Figure 1

(a) Two-dimensional (2D) hexagonal lattice for the Haldane tight-binding model, consisting of real nearest-neighbor hoppings (black lines) and complex direction-dependent next-nearest-neighbor hoppings (red and blue or gray lines). The imaginary components of the hopping matrix elements are generated by an effective vector potential that produces a “magnetic” field with zero total magnetic flux through the unit cell [i.e., the magnetic fluxes through the white and yellow (light gray) triangles have equal magnitudes and opposite signs]. (b) A three-dimensional (3D) realization of the model obtained by translating one sublattice (the blue spheres) along the direction perpendicular to the plane ($z$ direction). Staking such layers in the $z$ direction with the $A$ sublattice sites on top of $B$ sites generates a 3D generalization of the Haldane model on a diamond lattice. Alternatively, we can treat the model as a quasi-2D lattice of triangular pyramids. Neglecting the hopping between the apex sites (blue lines) does not change the topological properties of the model. Pyramids with a different base will generate similar models with nontrivial topological properties.Reuse & Permissions

Figure 2

(a) Quasi-2D model of a topological insulator with broken time-reversal symmetry on a lattice of square pyramids. Expanding the structure in the $z$ direction will generate 3D models of topological insulators. Alternatively, one can move the apex site into the base plane and generate a 2D square superlattice model. (b) Topological-insulator model on a 2D square superlattice. Instead of extra apex sites, as in (a), we consider different next-nearest-neighbor hoppings $t_2^{\prime }\ne t_2$. A vector potential $\mathbf{A}(\mathbf{r})$ produces an effective “magnetic” field with opposite flux through the yellow (light gray) and white squares and generates a complex direction-dependent nearest-neighbor hopping $t_1$. Hoppings with a given sign of the phase are marked by arrows.Reuse & Permissions

Figure 3

(Left) Optical superlattice potential corresponding to $V_1=3.4E_r$, $V_2=1.7E_r$, and $\alpha =2\hslash /a$ (see main text). A unit cell consisting of two squares with side length $a$ is marked with light blue (light gray) lines. Notice the $\pi /2$ rotation of the axes relative to lattice in Fig. 2b. (Right) Effective “magnetic” field generated by the vector potential $\mathbf{A}(\mathbf{r})$. The total flux through the unit cell is zero.Reuse & Permissions

Figure 4

Spectrum of the square superlattice model with periodic boundary conditions (no boundaries). Only the lowest two bands are shown, although four other bands were considered in the calculation. The wave vector takes values in the first Brillouin zone, $0⩽k_x⩽2\pi /\sqrt{2}a$, $0⩽k_y⩽2\pi /\sqrt{2}a$, and we make the choice of length units $a=1/\sqrt{2}$. (a) If $V_2=0$ (no superlattice structure), the two bands are degenerate at $\mathbf{k}=(\pi ,\pi )$. (b) If $\alpha =0$ (no vector potential), the gap closes at two Dirac points $(0,\pi )$ and $(\pi ,0)$. (c) For $V_1=3.4E_r$, $V_2=1.7E_r$ and $\alpha =2\hslash /a$ a full gap opens. The bands are shown along the $k_x$ direction (with $k_y$ out of the plane).Reuse & Permissions

Figure 5

Energy dispersion for the first two bands of the square superlattice model along the $(0,0)\to (\pi ,\pi )\to (\pi ,0)\to (0,0)$ path in the Brillouin zone. The full lines correspond to the parameters from Fig. 4c. The dashed lines show the energy dispersion obtained if we neglect the hybridization with higher-energy bands.Reuse & Permissions

Figure 6

Momentum dependence of the Berry curvature of the lowest-energy bands for the same parameters as in Fig. 4c. The integral of the Berry curvature over the first Brillouin zone (i.e., the total flux) is $2\pi$ for the lowest-energy band (a) and $-2\pi$ for the second band (b), corresponding to the Chern numbers $1$ and $-1$, respectively. The nonvanishing Chern numbers reveal the nontrivial topological properties of the system.Reuse & Permissions

Figure 7

The band structure of Eq. (3) in the stripe geometry. The corresponding lattice is schematically shown in the inset. Notice the edge state modes that populate the gap and merge with the bulk states. The edge modes cross the gap connecting the lower and upper bands and intersect at $k_x=0$ due to Kramers degeneracy. Small perturbations can modify the dispersion of the edge modes and the location of the Kramers degeneracy points but cannot open a gap for the edge states. Notice that the bulk bands can be obtained by projecting the two-dimensional spectrum shown in Fig. 4c on a plane perpendicular to the $y$ axis. The same set of parameters as in Fig. 4c was used.Reuse & Permissions

