_{\text{NN}}$, as indicated by the green region where the density of states is strongly reduced. (b) Amplitude of a representative eigenstate $|{\psi }_{\text{edge}}{\left(\mathit{x}\right)|}^2$, with energy $E_{\text{edge}}\approx 0$ in the bulk gap. This state is localized on the topological interfaces located at $x=x_{R,L}$, with a localization length of about five lattice sites [in agreement with Fig. 6]. System parameters are $t_{\text{NNN}}=0.4t_{\text{NN}}, {\delta }_{\text{max}}=20t_{\text{NN}}, L_x=100a$, and $L_y=56a$.

Figure 9

(a) In the absence of NNN hopping, $t_{\text{NNN}}=0$, the system is invariant under time reversal: the topological region disappears, which is indicated here by the equality $x_R=x_L$ for $t_{\text{NNN}}=0$; see Eq. (5). The trivial “interface” at $x=L_x/2$ hosts localized propagating states, with opposite chirality: In this scheme, the chirality of localized modes can only be spatially resolved when setting $t_{\text{NNN}}\ne 0$, i.e., when time-reversal symmetry is broken (see Figs. 5 and 6). (b) Energy spectrum $E=E\left(k_y\right)$, as a function of quasimomentum $k_y$, and amplitude of two localized states (green) and one delocalized state (gray) for the time-reversal-invariant case $t_{\text{NNN}}=0$. The energy of these depicted states is indicated in the spectrum by two green circles and one gray circle, respectively. All other system parameters are the same as in Fig. 6. We point out that only the states located within the bulk gap are well (spatially) localized (see the bulk state depicted in gray versus the localized states in green).

Figure 10

Comparison between the shape of the harmonic potential $V_{\text{trap}}\left(\mathit{x}\right)$ in the 2D plane (left) and the amplitude $|{\psi }_{\text{edge}}{\left(\mathit{x}\right)|}^2$ of an eigenstate whose energy is located within the bulk gap (right). The trap parameters are given by (a), (b) $V_x=V_y=0.2t_{\text{NN}}$, (c), (d) $V_x=V_y=0.5t_{\text{NN}}$, and (e), (f) $V_x=V_y=1.0t_{\text{NN}}$. In both cases, $L_{\text{obs}}=20a\sqrt{V_y/t_{\text{NN}}}$, so that the available propagation distance along the central interface is of about 40 sites. The “edge” state is indeed well localized along the topological interfaces, but only within regions where $V_{\text{trap}}\left(\mathit{x}\right)<{\mathrm{\Delta }}_{\text{gap}}\approx 0.9t_{\text{NN}}$ (see the white dotted ellipses on the right panels). System parameters are $t_{\text{NNN}}=0.4t_{\text{NN}}, {\delta }_{\text{max}}=20t_{\text{NN}}, L_x=100a$, and $L_y=50a$. On the left panels, the trap $V_{\text{trap}}\left(\mathit{x}\right)$ is measured in units of $t_{\text{NN}}$.

Figure 11

(a) Energy spectrum for the radial-interface (red) configuration [Eq. (8)]; this spectrum is compared with the linear-interface case (blue). Here, $\lambda$ is an integer labeling the eigenvalues by increasing order. In both cases, the bulk gap has a size ${\mathrm{\Delta }}_{\text{gap}}\approx t_{\text{NN}}$, as indicated by the green region where the density of states is strongly reduced. (b) Amplitude of a representative eigenstate $|{\psi }_{\text{edge}}{\left(\mathit{x}\right)|}^2$, with energy $E_{\text{edge}}\approx 0$ within the bulk gap, for the radial-interface configuration. This state is localized around the radial topological interface, defined by the radius $r=R_{\text{inter}}$. System parameters are $t_{\text{NNN}}=0.4t_{\text{NN}}, R_{\text{inter}}=17a$, and $L_x=L_y=75a$.

