Physical rendering of synthetic spaces for topological sound transport

We describe here the mechanism behind the physical rendering of a synthetic space and we start by describing the general setting. It consists of a generic acoustic structure of discrete resonators such that each resonator has an address $(\bm{n},m)$, with $\bm{n}$ being a horizontal label and $m$ a vertical one. The couplings in the horizontal plane occur through channels whose modulated widths are specified by a phason $\bm{\phi}$, which lives on a phason space such as the $d_{s}$-torus. Here, $d_{s}$ represents the dimension of the synthetic space. The couplings in the vertical direction are established through uniform channels. Now, we consider any curve $\bm{\phi}(z)$ in the phason space with bounded first derivative $\bm{\phi}^{\prime}(z)$, which we parametrize by the continuous $z$-coordinate. We use this curve to specify the phason values for each horizontal layer, specifically, $\bm{\phi}_{m}=\bm{\phi}(\epsilon z_{m})$, where $z_{m}=ma_{z}$ is the physical coordinate of the $m$-th layer. The parameter $\epsilon$ is small and is there to ensure that the variations of the phason from one layer to another are small. Compared with the wave-guide setting, the major differences are the discrete character of the $z$ coordinate and the strong coupling in the horizontal direction.

The resulting dynamical matrix $D_{\bm{\phi}}$ governing the collective resonant modes depends on the chosen path $\bm{\phi}(z)$ and, while $D_{\bm{\phi}}$ is not periodic in the $z$-direction, it displays the following covariance property

$$T_{v}^{\dagger}D_{\bm{\phi}}T_{v}=D_{\bm{\phi}\circ\tau},\quad\tau(z)=z+\epsilon a_{z},$$

(1)

where $T_{v}$ is the vertical translation by $a_{z}$. If $Q_{\bm{n},m}$ is the acoustic resonant mode supported by the $(\bm{n},m)$-cavity, then the pressure field of the resonant collective modes can always be sought in the form $\int dk_{z}\,\Psi_{k_{z}}(\bm{r};\bm{\phi})$ ¹¹1See Supplementary Materials for multi-mode expansions., with

$$\Psi_{k_{z}}(\bm{r};\bm{\phi})=\sum_{\bm{n},m}e^{ik_{z}m}\varphi_{\bm{n},m}(\bm{\phi},k_{z})Q_{\bm{n},m}(\bm{r}),$$

(2)

and the covariance property (1) requires

$$\varphi_{\bm{n},m+1}(\bm{\phi},k_{z})=\varphi_{\bm{n},m}(\bm{\phi}\circ\tau,k_{z}).$$

(3)

Since $\bm{\phi}(z)$ is a smooth path, then, $\bm{\phi}\circ\tau\approx\bm{\phi}+\epsilon a_{z}\bm{\phi}^{\prime}$ and, as such, $\varphi_{\bm{n},m+1}(\bm{\phi},k_{z})\approx\phi_{\bm{n},m}(\bm{\phi},k_{z})$ to 0-th order in $\epsilon$. In these conditions, the vertical dispersion of the $\varphi$-coefficients can be ignored and the $\varphi_{\bm{n},m}$ coefficients for a fixed $m$ become the eigen-modes of the reduced Hamiltonian (see Supplementary Materials)

	$\displaystyle H_{k_{z}}(\bm{\phi}_{m})=$	$\displaystyle\sum_{\bm{n}}\nu(k_{z})^{2}\ket{\bm{n}}\bra{\bm{n}}$		(4)
		$\displaystyle\qquad+\sum_{\langle\bm{n},\bm{n}^{\prime}\rangle}\kappa_{\bm{n},\bm{n}^{\prime}}(\bm{\phi}_{m})\ket{\bm{n}}\bra{\bm{n}^{\prime}},$

where the last sum goes over the neighboring cavities and $\kappa_{\bm{n},\bm{n}^{\prime}}$ are the horizontal couplings, determined entirely by the phason value $\bm{\phi}_{m}=\bm{\phi}(\epsilon z_{m})$. Also, $\nu(k_{z})$ is the dispersion of the decoupled vertical channels and, if $\epsilon(\phi_{m})$ is the eigenvalue of the $\varphi$-mode, then the value of $k_{z}$ at layer $m$ is determined by the relation $f^{2}-\nu(k_{z})^{2}=\epsilon(\phi_{m})$. The conclusion is that, by examining the horizontal spatial profiles of the collective resonant modes, one layer at a time, we can visualize the states of the Hamiltonian $H(\bm{\phi})$ along arbitrary paths inside the phason space.