Figure 8

Band structure for a stripe with inequivalent edges (see main text). All the sites on one boundary belong to the $A$ sublattice, while all the sites on the other boundary are $B$ type. For the top panel the parameters are the same as in Fig. 7, the only difference being an extra line of lattice sites at one of the boundaries. Note the completely different dispersion of the edge modes and the different location of the degeneracy point. The symmetry between left-moving and right-moving states is broken. A mirror image of the dispersion lines can be obtained by adding the extra line of lattice sites to the opposite edge or by reversing the direction of the vector potential, $\alpha \to -\alpha$ (lower panel). A topologically trivial edge mode develops at the top of the second band (state E belongs to this mode).Reuse & Permissions

Figure 9

Spatial dependence of the “amplitude” function $\left|{\Psi }_{k_x,\nu }\right|{}^2={\sum }_{(n,m)}|{\mathrm{\Psi ̃}}_{k_x,\nu }^{(n,m)}(y_j)|{}^2$ for the states marked by the letters A, B, and C in Fig. 7. The stripe has a width $d=252$ (in units of $a=1/\sqrt{2}$). State A, which is well inside the bulk gap, is localized near the upper edge and decays exponentially with a characteristic length scale of a few lattice constants. State B, which is at the gap edge, is a bulk state with a smoothly varying envelope function. Notice the multiplication factor of 30 introduced to make the function visible on the same scale as the edge states. State C belongs to the edge mode but is very close to the gap edge. It is localized near the bottom edge and decays exponentially but has a length scale significantly larger than state A.Reuse & Permissions

Figure 10

Spatial dependence of the “amplitude” function $\left|{\Psi }_{k_x,\nu }\right|{}^2={\sum }_{(n,m)}|{\mathrm{\Psi ̃}}_{k_x,\nu }^{(n,m)}(y_j)|{}^2$ for state E and for the sequence of states $D_1\to D_4$ from Fig. 8. $D_1$ is positioned in the middle of the gap, while $D_4$ is at the gap edge and has bulk character. During this transition the characteristic length scale for the exponential decay of the edge states increases continuously from a value of a few lattice sites to a value comparable to the width of the stripe. The state E has a very pronounced edge character but is not topologically protected. Note that the horizontal axis was translated for clarity.Reuse & Permissions

Figure 11

(Upper panel) Average position versus energy for the single-particle states that are solutions of Eq. (3) in the stripe geometry (with equivalent edges). The system is characterized by the parameters $V_1=3.4E_r$, $V_2=1.7E_r$, and $\alpha =2\hslash /a$ and has a width $d=100(\sqrt{2}a)$. The bulk states are characterized by $\langle y\rangle {}_{n{k}_x}\approx d/2$, while the edge states have $\langle y\rangle {}_{n{k}_x}\approx 0$ or $\langle y\rangle {}_{n{k}_x}\approx d$, corresponding to the positions of the two edges. (Lower panel) Average orbital momentum (in arbitrary units) versus energy for the same system. Notice the chiral nature of the topological edge states and the extra edge modes located in the upper band. The spectrum for this system is shown in Fig. 7 and the density of states (DOS) is shown in the right panels.Reuse & Permissions

Figure 12

Average position (upper panel) and average orbital momentum (lower panel) versus energy for the single-particle states that are solutions of Eq. (3) in the stripe geometry (inequivalent edges). The system is characterized by $V_1=3.4E_r$, $V_2=1.7E_r$, and $\alpha =2\hslash /a$ and has a width $d=101(\sqrt{2}a)$. The spectrum for this system is shown in Fig. 8.Reuse & Permissions

Figure 13

Density of states (a) and spectrum (b) showing the $s$ bands of a stripe with inequivalent edges calculated within a simplified tight-binding approximation that neglects hybridization with higher bands. For comparison, the density of states calculated within a three-orbital tight-binding approximation is also shown. The corresponding spectrum is given in Fig. 8 (top panel). Note that the topology of the edge modes is not altered, in spite of a significant redistribution in the density of states.Reuse & Permissions

