Abstract
In Hermitian topological systems, topological modes (TMs) are bound to interfaces or defects of a lattice. Recent discoveries show that nonHermitian effects can reshape the wavefunctions of the TMs and even turn them into extended modes occupying the entire bulk lattice. In this letter, we experimentally demonstrate such an extended TM (ETM) in a onedimensional (1D) nonHermitian acoustic topological crystal. The acoustic crystal is formed by a series of coupled acoustic resonant cavities, and the nonHermiticity is introduced as a nonreciprocal coupling coefficient using active electroacoustic controllers (AECs). Our work highlights the potential universality of ETMs in different physical systems and resolves the technical challenges in the further study of ETMs in acoustic waves.
Hermitian topological matters sustain topological modes (TMs) that are bound states localized at interfaces, edges, or defects of a specimen [1,2,3]. The TMs possess properties such as robustness against perturbations, which are highly desirable for many applications. The studies of topological matters have expanded beyond condensed matter physics and are nowadays pursued in photonics [4], acoustics, and mechanics [5, 6]. In all these systems, the existence of TMs rests on the nontrivial bulkband topology, which implies that a bulk lattice is indispensable. This requirement entails high costs, large footprints, and uneconomic use of space and materials for topological devices.
Recent studies of nonHermitian physics have unveiled a range of new physical phenomena [7,8,9,10,11], such as nonHermitian skin effect (NHSE) [12,13,14]. NHSE describes the appearance of a large number of skin modes, which are localized bulk eigenstates in openboundary lattices. This is in stark contrast to the extended Bloch waves observed in Hermitian systems. Recently, it has been found that NHSE can conversely affect the wavefunctions of TMs, and can even turn them into completely extended states occupying every unit cell in the entire lattice [15,16,17,18]. Such extended TMs (ETMs) have been experimentally demonstrated in 1D and twodimensional (2D) nonHermitian mechanical lattices [17, 18]. Because the theoretical models of ETMs are based on tightbinding theories that are universal, we anticipate similar phenomena to emerge in acoustics, electromagnetism, photonics, etc. However, so far ETMs have not been observed in all these areas. Here, we demonstrate an acoustic ETM in a nonHermitian Su–Schrieffer–Heeger (NHSSH) lattice formed by coupled acoustic cavities [5, 6, 19] with active electroacoustic controllers (AECs) controlled by acoustic feedbacks [20,21,22] generating nonreciprocal coupling. Our work contributes to nonHermitian acoustics and topological acoustics and opens new ways to manipulate acoustic waves.
Our experiment is based on a 1D NHSSH chain with nonreciprocal hopping, as depicted in Fig. 1a. The Hamiltonian of a finite lattice under open boundary condition (OBC) is
where \(a_i^\dagger\) and \({a}_{i}\) (\(b_i^\dagger\) and \({b}_{i}\)) are the creation and annihilation operators of \({A}_{i}\) site (\({B}_{i}\) site) in unit cell \(i\), \(N=15\) is the number of unit cells, and \(v=11.2 {\mathrm{s}}^{1}\) and \(w=34.6 {\mathrm{s}}^{1}\) are the intracell and intercell hopping, respectively. These values are retrieved from the experimental system. \(\delta\) represents the nonreciprocal hopping term from the site \({A}_{i}\) to \({B}_{i}\). In the Hermitian case, i.e., \(\delta =0\), the lattice is topological and the two bulk bands (Fig. 1b) are characterized by a quantized Zak phase of \(\pi\) [5, 23]. Because we keep only one site in the last unit cell (site \({A}_{N}\)), there is only one TM at zero energy localized at the left boundary, as shown in Fig. 1b, d. The bulk modes are extended states, which can be seen by the wavefunction average, defined as \(\overline{\Psi }\left(x\right)\equiv \frac{1}{2N1}{\sum }_{j=1}^{2N1}{\left{\psi }_{j}\left(x\right)\right}^{2}\) with \({\psi }_{j}\left(x\right)\) being the eigenfunction of the \(j\) th mode. (Our system has \(2N1\) sites in total, there are \(2N1\) modes.) In Fig. 1d, it is seen that \(\overline{\Psi }\left(x\right)\) is uniformly distributed over the entire lattice. When \(\delta \ne 0\), the system becomes nonHermitian, and its spectrum under periodic boundary condition (PBC) forms two loops in the complex plane (Fig. 1c), signifying the existence of NHSE [24]. This can be seen in Fig. 1e, where \(\delta =wv\). \(\overline{\Psi }\left(x\right)\) clearly collapses towards the right side of the lattice under OBC. Meanwhile, the TM is turned into a fully extended mode because \(\delta\) counters the spatial exponential decay that characterizes the Hermitian TM. This is the ETM that we aim to demonstrate in acoustics.
