Effects of smooth boundaries on topological edge modes in optical lattices

Since the experimental realization of synthetic gauge fields for neutral atoms, the simulation of topologically nontrivial phases of matter with ultracold atoms has become a major focus of cold-atom experiments. However, several obvious differences exist between cold-atom and solid-state systems, for instance the small size of the atomic cloud and the smooth confining potential. In this article we show that sharp boundaries are not required to realize quantum Hall or quantum spin Hall physics in optical lattices and, on the contrary, that edge states in a smooth confinement exhibit additional interesting properties, such as spatially resolved splitting and merging of bulk bands and the emergence of robust auxiliary states in bulk gaps to preserve the topological quantum numbers. In addition, we numerically validate that these states are robust against disorder. Finally, we analyze possible detection methods, with a focus on Bragg spectroscopy, to demonstrate that the edge states can be detected and that Bragg spectroscopy can reveal how topological edge states are connected to the different bulk bands.

Figure 1

Integrated spectral function for a system described by Eq. (2) with flux $\alpha =1/6$ and stripe geometry with $100\times 100$ lattice sites [36]. Three experimentally relevant confinements of the form $V\left(x\right)=V_0{(x/L)}^{\delta }$ are shown: (a) hard-wall confinement, $\delta \to \infty$, (b) quartic confinement, $\delta =4$, and (c) harmonic confinement, $\delta =2$. The spectra in the upper row, ${\rho }_L(k_y,\omega )$, show the $k_y$ dependence along the periodic direction, integrated over the left half of the confinement direction (see text), and the real-space spectra of the lower row, $\tilde{\rho }(x,\omega )$, show the $x$ dependence along the confinement direction. To the right of the figure are the transport coefficients, calculated using (3) with the Fermi edge set to the corresponding dotted line. For hard-wall and quartic confinement there is an appreciable number of bulk bands and the edge states are clearly distinguishable, whereas within harmonic confinement we consider almost all of the states to be edge states. In each case we indicate left-edge states in red, while the remaining bulk states are shown in black, corresponding to the left half of the cylinder in real space (see lower plots). Two possible approaches exist for designation of edge and bulk states in softly confined systems. As seen in the upper row, energy regions with well-defined topological quantum numbers can be identified in the spectrum. The corresponding states can be designated as part of the edge, and the remaining ones as the bulk. Alternatively, we can define the edge as the point at which no states have energies within the range of energies covered by states at the center of the trap. We use the latter designation, although there is little difference between the two methods.

Figure 2

Integrated spectral function ${\rho }_L(k_y,\omega )$ for a system described by Eq. (2) with flux $\alpha =2/5$ and hard-wall (left) and quartic (right) confinement. Edge states are shown in red, while bulk states are shown in black.

Figure 3

False color diagram of the integrated spectral density ${\rho }_L(k_y,\omega )$ for a system described by Eq. (2) with flux $\alpha =1/6$, stripe geometry, and a confining potential $V\left(x\right)=V_0{(x/L)}^{\delta }$. Bulk bands are indicated in black and edge modes as colored curves, also marked as (a), (b), (c), (d). There exist several true crossings and avoided crossings in the spectra which combine to preserve the topological invariants for any confinement exponent $\delta$. Auxiliary states of the corresponding edge modes are shown with dashed curves. The auxiliary states do not influence the topological phases of the system, since they always come in pairs with opposing velocities.

Figure 4

Integrated spectral density $\tilde{\rho }(x,\omega )$ for a system described by Eq. (2) on a $100\times 100$ lattice with quartic confinement $V$. In the left figure, the flux is chosen to be $\alpha =2/5$, while in the right figure $\alpha =1/6$. In the left figure, the edge states leaving the second bulk band immediately merge with the edge states of the first bulk band to form a state spatially localized between the bulk bands with the correct transport coefficient. In the right figure, the states leaving the second bulk band split up, i.e., they localize to more than one point in space. The inner part of these states merges with the edge states of the third band at higher energies. Similar behavior is observed for the third, fourth, and fifth bands. As pointed out in the text, this nontrivial behavior is an indication of the topological origin of the edge states.