Figure 14

Diagram showing the average radial position versus energy for the single-particle states of a system described by Eq. (3) in a confining potential (12). The parameters of the system are $V_1=3.4E_r$, $V_2=1.7E_r$, and $\alpha =2\hslash /a$. The underlying lattice (i.e., the minima of the effective optical lattice potential) is shown in the inset. The edge states are characterized by values of $\langle r\rangle {}_n$ comparable with $R_0=29a$, the disk radius, as a result of their localization in the vicinity of the boundary. The clearly defined edge mode crosses the bulk gap and connects the lower and upper bands, thus revealing its topological nature. The density of states (right) is practically identical with that shown in Fig. 13a for a similar system with stripe geometry.Reuse & Permissions

Figure 15

“Amplitude” functions for single-particle states in the disk geometry. The top and middle panels show “densities” corresponding to the first four energy levels starting with the ground state (top left corner). We note that the actual density is obtained by multiplying these envelope functions with a sum of $s$-type orbitals centered at each lattice site. The lower panels show a typical edge state (left) and a typical bulk state (right). The edge state decays exponentially away from the boundary, with a characteristic length scale of a few lattice constants.Reuse & Permissions

Figure 16

Energy dispersion for the first six bands of the square superlattice model along the $(0,0)\to (\pi ,\pi )\to (\pi ,0)\to (0,0)$ path in the Brillouin zone. The parameters used in the calculation are $V_1=3.4E_r$, $V_2=1.7E_r$, and $\alpha =2\hslash /a$. The $s$ bands (red lines) are identical with those shown in Fig. 5.Reuse & Permissions

Figure 17

The band structure of the Hamiltonian given by Eq. (3) in the stripe geometry. The left panel corresponds to the parameters $V_1=3.4E_r$, $V_2=1.7E_r$, and $\alpha =2\hslash /a$, while the right panel corresponds to a slightly deeper lattice with $V_1=3.7E_r$. Note that there are no edge states between the second and the third bands, but an edge mode is clearly visible between the third and the fourth bands. Increasing $V_1$ opens a gap in the spectrum (right panel) and reveals the topological nature of that edge mode. By contrast, the edge states that populate the gap between the fourth and the fifth bands are topologically trivial, as they do not connect the two bands.Reuse & Permissions

Figure 18

The $s$-band gap as function of the vector potential strength for several values of the superlattice generating component $V_2$. The main amplitude of the optical lattice potential is fixed, $V_1=3.4E_r$. Notice that each curve has a singularity point and that the slope to the left (right) of the singularity point is positive (negative). The positive slope correspond to an indirect gap between the maximum of the lower band at $(\pi ,\pi )$ and the minima of the upper band at $(0,\pi )/(\pi ,0$), while the negative slope corresponds to a direct gap at ($\pi ,\pi )$. Note that for weak vector potentials with the strength $\alpha$ below a critical value ${\alpha }^{*}(V_2)$ the indirect gap becomes negative.Reuse & Permissions

Figure 19

Band structure for a system with stripe geometry described by the parameters $V_1=3.4E_r$, $V_2=1.7E_r$, $\alpha =2\hslash /a$ and different values of the staggered potential $\Gamma$ (given in units of $E_r$). $\Gamma =0$ corresponds to the case shown in Fig. 7. Applying a small staggered potential reduces the gap in the vicinity of $k_x=0$ (top right panel). At the critical value ${\Gamma }_c\approx 0.075E_r$ the gap closes at $k_x=0$, while for $\Gamma >{\Gamma }_c$ a full gap opens again. At large values of $\Gamma$ the system is a standard band insulator with no edge modes inside the gap. Notice that the spectra are shown for half of the one-dimensional Brillouin zone while the other half can be obtained by mirror symmetry with respect to $k_x=\pi$.Reuse & Permissions

Figure 20

Band structure for a stripe with inequivalent edges and with the same parameters as in Fig. 19. For $\Gamma =0.12E_r$ the system is a standard band insulator but, in contrast with the equivalent edge case (see Fig. 19), a well-defined edge mode populates the gap. A clean insulator with the chemical potential inside the bulk gap supports gapless edge excitations. However, any small perturbation (such as disorder or interactions) will open a gap in the edge mode.Reuse & Permissions