The experimental platform is an acoustic crystal consisting of 9 cuboid cavities (\(35\times 35\times 80 {\mathrm{mm}}^{3}\) in length, width, and height, respectively) connected by channels 40 mm in length with periodically alternating crosssectional areas (\(23.7 {\mathrm{mm}}^{2}\) and \(165 {\mathrm{mm}}^{2}\) for the intracell hopping \(v\) and intercell hopping \(w\), respectively), as shown in Fig. 2a. To observe the TM, we place a loudspeaker at cavity \({A}_{1}\) and measure spectral responses of every cavity. Figure 2b shows the normalized pressure responses at cavity \({B}_{3}\) and \({A}_{2}\), respectively. Because the TM only has support in sublattice \(A\), two bulk bands separated by a bandgap are seen in the results measured at cavity \({B}_{3}\). The TM is observed as the midgap response peak at 2145 Hz in the results measured at cavity \({A}_{2}\).
The next step is to add nonreciprocal coupling to the acoustic system. This is achieved by using AECs, as shown in Fig. 3a and b. The device consists of a microphone (Panasonic WMG10DT502), a loudspeaker, and a custommade printed circuit board (PCB) that integrates a phase shifter, an amplifier, and a microcontroller (MCU, ESP32 LoLin32 Lite). The MCU is programmed to function as two independent digital potentiometers that control the phase shifter and the amplifier, respectively. The phase shifter compensates for the phase delay induced by other electric components. This is to ensure that \(\delta\) is real. The amplifier determines the strength of the nonreciprocal coupling, \(\left\delta \right\). To verify the successful realization of acoustic nonreciprocity and to benchmark the AEC’s performance, we use a simple twolevel model
wherein \({\mathrm{I}}_{2}\) is a \(2\times 2\) identify matrix, \({\omega }_{0}\) is the resonant frequency of the cavity, and \(\gamma\) represents the dissipative rate of the cavities, \({\kappa }_{0}\) denotes the reciprocal part of the coupling. The eigenvalues of \({H}_{2}\) are \({\omega }_{\mathrm{1,2}}={\omega }_{0}i\gamma \pm \sqrt{{\kappa }_{0}({\kappa }_{0}+\delta )}\), which reach an exceptional point at \(\delta /{\kappa }_{0}=1\). The acoustic system for Eq. (2) is simply two coupled cavities, denoted \({C}_{1}\) and \({C}_{2}\), as shown in Fig. 3a. We excite at cavity \({C}_{1}\) with a loudspeaker and measure the response function at \({C}_{2}\) at different nonreciprocal coupling generated by an AEC. The results are shown in Fig. 3c as markers which agree well with the responses computed using Green’s function (see, e.g., refs [17, 25]. for more details of the method). Here, the retrieved parameters are onsite resonant frequency \({f}_{0}=\frac{{\omega }_{0}}{2\pi }=2145 \mathrm{Hz}\), dissipative rate \(\gamma =11.2 {\mathrm{s}}^{1}\), and reciprocal coupling strength \({\kappa }_{0}=8.8 {\mathrm{s}}^{1}\), respectively. In Fig. 3d, the retrieved \(\mathrm{Re}\left({\omega }_{\mathrm{1,2}}\right)\) are plotted as functions of \(\delta /{\kappa }_{0}\), in which the exceptional point is clearly identified at \(\delta /{\kappa }_{0}=1\). These experimental results confirm that our AECs can indeed realize nonreciprocal hopping.
The AECs are then combined with the acoustic crystal, as shown in Fig. 2a. We then use a loudspeaker to excite the system from cavity \({A}_{1}\) at 2145 Hz, which is the eigenfrequency of the TM, and measure the pressure across the entire acoustic crystal for different nonreciprocal hopping. The results are shown in Fig. 2c. We observe that the pressure responses in the bulk increase with \(\left\delta \right\). And when \(\delta =23.3 {\mathrm{s}}^{1}\), the response profile is almost flat while maintaining the bipartite characteristics, which is evidence of the emergence of ETM. The same pressure distributions are obtained in numerical simulation using finiteelement solver COMSOL Multiphysics, as shown in Fig. 2d. Both the experimental and simulation results agree well with theoretical predictions based on the tightbinding model (Eq. 1).
In conclusion, we report the realization and observation of an ETM in acoustics for the first time, wherein the nonreciprocal hopping is realized using the AECs. Together with the previous demonstration in mechanical lattices [17, 18], our results show that the ETM is, in principle, a universal nonHermitian phenomenon, which should also appear in other realms, such as electromagnetism, photonics, and so on. We envision ETM to have good application potential, and to be particularly valuable for largearea singlemode devices.
Availability of data and materials
Data and materials are available upon request.
References