Figure 5

Integrated partial spectral density ${\rho }_L(k_x,\omega )$ of a system described by Eq. (2) on a $120\times 60$ lattice with $\alpha =1/6$ [36]. Right: with an additional binary disordered potential, given by Eq. (7) and ${\Delta }_{\max }=0.5t$. Left: with the disordered potential being set to zero. The robust edge states are still clearly pronounced and gapless, while the former bulk bands are smeared out and show a mobility gap (not shown here but obtainable from the Anderson-localized bulk eigenstates).

Figure 6

Wave function ${\left|\Psi (x,y)\right|}^2$ as a function of the lattice spacing $a$ of different eigenstates of the system within complete quartic confinement. The real-space spectral density shown to the left is a cross section of the complete system: $\rho (x,y=0,\omega )$. Three particular eigenstates have been shown, and their energies indicated by arrows to the spectral density. The wave functions belong to (a) a bulk region and (b) and (c) to a pair of edge states splitting up after leaving the second bulk band.

Figure 7

As in Fig. 6 but with complete harmonic confinement. The three states are shown from (a) a bulk region and (b) and (c) edge states of states belonging to different bulk bands. Spatial coordinates in units of thelattice spacing $a$.

Figure 8

Dynamical structure factor for a system with quartic confinement and Fermi energy in the first bulk gap (${\epsilon }_F=-2t$). Left: $S(q,\omega )$ for fixed momentum $q$ as a function of $\omega$. The first peak belongs to $\text{edge}\to \text{edge}$ scattering and its position is sensitive to $q$ and can be written, for small $\omega$, as ${\omega }_q=v_{F,\mathrm{edge}}q$. The second and third peaks belong to $\text{edge}\to \text{bulk}$ scattering from edge states into the third and fourth bulk bands, located around $\epsilon =0$, and to $\text{bulk}\to \text{bulk}$ scattering from the first to the second bulk band, where the frequency is independent of $q$. No signal appears of scattering from edge states to the second bulk band, located at $\omega =0.5t$, indicating a disconnection between these states, i.e., these states have vanishing matrix elements of the Bragg operator. Right: $S(q,\omega )$ for fixed frequency as a function of momentum transfer $q$ for (a) $\text{edge}\to \text{edge}$ scattering at $\omega =0.2t$, and (b) $\text{bulk}\to \text{bulk}$ scattering processes at $\omega =1.5t$.

Figure 9

$S(q,\omega )$ for a fixed frequency $\omega =0.2t$ as a function of momentum transfer $q$, for a Fermi energy ${\epsilon }_F=-2t$. Left: $S(q,\omega )$ for the quartic confined system. The system shows a strong response when one component of $q=(q_x,q_y)$ has an absolute value $|q_{x,y}|=q_0=\omega /v_F$ because excitations along the $x$ and $y$ axes are most favorable (see Fig. 6). Right: $S(q,\omega )$ for the harmonically confined system. Here, the response is close to circularly symmetric in $q$ space, reflecting the shape of the eigenstates. No particular direction is favorable anymore, as long as the absolute value of $\left|q\right|=q_0$ is fixed.

Figure 10

Dynamical structure factor $S(q,\omega )$ for fixed momentum $q$ as a function of frequency, for a Fermi energy ${\epsilon }_F=-2t$, as seen in Fig. 8 but with artificially suppressed $\text{bulk}\to \text{bulk}$ scattering processes. The first peak belongs to edge-to-edge scattering processes and is sensitive to the momentum transfer $q$ with approximate frequency ${\omega }_q=v_{F,\mathrm{edge}}q$. The broadened peaks around $\omega =1.5t$ belong to $\text{edge}\to \text{bulk}$ scattering to the third and fourth bulk bands. It is clearly visible that there is no scattering from the edge states to the second bulk band, which is located at $\omega =0.5t$.