Figure 12

Comparison between (a) the shape of the harmonic potential $V_{\text{trap}}\left(\mathit{x}\right)$ in the 2D plane, and (b) the amplitude $|{\psi }_{\text{edge}}{\left(\mathit{x}\right)|}^2$ of an eigenstate whose energy is located within the bulk gap, for the radial-interface configuration [Eq. (8)]. The radius of the circular interface $R_{\text{inter}}$ is indicated by a white (resp. blue) dotted circle in (a) [resp. (b)]. The robustness of this localized edge state relies on that it is entirely located in a region where $V_{\text{trap}}\left(\mathit{x}\right)<{\mathrm{\Delta }}_{\text{gap}}$ [see the white dotted ellipse in (b)]; compare with Fig. 10. The system parameters are $V_0=0.25t_{\text{NN}}, t_{\text{NNN}}=0.4t_{\text{NN}}, R_{\text{inter}}=17a$, and $L_x=L_y=75a$. In panel (a), the trap $V_{\text{trap}}\left(\mathit{x}\right)$ is measured in units of $t_{\text{NN}}$.

Figure 13

Wave-packet dynamics in the 2D honeycomb lattice. (a) A small Gaussian wave packet is initially prepared at the center of the system (i.e., on the central interface at $x=x_R$), and its mean quasimomentum ${\mathit{k}}^0=(0,2.1/a)$ is adjusted so as to maximize the projection unto the chiral localized modes. Due to its small mean deviation ${\sigma }_x=3a$, which is of the order of the localization length of the central chiral mode, this wave packet projects unto this chiral mode with about $90\%$ efficiency, leading to a clear chiral motion along the central topological interface. (b) Same wave packet, but shifting its initial mean position away from the interface $x^0=x_R+20a$: this wave packet mainly projects onto the (nonchiral) delocalized states and diffuses into the bulk. (c) The wave packet is prepared at the center of the system, but with an opposite initial mean quasimomentum ${\mathit{k}}^0=(0,-2.1/a)$: as in (b), the wave packet projects onto delocalized states and diffuses into the bulk. (d) Initially preparing the wave packet on the other interface ($x=x_L$), with mean quasimomentum ${\mathit{k}}^0=(0,-2.1/a)$: this wave packet projects onto the other localized mode and undergoes the opposite chiral motion. In all cases, the system parameters are $t_{\text{NNN}}=0.4t_{\text{NN}}, {\delta }_{\text{max}}=20t_{\text{NN}}, L_x=100a$, and $L_y=150a$, and the trap parameters are $V_x=V_y=0.2t_{\text{NN}}$ and $L_{\text{obs}}=(L_y/2)\sqrt{V_y/t_{\text{NN}}}$. In all figures, time is expressed in units of $\hslash /t_{\text{NN}}$.

Figure 14

Center-of-mass evolution along the $y$ direction, $\langle y\rangle =\langle y\rangle \left(t\right)$, for the four situations [(a)–(d)] illustrated in Fig. 13. The cases (a) and (d) correspond to opposite chiral motion along the interfaces $x_R$ and $x_L$, respectively. The cases (b) and (c) illustrate nonchiral motion generated by the bulk (delocalized) states. Time is expressed in units of $\hslash /t_{\text{NN}}$.

Figure 15

Dynamics in the 2D honeycomb lattice for a large wave packet, of mean deviation ${\sigma }_x=20a$. The system parameters are the same as in Fig. 13, except $L_y=180a$. (a) The large wave packet is prepared on the central topological interface, with mean quasimomentum ${\mathit{k}}^0=(0,2.1/a)$, so as to maximize the projection unto the localized modes [as in Fig. 13]. However, due to its large width, this state only projects unto the localized mode with about $30\%$ efficiency, which leads to significant bulk noise in the background of the moving cloud [compare with Fig. 13]. (b) The time-reversal (TR) counterpart of the configuration depicted in (a), namely, when reversing the TR-breaking term $t_{\text{NNN}}\to -t_{\text{NNN}}$ and the mean quasimomentum of the initial state ${\mathit{k}}^0\to -{\mathit{k}}^0$. (c) The differential measurement, obtained by subtracting the TR-related situations in (a) and (b), allows one to significantly reduce the noise associated with the bulk contributions and, hence, highlight the motion of topological chiral modes. In all figures, time is expressed in units of $\hslash /t_{\text{NN}}$.

Figure 16

Wave-packet dynamics in the 2D honeycomb lattice for the radial-interface configuration [Eq. (8)], with an interface radius $R_{\text{inter}}=17a, t_{\text{NNN}}=0.4t_{\text{NN}}$, and a harmonic potential of strength $V_0=0.25t_{\text{NN}}$ [see also Eq. (10) and Fig. 12]. In all figures, time is expressed in units of $\hslash /t_{\text{NN}}$.

