Complete absorption of topologically protected waves

Chiral edge states can transmit energy along imperfect interfaces in a topologically robust and unidirectional manner when protected by bulk-boundary correspondence. However, in continuum systems, the number of states at an interface can depend on boundary conditions. Here we design interfaces that host a net flux of the number of modes into a region, trapping incoming energy. As a realization, we present a model system of two topological fluids composed of counter-spinning particles, which are separated by a boundary that transitions from a fluid-fluid interface into a no-slip wall. In these fluids, chiral edge states disappear, which implies non-Hermiticity and leads to an interplay between topology and energy dissipation. Solving the fluid equations of motion, we find explicit expressions for the disappearing modes. We then conclude that energy dissipation is sped up by mode trapping. Instead of making efficient waveguides, our paper shows how topology can be exploited for applications towards acoustic absorption, shielding, and soundproofing.

Figure 1

Completely absorbed interface mode in a chiral active fluid. (a) Schematic of a two-dimensional system of two chiral active fluids composed of counter-rotating particles. Each particle spins with rotational speed $\pm {\mathrm{\Omega }}_A$ and is subject to a Coriolis force density $\pm {\mathrm{\Omega }}_B{\mathit{v}}^{*}$, where ${\mathit{v}}^{*}$ is the velocity $\mathit{v}$ rotated by ${90}^{\circ }$. The bulk is gapped with topological invariant $C=\mathrm{sgn}\left({\mathrm{\Omega }}_A\right)+\mathrm{sgn}\left({\mathrm{\Omega }}_B\right)$. The interface between the two fluids supports right-moving edge modes, one of which is shown in (b)–(d). (b) The right-moving pulse along the fluid-fluid interface (dotted line). (c) The pulse hits the no-slip wall (solid line). (d) The pulse is completely absorbed. Density differences $\delta \rho$ around an equilibrium density ${\rho }_0$ are shown in color, and the velocity field is portrayed by white arrows. Time $\tilde{t}=t/T$ and space $\tilde{x}=x/L$ are nondimensionalized by $L\equiv |{\nu }^o|/c$ and $T\equiv |{\nu }^o|/c^2$, using the odd viscosity parameter ${\nu }^o\sim {\mathrm{\Omega }}_A$ (see text). The incoming wave has parameters ${\mathrm{\Omega }}_{B}T=1/5$ and $\omega T=qL=\pi /30$.

Figure 2

Mode absorption and non-Hermiticity. In a region of space (gray circle), we count the number of in- and out-flowing modes (black arrows) and apply the pigeonhole principle. Insets in (a)–(d) show different bulk (green) and interface (red) dispersion relations. (a) Ungapped time-reversal-symmetric system. Every in-flowing mode is paired with a mode flowing out by time-reversal symmetry (TRS). The net mode flux $\mathrm{\Phi }$ is zero. (b–d) Gapped systems, for which $\mathrm{\Phi }={\mathrm{\Phi }}_L+{\mathrm{\Phi }}_R$, where ${\mathrm{\Phi }}_{L/R}$ counts modes at the two points where the interface (dotted/solid lines) crosses the region's boundary. (b) Gapped system with TRS. ${\mathrm{\Phi }}_L={\mathrm{\Phi }}_R=0$, even if BCs change the number of interface modes. (c) System for which TRS is broken, but bulk-interface correspondence holds. Topological protection guarantees one out-flowing mode for each in-flowing mode, even along complex interfaces, ${\mathrm{\Phi }}_L+{\mathrm{\Phi }}_R=0$. (d) When both TRS and bulk-interface correspondence are broken, the in-flowing modes can outnumber the out-flowing modes, so that $\mathrm{\Phi }>0$ and non-Hermiticity is guaranteed.

Figure 3

Interface solutions for fluids with odd viscosity and no dissipation. (a), (b) Mode spectrum for two different interfaces between chiral active fluids [Fig. 1], showing the dispersion relations for bulk modes in green and for interface modes in red ($m=1/5$). (a) For a fluid-fluid interface, any frequency $\omega$ in the gap corresponds to four interface modes (blue points): two degenerate Kelvin modes ($K_1,K_2$) and two nondegenerate Yanai ($Y_1,Y_2$) modes. (b) For the no-slip interface, only two degenerate Kelvin modes ($K_1,K_1^a$) exist. No Yanai modes are present [cf. Fig. 2]. (c) Density (color) and velocity (arrows) of the labeled interface modes. The modes shown in the right column ($K_1$ and $K_1^a$) are the only modes on the no-slip interface, while all except the $K_1^a$ mode are present on the fluid-fluid interface. The fourth mode on the fluid-fluid interface is the Yanai mode $Y_2$, which is similar in shape to the depicted $Y_1$ mode but with a longer wavelength.