So far we have established that the coefficients $\varphi_{\bm{n},m}$ for fixed $m$ are eigen-modes of Hamiltonian (4), but what is the weight and the phase of these modes in Eq. (2)? An expansion in the first order in $\epsilon$ (see Suplementary Materials) reveals that the answer is supplied by the equation

$$i\epsilon\Gamma(\phi)\partial_{\phi}|\varphi(\bm{\phi})\rangle=[H(\bm{\phi})-\epsilon(\bm{\phi})]|\varphi(\bm{\phi})\rangle,$$

(5)

which governs the evolution of the $\varphi$-coefficients along the $\bm{\phi}$-trajectory. The function $\Gamma(\bm{\phi})$ is determined entirely by the vertical mode dispersion. After a proper change of variable, Eq. (5) becomes the classic equation of adiabatic evolution TeufelBook , hence the amplitudes of the modes along the $z$ direction are all equal but a non-trivial phase $e^{i\alpha(z)}$ does develop along the stacking direction. It can be computed as the Berry phase of the Wilczek-Zee connection FrankPRL1984 along the path $\bm{\phi}(z)$. The important conclusion is that we not only can control the spatial profiles of the modes but also their phases. The latter have been already proposed as vehicles for certain forms of information processing with classical meta-materials BarlasPRL2020 .

Figure 1a shows a planar array of acoustic cavities coupled horizontally and vertically through channels. Each cavity has an address $(n,m)\in\mathbb{Z}^{2}$ and the thickness of the horizontal channel connecting $(n,m)$ and $(n+1,m)$ resonators is modulated according to the protocol $h^{x}_{nm}=h_{0}[1+\delta\cos(b_{n\,{\rm mod}\,3}+\phi_{m})]$, where $h_{0}$ is the average thickness of the horizontal channels, $\delta$ is the modulation amplitude, and $b_{j}$’s are free parameters. The values of phason for each layer are set by $\phi(z)=\phi_{i}+(\phi_{f}-\phi_{i})\frac{z}{L_{z}}$, where $\phi_{i}=-0.2\pi$, $\phi_{f}=0.2\pi$ and $L_{z}=16a_{z}$. This results in a variation $\Delta\phi=0.026\pi$ from one layer to another, hence within the conditions of Sec. II.1.

By design, when $b_{j}=(j-1)\frac{2\pi}{3}$, the effective Hamiltonian $H_{k_{z}}(\phi)$ in Eq. (4) is just the 1D Aubry-André-Harper (AAH) model Aubry1980 associated with the 2D Hofstadter model at magnetic flux $\pi/3$, with $\phi$ playing the role of a quasi-momentum. Figure 1b shows the resonant spectrum of $H_{k_{z}}(\phi)$ as function of $\phi$ and $k_{z}$. The computation was carried out with COMSOL Multiphysics on the domain of a finite horizontal stack with $k_{z}$-twisted Bloch boundary conditions in the vertical direction. The bulk and the boundary spectra are shown with distinctive colors and, as expected from the Hofstadter butterfly HofstadterPRB1976 , two bulk spectral gaps are observed. Also from the Hofstadter butterfly, one can read that the lower and upper band gaps carry first Chern numbers $C_{1}=\{-1,1\}$. This is confirmed in the Supplementary Materials by direct calculations of the Chern numbers for an entire phase diagram computed for various values of $b_{j}$’s. Lastly, as expected from the bulk-boundary correspondence for the 2D IQHE, topological edge modes are observed in the spectrum reported in Fig. 1b (see the blue sheets). At fixed $k_{z}$, there are precisely one chiral edge band per edge and the slopes of these bands are consistent with the values of the Chern numbers (see Supplementary Materials).