Figure 21

Energy dispersion curves for a system with $V_1=3.7E_r$, $\alpha =2\hslash /a$, and different values of $V_2$ along the $(0,0)\to (\pi ,\pi )\to (\pi ,0)\to (0,0)$ path in the Brillouin zone. The numbers inside the yellow circles represent the Chern numbers of the bands. Note the closing of the direct gap at certain critical values of $V_2$ (panels b and d) and the corresponding change of the Chern numbers. The spectra shown in panels a, c, and e are consistent with different insulating states: (a) topological insulator (with the first three bands filled) and conventional insulator (four bands filled); (c) topological insulator (one band filled) and conventional insulator (four bands filled); and (e) topological insulator I (one band filled), conventional insulator (two bands filled), and topological insulator II (four bands filled).Reuse & Permissions

Figure 22

Detail of the band structure for a system in the stripe geometry with the same parameters as in Fig. 21a (left) and 21e (right). In the left panel notice one pair of topological edge modes inside the gap above the third band and the topologically trivial edge states above the fourth band, consistent with the Chern numbers shown in Fig. 21a. On the right we observe four pairs of edge modes inside the gap above the fourth band. This number of pairs of edge modes is equal to the total Chern number of the bands below the gap [see Fig. 21e].Reuse & Permissions

Figure 23

(Upper panel) Spectrum for the square superlattice model (1) with parameters $t_1=(1+0.2i)E_r$, $t_2=-0.3E_r$, and $t_2^{\prime }=-0.06E_r$ in the stripe geometry. The system is finite in the $y$ direction and infinite in the $x$ direction, with the axes are oriented along the next-nearest-neighbor directions (see Fig. 3). For an odd number of layers (i.e., equivalent edges) the edge states (red/dark gray lines) with $k_x<\pi$ are localized near $y=0$, while the edge states with $k_x>\pi$ are localized near the opposite boundary. For an even number of layers (i.e., inequivalent edges) the mode localized near $y=0$ remains unchanged, while the other mode is now characterized by $k_x<\pi$ (green/light gray line). (Lower panel) Edge state amplitudes multiplied by exponential factors to reveal the oscillatory behavior. Blue (dark gray): edge state corresponding to $k_x=0.97\pi /\sqrt{2}a$ and $\xi =1.441\sqrt{2}a$. Orange (light gray): edge state with $k_x=0.7\pi /\sqrt{2}a$ and $\xi =1.351\sqrt{2}a$.Reuse & Permissions

Figure 24

(Upper panel) Spectrum for the square superlattice model (1) in the stripe geometry for the same parameters as in Fig. 23 but with the symmetry axes oriented along the nearest-neighbor directions. The dispersion of the edge modes is independent of the parity of the number of layers. (Lower panel) Edge state amplitude multiplied by an exponential factor with $\xi =2.396a$ for the state with $k_x=0.97\pi /a$. Note the even-odd oscillations and the $k$-dependent long wavelength oscillatory component. At $k_x=\pi /a$ the extra oscillatory component is absent.Reuse & Permissions

Figure 25

Opening of a gap in the edge state spectrum due to finite size effects. The system is a thin stripe with the same geometry and hopping parameters as in Fig. 24. The size of the gap is determined by the overlap of edge states localized on opposite boundaries and, consequently, varies roughly exponentially with the width of the stripe and inherits the oscillatory behavior of the edge states. For large $N$ the edge states are gapless and independent of parity. For thin films, a finite gap opens in systems with even number of layers, while systems with odd number of layers remain gapless. Note that the “bulk” spectrum varies dramatically with the system size, in contrast with the edge modes that are practically unaffected if the system size is much larger than their characteristic length scale.Reuse & Permissions

Figure 26

Gap dependence on the stripe thickness ($w$) for a system with $t_1=(1+0.05i)E_r$, $t_2=-0.05E_r$, and $t_2^{\prime }=0.05E_r$ and the symmetry axes oriented along the nearest-neighbor directions. (Upper panel) Edge mode dispersions for stripes with $w=(N-1)a$, where $N$ is the number of layers. The gap at $k_x=\pi$ is almost zero for $N=48$, $0.1E_r$ for $N=24$, and $0.36E_r$ for $N=12$. Edge states with $k_x\ne \pi$ have extra oscillatory components (similar to that shown in the lower panel of Fig. 24) that may generate a vanishing overlap between edge states located on opposite edges. Consequently, for $N<44$ the gap minima shift away from $k_x\ne \pi$. (Lower panel) Dependence of the minimum gap amplitude on the stripe thickness. Note the even-odd oscillations. The long period extra oscillatory components of the edge states (see Fig. 24) determine a deviation from an exponential dependence of the gap amplitude on the stripe thickness for $N<44$ in stripes with even number of layers (red/smooth line).Reuse & Permissions