Figure 4

Simulations for a change of boundary conditions for fluids with odd viscosity and small dissipation. Density (color) and velocity (arrows) of interface modes coming from the left along the fluid-fluid boundary (dashed line) and interacting with a finite-length segment of no-slip wall (solid black line). For a video, see Supplemental Material [83]. (a) The $K_2$ Kelvin wave is orthogonal to all waves supported on the no-slip wall and thus cannot be transmitted. The inset shows a zoom-in, including streamlines, near the change in boundary. (b) The $K_1$ Kelvin wave is supported on both interfaces and traverses the no-slip wall unaltered. (c) The antisymmetric Yanai waves map to the antisymmetric Kelvin mode $K_1^a$ on the no-slip wall. Simulations were performed with parameters $m=0.2$, $D=0$, $\nu /{\nu }^o=0.01$.

Figure 5

Plot of (a) the total energy $E$ and (b) dissipation rate $\partial E/\partial t$ of a single pulse composed of the $K_2$ Kelvin wave, for various values of $\nu$ ($=D$). The pulse travels at constant speed along the fluid-fluid interface with (solid lines) or without (dashed lines) a change in boundary conditions to a no-slip wall. In the former case, the front (tail) of the wave hits the no-slip wall at $t=10$ ($t=40$). We observe that dissipation rate through absorption increases towards a finite value as $\nu$ tends to zero.

Figure 6

Schematic of a beam-splitting setup with topological interfaces. The infinite plane is divided into four quadrants with alternating signs for the Chern numbers originating, e.g., from microscopic rotations as illustrated by the arrows. The horizontal interface supports two independent unidirectional modes (indicated by the green and red triangles), which can flow either from the right or from the left in accordance with the bulk-boundary correspondence principle. Each of the two different vertical interfaces breaks bulk-boundary correspondence by supporting only one of the two incoming unidirectional modes.

Figure 7

The total Berry curvature (black) integrated over the two-dimensional space of wave numbers $\mathit{q}$, plotted against the dissipative difference parameter $\mathrm{\Delta }=D-\nu$. Around $\mathrm{\Delta }=2.8$ we can split the integral in three sections as seen from the origin of the $\mathit{q}$ plane: before (blue), in between (yellow), and beyond (green) the exceptional points. The yellow line is zero everywhere. The blue and green lines are directly on top of each other, indicating that total Berry curvature is equally shared between the two sections outside the exceptional points.

Figure 8

Invertibility of submatrices that determine the interface conditions. The determinants are real between the bulk curves $\omega ={\omega }_{\pm }\left(q\right)$. Interface waves exist along the curves where the determinant changes from negative (blue) to positive (yellow). (a) For the no-slip interface, $detM=0$ has only one doubly degenerate Kelvin solution for every $\omega$ in the gap (dashed lines). (b) For the fluid-fluid interface, $det\underline{U}=0$ gives two singly degenerate Yanai curves, in addition to the doubly degenerate Kelvin curve $\omega =q$ (not depicted) corresponding to $det\underline{V}=0$.

Figure 9

Plots of the band structure $\omega \left(\mathit{q}\right)$ and Berry curvature of the top band ${\mathcal{B}}_{+}\left(\mathit{q}\right)$, for different values of the dissipative difference parameter $\mathrm{\Delta }\equiv D-\nu$. We take a cut $\mathit{q}=(q_x,0)$ in wave-number space, which will look like any other cut due to radial symmetry. As $\mathrm{\Delta }$ increases, the imaginary parts of the eigenvalues become nonzero and develop geometric structure. The middle band (blue) retains a zero real part, whereas the upper and lower bands (red) have reflection symmetry about the imaginary axis. While the viewing angle might suggest that red lines intersect, in three dimensions these lines do not touch. At $\mathrm{\Delta }=2.5$, we see that the red bands have developed a secondary minimum in their real part which is pushed towards zero until the bottom and top bands meet the (blue) middle band, resulting in exceptional points. These exceptional points immediately separate as seen at $\mathrm{\Delta }=3$. Finally, the blue middle band pinches off to form rings in the imaginary plane, which are stable even beyond $\mathrm{\Delta }=8$. In the bottom subfigures, we see a bump emerging in the Berry curvature, which diverges at the exceptional points. In between the exceptional points the curvature is zero. The colors for the curvature sections correspond to those in Fig. 7. These plots were calculated for $\nu =0$, which gives $\mathrm{\Delta }=D$. A nonzero $\nu$ adds a further imaginary component to each band, which keeps the essential features (such as exceptional points) invariant.