Revealing the Boundary Weyl Physics of the Four-Dimensional Hall Effect via Phason Engineering in Metamaterials

Quantum Hall physics has been theoretically predicted in four dimensions and higher. In hypothetical 2n dimensions, the topological characters of both the bulk and the boundary are manifested as quantized nonlinear transport coefficients that respectively connect to the $n\mathrm{th}$ Chern number of the bulk gap projection and to the $n\mathrm{th}$ winding number of the Weyl spectral singularities on the ($2n-1$)-dimensional boundaries. Here, we introduce the concept of phason engineering in metamaterials and use it as a vehicle to access and apply the quantum Hall physics in arbitrary dimensions. Using these specialized design principles, we fabricate a reconfigurable two-dimensional aperiodic acoustic crystal with a phason living on a 2-torus, giving us access to the four-dimensional quantum Hall physics. Also, we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as a function of the quasimomenta. We also demonstrate topological wave steering enabled by the Weyl physics of the three-dimensional boundaries.

Figure 1

Four-dimensional topological quantum Hall effect. (a) Photograph of a fully assembled acoustic 2D sinusoidal pattern consisting of top and bottom cylindrical resonators. The middle red bar indicates the presence of an inner chamber, which connects the top and bottom resonators and is referred to as the spacer. (b) Photograph of the inner structure, with the spacer now fully visible. (c) The wave propagation domain, together with relevant parameters. (d) Illustration of the algorithm that supplies the position of the resonators. (e) comsol simulated bulk resonant spectrum against $\theta$, together with labels for the topological and nontopological gaps. (f) Left: comsol simulated resonant spectrum for hard wall termination, shown against the phason parameters ${\varphi }_1$ and ${\varphi }_2$. The Weyl singularity is the spectral surface connecting the indicated bulk bands. Right: experimental measurement of the density of states, with the phason space sampled in several directions. (g),(h) Comparison between the experimentally measured density of states and the simulated spectrum (blue dots) for the traces ${\varphi }_1=0.5$ and ${\varphi }_1={\varphi }_2$, respectively. The theoretical spectra have been stretched by a small factor to overlay the experimental data. The bright dispersive modes on the left side are part of the spectral dome. The right side displays the raw density of states data for the bulk, from where we read the edges of the bulk gap.

Figure 2

Bulk measurements. (a) comsol simulated resonant spectrum for the experimental setup from Fig. 1, with arrows indicating the topological gaps. The vertical box identifies $\theta =0.25$, used in experiments. (b) Measured local density of states, assembled from microphone readings on 42 bulk resonators. (c) Collapse on the frequency axis of the intensity plot reported in panel (b). Two spectral gaps can be clearly identified and seen to be well aligned with the theoretical predictions.

Figure 3

Weyl singularity and mode steering. (a) comsol simulated spectrum as a function of the phason $\mathit{\varphi }=({\varphi }_1,{\varphi }_2)$, revealing the Weyl singularity. (b) Cross section of the Weyl singularity at 5910 Hz. (c) Acoustic pressure field distribution for the eight phasons marked in panel (b), revealing circular mode steering around the crystal’s boundary.

Figure 4

Dynamically generated patterns. (a) Same as the pattern of resonators in Fig. 1 but with the simplified $F(\mathit{\varphi })=ϵD[\sin (2\pi {\varphi }_1),\sin (2\pi {\varphi }_2)]$. (b) Same as (a) but with the lattice ${\mathcal{L}}^{\prime }$ rotated by ${45}^{\circ }$ relative to $\mathcal{L}$. (c) Same as (a) but with $F$ replaced by $G\circ F$ with $G(x,y)=(x+y,y-x)$. All three patterns are generated with $\theta =1/2\sqrt{2}$.

Figure 5

Spectral butterflies. Panels (a), (b), and (c) correspond to the patterns in Figs. 4–4, respectively. The spectra are computed with the model dynamical matrix $D_{\mathcal{P}}=\sum _{\mathbf{x},\mathbf{y}\in \mathcal{P}}{e}^{-1.5|\mathbf{x}-\mathbf{y}{|}^2}|\mathit{x}⟩⟨\mathbf{y}|$, with the distance measured in units of $D$. The marked spectral gaps are related to bulk spectral gaps mapped in the experiments.

Figure 6

Stability of the spectral dome. (a) A crystal cut as a square and (b) its resonant energy spectrum. (c) A crystal cut as an octagon and (d) its resonant energy spectrum. In both cases, the spectra have been resolved by ${\varphi }_1$ and ${\varphi }_2$ and a spectral dome can be clearly identified. The spectra are computed with model (a) from Fig. 5.

Figure 7

Visualizing the topological invariants. (a),(b),(c) Integrated density of states (IDS) as computed from the spectra in Figs. 5–5, respectively. The IDS values inside the spectral gaps can be identified by the abrupt changes in color. (d),(e),(f) Fittings of the IDS values inside the spectral gaps, seen in panels (a), (b), and (c), with Eq. (18). Each curve is determined by the six topological invariants associated with each spectral gap. The marked curves correspond to the marked gaps in Fig. 5 and they can be fitted with $1-{\theta }^2$, indicating $n_{\{1,2,3,4\}}=-1$; hence, a second Chern number $-1$.