We now focus on the spatial profiles of the modes, as excited at frequency $f=4960\ \mathrm{Hz}$, indicated by the red horizontal sheet in Fig. 1b. It was chosen to intersect the dispersion surface of the edge modes such that we can visualize a topological pumping of sound. All modes along the curve resulted from the intersection of the $f=4960\ \mathrm{Hz}$ plane and the dispersion surfaces will be excited. As argued in the previous section, $\phi$ is resolved by the $z$ coordinate, hence this pumping curve, shown again in Fig. 1c, can be parametrized by the physical coordinate along the stacking, $(\phi(z),k_{z}(z))$. In other words, the spectral data from Fig. 1c has been rendered in the physical dimension, for us to observe. We now can understand the spatial profiles of the excited modes, when examined one stack a time. Along the horizontal stack at a given coordinate $z$, one should observe the mode of $H(\phi(z))$ at energy $f^{2}-\nu(k_{z}(z))^{2}$. The evolution of this mode along the pumping curve is shown in Fig. 1c, which confirms that sound is indeed pumped from one edge to the other. For example, the left edge state denoted in the inset (1) is selected as an initial state with a negative pumping value, which remains localized on the left boundary with the adiabatic increase of the pumping parameter and the corresponding wave number. When the pumping parameter approaches $\phi=0$, the left edge state becomes the bulk state as inset (2) depicted. As the pumping parameter increases further, the bulk state is then transformed into an edge state (3) localized at the right side. Then our prediction is that sound is transported from one side to the opposite side of the structure and this topological sound steering can be witness by walking along vertical coordinate. Experimental observation and confirmation of the adiabatic pumping via topologically protected boundary states is reported in Fig. 1d. Here, the sound was injected into the bottom-left corner of the structure and the pressure field was mapped by a microphone for each site (see the inset in Fig. 1(a)). As one can see, the pressure distribution indeed renders the topological pumping process in the physical dimensions. The experimental observation is also verified by the numerical simulation based on the exact geometry (Fig. 1e, see also Supplementary Movie S1 for 1D transient topological edge pumping). The minor difference between experiment and simulation may be attributed to manufacturing deviations from connecting adiabaticity and perfect coupling to edge states (Details of the sample manufacturing, numerical simulation and experimental testing can be found in Methods). Let us also point out that there is substantial region where the wave has a bulk character and where dissipation mostly occurs. This region can be reduced by optimizing the function $\phi(z)$ from a linear to a tangent hyperbolic profile.

We now investigate the three-dimensional (3D) acoustic structure shown in Fig. 2a, engineered to have a phason $\bm{\phi}=(\phi^{x},\phi^{y})$ living on 2-torus. In this case, each cavity has an address $(\bm{n},m)\in\mathbb{Z}^{3}$ and the thicknesses of the horizontal connecting channels in the $\alpha=x,y$ directions are modulated according to the protocol $h^{\alpha}_{\bm{n},m}=h_{0}[1+\delta\cos(b^{\alpha}_{n_{\alpha}\,{\rm mod}\,3}+\phi_{m}^{\alpha})]$, while the vertical connecting channels are uniform. By design, the effective horizontal Hamiltonian (4) is just a sum of two copies of the Hamiltonian from Sec. II.2, $H(\bm{\phi})=H(\phi^{x})\otimes I+I\otimes H(\phi^{y})$, which is known to host 4D QHE physics zilberberg2017photonic .

The system can be pumped along any orbit inside the phason space by using the strategy described in Sec. II.1. The particular crystal in Fig. 2a was designed to pump along the diagonal orbit $\bm{\phi}(z)=(\phi(z),\phi(z))$, with $\phi(z)=\phi_{i}+(\phi_{f}-\phi_{i})\frac{z}{L_{z}}$, where $L_{z}=15a_{z}$ (see the inset of Fig. 2c). In the Supplementary Materials, we present additional 3D acoustic structures, engineered to pump along the orbit $(\phi(z),\phi_{i})$. Figure 2b shows the dispersion surfaces of the resulting effective Hamiltonian (4) as function of the pumping parameter $\phi$ and $k_{z}$. The computation, which was carried as for the 2D structure, reveals two bulk gaps and, in addition, two spectral sheets highlighted in blue for which the modes are localized along two edges, as well as one sheet highlighted in green for which the modes are localized at the corners. In the Supplementary Materials, we demonstrate that both bulk gaps carry non-trivial second Chern numbers. Figure 2c (upper panel) shows the pumping curves resulted from the intersection of the dispersion diagram with the plane at frequency $f=7380\ \mathrm{Hz}$, with the latter highlighted in red in Fig. 2b. In contrast to the 1D topological pumping, there are more than one such pumping curve. Similarly, Fig. 2c (lower panel) shows the pumping curve resulted from the intersection with the plane at frequency $f=6020\ \mathrm{Hz}$, with the latter highlighted in purple in Fig. 2b. All these three pumping curves are parametrized by the $z$-coordinate and, as in the 1D case, we can predict that, if one examines the horizontal stack at coordinate $z$, one should observe the eigen-modes of $H(\bm{\phi}(z))$ at energy $f^{2}-\nu(k_{z}(z))^{2}$. Samples of these modes are rendered in the inset of Fig. 2c and, as one can see, all three pumping curves that we engineered are very special. Indeed, as one pumps along the blue contours, the mode is pumped from one pair of edges to the opposite pairs of edges, while if one pumps along the green curve the mode is pumped from one corner to the opposite corner. Both pumping processes proceed through a bulk delocalization transition.