Figure 27

In-gap states localized along a line defect in a system with no boundaries described by Eq. (1) with $t_1=(1+0.3i)E_r$, $t_2=-0.3E_r$, and $t_2^{\prime }=0.3E_r$. The one-dimensional defect is oriented along the nearest-neighbor direction and is characterized by (see main text): $\delta t=0.5E_r$ and no lattice mismatch (green or light gray lines), $\delta t=0.5E_r$ and a lattice mismatch (red lines and circles), $\delta t=\mathrm{Re}[t_1]=E_r$ and a lattice mismatch (black lines and triangles). In thin stripes the edge states can overlap with states localized along extended defects and increase the size of the finite size induced gap.Reuse & Permissions

Figure 28

Spectrum for a system with a soft boundary described by Eq. (1) with $t_1=(1+0.3i)E_r$, $t_2=-0.3E_r$, and $t_2^{\prime }=0.3E_r$ in the stripe geometry. The stripe is oriented along the nearest-neighbor directions and is finite along the $y$ direction. The confinement at $y\approx 0$ is given by the exponential potential $V_c(y)=5.0\exp [-y/(2a)]$, while the confinement at the opposite boundary is given by an infinite potential wall. The dispersion of the edge mode near the soft boundary (orange or light gray lines inside the gap) differs significantly from the dispersion of the hard boundary edge mode (red or dark gray line inside the gap) but retains the fundamental property of a topological-insulator edge mode, i.e., it connects the two bulk bands (note that $k_x=2\pi$ has to be identified with $k_x=0$). Notice the topologically trivial edge modes that are generated at high energies (light green or light gray lines).Reuse & Permissions

Figure 29

Detail of the band structure shown in Fig. 28. For comparison the dispersion of an edge mode corresponding to a softer boundary, $V_c(y)=5.0\exp [-y/(6.5a)]$, is also shown (black lines).Reuse & Permissions

Figure 30

Amplitudes for the edge states localized near a soft boundary marked in Fig. 29. The corresponding confining potential, $V_c(y)$ is also shown. For comparison we also show $|{\Psi }_E(W-y)|{}^2$ for the state E which is localized near the hard boundary.Reuse & Permissions

Figure 31

Average position (left panel) and average angular momentum (right panel) for a stripe with a soft boundary and same parameters as in Fig. 28.Reuse & Permissions

Figure 32

Spectrum of a system with harmonic confining potential. Panel (a) corresponds to a square superlattice topological “insulator” model with the same parameters as in Fig. 28, while panel (b) represents a standard-insulator model on the square superlattice with $t_1=E_r$, $t_2=-0.3E_r$, $t_2^{\prime }=0.3E_r$ and a staggered on-site potential $\pm \Gamma$ with $\Gamma =E_r$. The lower panels show the low-energy spectra that characterize the inhomogeneous topological (left) and normal (right) metals. Note that the topological metal has a spectrum consisting of two modes that are qualitatively the same as the soft mode shown in Fig. 29 (in the present case both boundaries are soft). By contrast, the spectrum of the normal metal contains many disconnected modes.Reuse & Permissions

Figure 33

Density profiles for a fermionic system with harmonic confinement and different values of the chemical potential. Panel (a) corresponds to a topological-insulator model, while panel (b) shows the density of a standard insulator. The parameters for the models are the same as in Fig. 32. Note the plateaus characterized by $\rho =1$ (particles per lattice site), which correspond to the insulating phase. No qualitative difference can be observed between the two types of models.Reuse & Permissions

Figure 34

Dynamical structure factor for a system described by Eq. (1) with $t_1=(1+0.3i)E_r$, $t_2=-0.3E_r$, and $t_2^{\prime }=0.3E_r$. The blue (dark gray) line corresponds to a sharp boundary, while the red (gray) line is for a soft boundary with the same profile as in Fig. 28. The extra in-gap contribution represents transitions between different branches of the soft edge mode. The green (light gray) line corresponds to the same parameters as the red curve and opposite scattering wave vector $-\mathbf{q}$. Note the absence of the low-energy peak. The inset is a schematic illustration of the intersecting laser beams illuminating a portion of the system.Reuse & Permissions