Figure 17

Wave-packet dynamics in the 2D honeycomb lattice for the same configuration as in Figs. 13 and 13, but including a disordered potential of strength $D=2t_{\text{NN}}$, and averaging over 10 realizations of disorder. (a) The edge mode localized on the central topological interface is immune to disorder, and freely propagates along the topological interface. (b) A wave packet associated with (nonchiral) bulk states is localized by the disorder [compare with Fig. 13].

Figure 18

Dynamics of a large wave packet, in the same configuration as in Fig. 15, but including a disordered potential of strength $D=2t_{\text{NN}}$, and averaging over 30 realizations. (a) The chiral motion of the localized mode is clearly identified in the presence of disorder, as the latter annihilates the propagation of the bulk states. (b) The differential measurement in the presence of disorder reveals clean edge-mode dynamics at the central topological interface, by removing the contribution of the (dephased) bulk states to the density [compare with Fig. 15].

Figure 19

Experimental implementation of the linear scheme. (a) Beam setup. The retroreflected laser beams $X_1$ and $Y_1$, with wavelength ${\lambda }_1$ interfere with each other at the position of the atomic cloud (red) and are phase stabilized. The additional beam $\overline{X}$ (blue) has a different wavelength $\overline{\lambda }$ and has the same transverse spatial mode as $X_1$. (b) Cut through the optical lattice potentials at $y=0$. The standing wave created by $\overline{X}$ (blue) has a lattice spacing of $\overline{\lambda }/2$, which is slightly more than half of ${\lambda }_1$, the $x$-direction spacing of the interference lattice created by $X_1$ and $Y_1$ (red). Therefore, their extrema only coincide at a particular point in the system. The resulting total potential (green) then has a spatially varying site offset ${\mathrm{\Delta }}_{\text{AB}}$. (c) Sketch of the resulting potential, with the tight-binding lattice structure (black) superimposed. Minima are dark green, maxima light green. The variation of ${\mathrm{\Delta }}_{\text{AB}}$ in the $x$ direction is exaggerated for better visibility.

Figure 20

Experimental implementation of the radial-symmetric scheme. (a) Beam setup. Lasers $X_1, Y_1$ (red), and $\overline{X}$ (dark red, dashed) are similar as in Fig. 20, except that now $\overline{\lambda }\approx {\lambda }_1$. In addition, a pair of phase-stabilized beams $X_2$ and $Y_2$ with a smaller beam waist are focused onto the center of the atomic cloud. (b) Laser beams $X_1$ and $Y_1$ together with $\overline{X}$ form an imbalanced honeycomb lattice potential, which is nearly uniform over the extent of the atomic cloud (dark blue minima, light blue maxima, tight-binding structure indicated by black lines). The minima of the additional potential (orange) created by $X_2$ and $Y_2$ are aligned with every second site of the honeycomb lattice. Their intensity decreases away from the center. (c) The resulting potential has a radially varying site offset (dark green minima, light green maxima). The spatial variation is exaggerated for better visibility.

Figure 21

Wave-packet dynamics in the time-reversal case $t_{\text{NNN}}=0$, and ${\delta }_{\text{max}}=20t_{\text{NN}}$. (a) By setting the mean position of the Gaussian wave packet on the interface ($x=x_R$), and the mean quasimomentum along $y$ to be $k_y^0\approx +2.1/a$, we find that the Gaussian wave packet essentially projects onto the “green” states with positive group velocity in Fig. 9: the wave packet goes up along the central interface ($x=L_x/2$) where these states are localized. (b) By setting the mean quasimomentum along $y$ to be opposite, $k_y^0\approx -2.1/a$, the wave packet essentially projects onto the states with negative group velocity, which are also localized at $x=L_x/2$: the wave packet goes down along the central interface. (c) Keeping $k_y^0\approx +2.1/a$, but now setting the mean position of the Gaussian wave packet away from the interface. We find that the wave packet mainly projects unto standard (2D) bulk states and, hence, undergoes a nonchiral motion in the plane. Time is expressed in units of $\hslash /t_{\text{NN}}$.