Observation of topological edge states induced solely by non-Hermiticity in an acoustic crystal

Non-Hermiticity alters band topology in the presence of loss and/or gain in topological systems, which not only introduces new definitions in topological classifications, topological invariants, and the bulk-boundary correspondence, but also gives rise to unprecedented applications such as topological insulator lasers. Most existing non-Hermitian topological systems derive their topological phases from Hermitian components, rather than being driven by non-Hermiticity itself. Here we report on the experimental observation of topological edge states induced solely by non-Hermiticity in an acoustic crystal. The acoustic crystal consists of a periodic one-dimensional chain of coupled acoustic resonators with tunable loss. In the Hermitian limit, or when the loss is negligible, the crystal exhibits no band gap and hosts no topological edge states. By introducing loss, we show that a band gap is induced, which can be either topological or trivial, depending on the loss configuration. In the topological case, topological edge modes are found inside the band gap. These results demonstrate that non-Hermiticity is able to drive a topological phase transition from a trivial system to a topological one, offering the possibilities for actively steerable topological wave manipulations in applications ranging from acoustics to photonics.

Figure 1

(a) Schematic of a 1D acoustic crystal of coupled acoustic resonators with ${\gamma }_1={\gamma }_2=0$. A large cell containing four resonators is shown. (b) Schematic of a 1D acoustic crystal of coupled acoustic resonators with ${\gamma }_1={\gamma }_2\ne 0$. Here one unit cell consists of four resonators. The resonators which are colored in blue have additional loss. (c) and (d) Corresponding dispersions of the lattices in (a) and (b), respectively. The color denotes the imaginary parts of the eigenfrequencies. (e) Schematic of the loss-induced topological edge states in a finite-sized crystal.

Figure 2

(a)–(c) Simulated eigenfrequencies of a finite-sized acoustic crystal with ${\gamma }_1={\gamma }_2=0$ (a), ${\gamma }_1={\gamma }_2\ne 0$ (b), and ${\gamma }_1=-{\gamma }_2\ne 0$ (c). Each finite-sized crystal consists of 12 resonators. (d) Sum of the amplitudes for the two in-gap states in (b).

Figure 3

(a) and (b) Photos of 3D printed single resonators with only background loss (a) and with additional loss (b). Values for the parameters are: $L=92\mathrm{mm}$, $W=52\mathrm{mm}$, $H=22\mathrm{mm}$, $r=1\mathrm{mm}$, $R=4\mathrm{mm}$, $d=2.4\mathrm{mm}$, $D=4\mathrm{mm}$. The wall thickness is 6 mm. (c) and (d) Measured spectra (red dots) for the sample in (a) and (b), which are normalized to the peak value in respective spectrum. Blue solid lines are corresponding simulation results.

Figure 4

(a) Photo of the 3D printed coupled acoustic resonator chain with ${\gamma }_1={\gamma }_2=0$. (b) Measured bulk spectrum for the sample shown in (a). (c) Measured edge spectrum for the sample shown in (a). (d)–(i) Similar to (a)–(c) but with ${\gamma }_1={\gamma }_2\ne 0$ [(d)–(f)] and with ${\gamma }_1=-{\gamma }_2\ne 0$ [(g)–(i)]. In the bulk measurement, a speaker and a microphone are respectively placed at the fifth and the ninth resonator to observe the bulk transmission. In the edge measurement, both the speaker and the microphone are placed at the first resonator. In the spectrum plots, red dots are measured from experiments and solid blue lines are obtained from simulations, which are respectively normalized. In the simulations, the loss is considered as imaginary part of sound speed, whose values are determined from single resonator measurements [41].