F.D.M. Haldane, Nobel lecture: topological quantum matter. Rev. Mod. Phys. 89(4), 040502 (2017)

M.Z. Hasan, C.L. Kane, Colloquium : topological insulators. Rev. Mod. Phys. 82(4), 3045–3067 (2010)

J.E. Moore, The birth of topological insulators. Nature. 464(7286), 194–198 (2010)

T. Ozawa, H.M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M.C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, I. Carusotto, Topological photonics. Rev. Mod. Phys. 91(1), 015006 (2019)

G. Ma, M. Xiao, C.T. Chan, Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1(4), 281–294 (2019)

H. Xue, Y. Yang, B. Zhang, Topological acoustics. Nat. Rev. Mater. 7(12), 974–990 (2022)

C.M. Bender, Making Sense of NonHermitian Hamiltonians. Rep. Prog. Phys. 70(6), 947–1018 (2007)

Y. Ashida, Z. Gong, M. Ueda, NonHermitian physics. Adv. Phys. 69(3), 249–435 (2020)

E.J. Bergholtz, J.C. Budich, F.K. Kunst, Exceptional topology of nonHermitian systems. Rev. Mod. Phys. 93(1), 015005 (2021)

K. Ding, C. Fang, G. Ma, NonHermitian topology and exceptionalpoint geometries. Nat. Rev. Phys. 4(12), 745–760 (2022)

C. Wu, A. Fan, S.D. Liang, Complex Berry curvature and complex energy band structures in nonHermitian graphene model. AAPPS Bulletin 32, 39 (2022)

S. Yao, Z. Wang, Edge States and Topological Invariants of NonHermitian Systems. Phys. Rev. Lett. 121(8), 086803 (2018)

F.K. Kunst, E. Edvardsson, J.C. Budich, E.J. Bergholtz, Biorthogonal BulkBoundary Correspondence in NonHermitian Systems. Phys. Rev. Lett. 121(2), 026808 (2018)

X. Zhang, T. Zhang, M.H. Lu, Y.F. Chen, A review on nonHermitian skin effect. Adv. Phys. X. 7(1), 2109431 (2022)

P. Gao, M. Willatzen, J. Christensen, Anomalous Topological Edge States in NonHermitian Piezophononic Media. Phys. Rev. Lett. 125(20), 206402 (2020)

W. Zhu, W.X. Teo, L. Li, J. Gong, Delocalization of topological edge states. Phys. Rev. B. 103(19), 195414 (2021)

W. Wang, X. Wang, G. Ma, NonHermitian morphing of topological modes. Nature. 608(7921), 50–55 (2022)

W. Wang, X. Wang, G. Ma, Extended State in a Localized Continuum. Phys. Rev. Lett. 129(26), 264301 (2022)

Z.G. Chen, L. Wang, G. Zhang, G. Ma, Chiral Symmetry Breaking of TightBinding Models in Coupled AcousticCavity Systems. Phys. Rev. Applied. 14(2), 024023 (2020)

B.I. Popa, S.A. Cummer, Nonreciprocal and highly nonlinear active acoustic metamaterials. Nat. Commun. 5(1), 3398 (2014)

F. ZangenehNejad, R. Fleury, Active times for acoustic metamaterials. Rev. Phys. 4, 100031 (2019)

L. Zhang, Y. Yang, Y. Ge, Y.J. Guan, Q. Chen, Q. Yan, F. Chen, R. Xi, Y. Li, D. Jia, S.Q. Yuan, H.X. Sun, H. Chen, B. Zhang, Acoustic nonHermitian skin effect from twisted winding topology. Nat. Commun. 12(1), 6297 (2021)

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z.Q. Zhang, C.T. Chan, Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11(3), 240–244 (2015)

K. Zhang, Z. Yang, C. Fang, Correspondence between Winding Numbers and Skin Modes in NonHermitian Systems. Phys. Rev. Lett. 125(12), 126402 (2020)

W. Tang, X. Jiang, K. Ding, Y.X. Xiao, Z.Q. Zhang, C.T. Chan, G. Ma, Exceptional nexus with a hybrid topological invariant. Science. 370(6520), 1077–1080 (2020)
Funding
This work is supported by the Hong Kong Research Grants Council (RFS22232S01, 12301822, 12302420, 12302419) and the Ministry of Science and Technology of China (2022YFA1404400).
Author information
Authors and Affiliations
Contributions
X. W. performed numerical calculations and experiments. All authors discussed and analyzed the data. X. W. and G. M. prepared the manuscript. G. M. led the research.
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.