In Fig. 3a, we demonstrate experimentally that the predicted edge-bulk-edge pumping is indeed rendered in the physical space of the 3D structure. The acoustic wave is excited by a sound speaker along the bottom edge at $f=7380\ \mathrm{Hz}$ and the pressure distribution is measured by the microphone, layer by layer along the stacking direction (see the inset in Fig. 2a). As seen in Fig. 3a, the pressure field does evolve from the left to the right edge as one walks along the stacking direction. It is interesting to note that the edge states that we excite have the same energies as bulk states, which are also seen being excited in the very bottom layer. However, since the bulk states extend throughout the horizontal layer, they experience increased dissipation and, as such, they fade away along the stacking direction. On the other hand, the topologically pumped states are long-propagated, which further demonstrates the advantages of topological steering of sound. A numerical simulation was also conducted to validate our experimental observation (Fig. 3b, see also Supplementary Movie S2 for 2D transient topological edge pumping).

In Fig. 3c, we demonstrate experimentally that also the predicted corner-bulk-corner pumping can be rendered in the physical space of our 3D structure. Indeed, with source placed at a bottom corner and with the frequency adjusted at $f=6020\ \mathrm{Hz}$, one sees the measured pressure field evolving towards the opposite corner as one walks along the staking direction. This pumping is resolved much better than in the previous case, because the source couples less effectively to the bulk modes, hence the latter are not excited in this setup. Also, due to reduced dimensionality of the mode, the dissipation is weaker and the pressure field can be sustained longer along the stacking direction. Numerical simulations for the entire 3D structure are reported in Fig. 3c and they validate our experimental findings (see also Supplementary Movie S3 for 2D transient topological corner pumping).

The emergence of the chiral boundary spectrum that makes possible the topological pumpings observed in Fig. 3 is due to the second Chern number of the gaps, which is the strong topological invariant in 4D. As it is the case for any strong topological invariant, topological boundary spectrum emerges regardless of how the boundary of the crystal is cut, provided the available quasi-momenta are properly sampled. In our case, the phason plays the role of synthetic momenta and this implies that the pumping process along a given phason orbit manifests only in a particular space direction. To demonstrate that the pumping processes are indeed highly directional, we simulated the acoustic characteristics of the 3D structure for different phason orbits and with the source placed at different space locations and encountered the following scenarios: (1) propagation along a facet; (2) propagation along an edge; (3) pumping in the $x$ but not in the $y$ direction; (4) pumping in the $y$ but not in the $x$ direction; (5) pumping along the first diagonal $x=y$ but not along the second diagonal $x=-y$. (6) pumping along the second diagonal $x=-y$ but not along the first diagonal (see Supplementary Materials). We also demonstrate what we call an “antagonistic” effect, which manifests as follows: If edge-to-edge or corner-to-corner pumping is observed with the source placed on one edge or corner, respectively, then the pumping or any propagation are completely absent if the source is moved to the opposite edge or corner. In this respect, the structure acts like a perfect “transistor” because the pumping can be turned on and off by $180^{\degree}$ rotations. In fact, by combining all the effects listed above, our 3D structure can be transformed into a multi-functional an programmable acoustic device for sound transport and distribution. We stress that our structure does not possess any crystalline symmetry so that the bulk polarizations are not quantized Petrides2020 . Therefore, the higher-order topological corner modes in our system are fundamentally different from previous realizations based on quantized quadrupole polarization Serra2018 ; Peterson2018 ; Imhof2018 or quantized Wannier centres NiX2019 ; Xue2019a ; Xue2019b ; Weiner2020 .

Topological phases with non-zero second Chern numbers display intriguing wave transport characteristics, such as robustness against impurities or defects ProdanSpringer2016 . It is of interest to quantify the extent of the topological protection in such conditions. To evaluate that, a 3D hollow structure is constructed by removing nine cavities at the center of the system and the topological edge-bulk-edge and corner-bulk-corner pumpings in the defected structure are experimentally measured, as shown in Figs. 4a and 4c, respectively. When comparing performance with the system without defects (Figs. 3a and 3c), both edge and corner modes can be smoothly pumped despite of the presence of defects in a relatively large scale. This confirms that the topological pumping is immune against back reflections from defects or discontinuity. The experimental observation is also verified by the numerical simulation based on the exact geometry (Figs. 4b and 4d). In addition, a robust topological pumping due to disorder in the pumping parameters is also numerically evaluated in the Supplementary Materials.