Anomalous Hall effects of light and chiral edge modes on the Kagomé lattice

We theoretically investigate a photonic Kagomé lattice which can be realized in microwave cavity arrays using current technology. The Kagomé lattice exhibits an exotic band structure with three bands, one of which can be made completely flat. The presence of artificial gauge fields makes it possible to emulate topological phases and induce chiral edge modes which can coexist inside the energy gap with the flat band that is topologically trivial. By tuning the artificial fluxes or in the presence of disorder, the flat band can also acquire a bandwidth in energy allowing the coexistence between chiral edge modes and bulk extended states; in this case the chiral modes become fragile towards scattering into the bulk. The photonic system then exhibits equivalents of both a quantum Hall effect without Landau levels and an anomalous Hall effect characterized by a nonquantized Chern number. We discuss experimental observables such as local density of states and edge currents. We show how synthetic uniform magnetic fields can be engineered, which allows an experimental probe of Landau levels in the photonic Kagomé lattice. We then draw on semiclassical Boltzmann equations for transport to devise a method to measure Berry's phases around loops in the Brillouin zone. The method is based solely on wave-packet interference and can be used to determine band Chern numbers or the photonic equivalent of the anomalous Hall response. We demonstrate the robustness of these measurements towards on-site and gauge-field disorder. We also show the stability of the anomalous quantum Hall phase for nonlinear cavities and for (artificial) atom-photon interactions.

Figure 1

The photonic Kagomé lattice with three sites per unit cell $A,B,C$. Artificial fluxes can be threaded through the triangular plaquettes [21]. (Top left) Lattice periodic in the $x$ direction, with “line” boundaries composed of $A,B$ sites only. The blue parallelogram encloses a one-dimensional superunit cell which generates the entire lattice. (Top right) A projection of the Brillouin zone onto the $k_x\equiv k_1$ direction. (Bottom left) At $\phi =\pi /6$ the system exhibits a flat, topologically trivial middle band, while the lower and upper bands are characterized by finite Chern numbers. (Bottom right) The middle band becomes dispersive if one deviates from $\pi /6$, for example, for $\phi =\pi /4$. Here, we have considered a cylinder geometry which allows for two chiral edge modes: right-moving at the lower edge (purple), left-moving at the upper edge (orange); corresponding chiral currents depicted on the lattice. As discussed in Secs. 4b  and 4c, the chiral edge modes can also be detected in the square geometry [22].

Figure 2

Two-dimensional phase diagram of a particle (photon) with energy $E$ on the Kagomé lattice defined with hopping phase $\phi$. By tuning $(E,\phi )$ the photon system is either characterized by extended bulk states (“bulk band”) or by a “band gap” where the system exhibits unidirectional edge modes but no bulk states. The $F$ points represent states in the flat band (the blue points at $E-\hslash \omega =0$ correspond to a middle flat band separated by finite energy gaps from the lower and upper bands; black points at $E-\hslash \omega =\pm 2\left|t\right|$ correspond to flat upper or lower band). The $D$ points are degeneracy points between pairs of bands at the Dirac points $K^{\pm }$. The dashed lines are explained in the main text.

Figure 3

Effect of different boundary conditions on the dispersion of edge modes in the proximity of the dispersive middle band for a system with $\phi =\frac{\pi }{4}$: left, two “line” boundaries; middle, one line edge, the other “armchair”; right, both boundaries are armchair. The bottom panels exhibit the minimal ladder lattices that exhibit the three boundary conditions. All three ladders shown support edge modes.

Figure 4

The quantities ${\lambda }_{+}^2$ (solid line) and ${\lambda }_{-}^2$ (dashed line) defined in Eqs. (20) describe how the wave function of the edge mode decays into the bulk. The reference line is at ${\lambda }_{\pm }^2=1$. For $|{\lambda }_{\pm }|>1$ the respective branch $E_{\pm }\left(k_1\right)$ is located at the bottom edge. Conversely, if $|{\lambda }_{\pm }|<1$, it is located at the top edge. This results in the color scheme for the edge modes on Fig. 1.

Figure 5

Local density of states. States are localized in the hexagonal plaquettes when the energy overlaps with the flat band at $\phi =\frac{\pi }{6}$ (a); for the same system, intragap states are confined at the edges of the sample (b); dispersive middle band for $\phi =\frac{\pi }{4}$ contains extended states (c); for comparison, local density of states for the disordered $\phi =\frac{\pi }{6}$ system is shown in (d) for an energy $E$ within the proximity of the middle band.

Figure 6

Plots of the expectation value of the lattice current operator in two implementations of the $\phi =\frac{\pi }{4}$ system. (Left) The system has a dispersive middle band overlapping in energy with the edge mode (see Fig. 3). In the presence of a $\delta$-function impurity the edge state current (red) will deviate around the impurity, maintaining its chirality, if the energy of the edge state lies in the gap and does not overlap with the middle band. This corresponds to the anomalous quantum Hall phase. As opposed to this, the current (green) for a state whose energy is within the overlap region with the middle band is not chiral and has non-zero expectation value in the bulk. (Right) In the presence of a dispersive middle band (with $\phi =\frac{\pi }{4}$) and disorder, the edge state can leak into the bulk with a finite probability.

Figure 7

Constant energy trajectories in momentum space, overlayed on the constant energy contours of the lowest band at $\phi =\frac{\pi }{6}$. Trajectories can be closed within the first Brillouin zone resulting in closed real-space trajectories or, if the initial momentum lies on the separatrix lines (see, for example, the dashed lines), then the trajectory in momentum space can only close in the extended Brillouin zone, and in real space the particle traces a straight line.

Figure 8

Low-magnetic-field ${\mathcal{B}}_s$ spectrum of the Kagomé lattice with $\phi =\frac{\pi }{4}$. At 0 external field, we recover the original bandstructure of Fig. 1. At flux $p/q$ there are $3q$ subbands with possible degeneracies. For example, at $p/q=1/5$, the density of states reveals that there are six separated subbands in the upper band, five degenerate subbands in the middle band, and four separated subbands in the lower band, in agreement with Eq. (44) and consistent with the fact that the lower and upper Bloch bands have Chern numbers $-$1 and 1, respectively. The bands become degenerate (no energy gaps in the spectrum) at $p/q=1/4$. See text for details.

Figure 9

(a) Dispersive middle band for $\phi =\frac{\pi }{4}$. The Chern number, computed with the formula of Eq. (50) gives a nonquantized value in the overlap region which can be measured directly in an interference experiment. (b) Chern number calculation for the flat-band system $\phi =\frac{\pi }{6}$ with increasing amplitude of phase disorder: $W_{\phi }=0,\frac{\pi }{6}\times \frac{2}{5},\frac{\pi }{6},\frac{\pi }{6}\times \frac{9}{5}$. (c) Flat-band system with increasing amplitude of on-site disorder $W_{\mathrm{site}}=0,\frac{\left|t\right|}{2},\left|t\right|,\frac{3\left|t\right|}{2}$. The disorder potential has no spatial correlation and is drawn from a uniform distribution (see text).

Figure 10

(Left) Deviation of the effective quasiparticle dispersion ${\xi }_n\left(\mathbf{k}\right)$ (black line) from the free-particle dispersion $E_n\left(\mathbf{k}\right)$ (dashed red line) for the $\phi =\frac{\pi }{6}$ system with $U/\left|t\right|=0.13$ (the plot is along the line $k_x=0$). (Right) Low-energy quasiparticle spectrum exhibits linear dispersion (magenta) at the $\Gamma$ point $\mathbf{k}=0$; plotted for comparison is the free-particle dispersion (light gray).