Acoustic non-Hermitian skin effect from twisted winding topology

The recently discovered non-Hermitian skin effect (NHSE) manifests the breakdown of current classification of topological phases in energy-nonconservative systems, and necessitates the introduction of non-Hermitian band topology. So far, all NHSE observations are based on one type of non-Hermitian band topology, in which the complex energy spectrum winds along a closed loop. As recently characterized along a synthetic dimension on a photonic platform, non-Hermitian band topology can exhibit almost arbitrary windings in momentum space, but their actual phenomena in real physical systems remain unclear. Here, we report the experimental realization of NHSE in a one-dimensional (1D) non-reciprocal acoustic crystal. With direct acoustic measurement, we demonstrate that a twisted winding, whose topology consists of two oppositely oriented loops in contact rather than a single loop, will dramatically change the NHSE, following previous predictions of unique features such as the bipolar localization and the Bloch point for a Bloch-wave-like extended state. This work reveals previously unnoticed features of NHSE, and provides the observation of physical phenomena originating from complex non-Hermitian winding topology.


Supplementary Note 1: More complex unconventional topological windings
Our acoustic platform can generate more complex unconventional topological windings, as the nonreciprocity can be applied between arbitrary two sites. Here, we present two models. As shown in Fig. S1(a), the first example has reciprocal nearest-neighbor coupling κ1 = 6 Hz (yellow line) and unidirectional next-nearest-neighbor coupling a  = -11 + 3.9i Hz (blue line). The corresponding lattice Hamiltonian is

Supplementary Note 3: Coupled-mode theory (CMT) for two-resonator model
As shown in Fig. 2(a) in the main text, the two-resonator model is composed of two acoustic cavities connected via dual crossed waveguides, an active component amplifying the sound from cavity 1 to cavity 2 unidirectionally, a source, and a detector. As shown in Fig. S3(a), the two cavities have the same resonance frequency ω0 and intrinsic loss γ0, and two crossed waveguides provide the reciprocal coupling coefficient κ1. The source and the detector are inserted into the cavities with the same coupling strength γ1 (considering the source tube and the detector tube have the same size).
The active component provides complex unidirectional coupling a  . According to the CMT 1 , when the wave is incident to cavity 1 (Fig. S3), the dynamic equation can be described as where a1 (a2) is the mode in cavity 1 (2), s1+ represents the incident wave from cavity 1. For the case when the wave is incident to cavity 2, change The corresponding effective Hamiltonian is, It can be seen that in this Hamiltonian, the Hermiticity, time-reversal symmetry, and reciprocity are broken simultaneously 2 .
Supplementary Fig. 3 The CMT model for the two-resonator model. The CMT model for the two-resonator model composed of two acoustic cavities.

Supplementary Note 4: Transmission coefficients for the two-resonator mode
According to Eq. 1, the transmission coefficients are In the reciprocal model without the unidirectional coupling, the transmission coefficients are

Supplementary Note 9 Experimental results of the reciprocal acoustic crystal
To probe the field distribution, we place the source at site 10 and detect the response at each site of the chain (the same sample shown in Fig. 3(a) and the active components are not in operation).
As shown in Fig. S8(a), the wave is concentrated on the input site 10 and propagates toward both directions with almost identical amplitude at ω = 1713 Hz. By applying Fourier transform to the measured field distributions in the chain, we obtain the dispersion, as depicted in Fig. S8(b).
Supplementary Fig. 8 Experimental results of the reciprocal acoustic crystal. a Field distribution at frequency

Supplementary Note 10: Measured field intensity distributions in the acoustic crystal with the next-nearest-neighbor non-reciprocal coupling
In this section, we have plotted Fig. S9 to show the measured field distributions at different frequencies for both winding numbers. One can see that below the frequency of the Bloch point (around 1696 Hz), the wave excited from site 10 is strongly suppressed towards the left boundary, but is dramatically amplified towards the right boundary, implying the winding number of v = -1.
On the contrary, the phenomenon is reversed above the Bloch point, indicating the flip of the sign of the winding number. Supplementary Fig. 9 Measured field intensity distributions in the acoustic crystal with the next-nearestneighbor non-reciprocal coupling. a-h Field intensity distributions at different frequencies. The measured results are normalized by the amplitude at the source location (indicated by little arrows).

Supplementary Note 11: Theoretical calculation of field intensity distributions in the acoustic crystal with the next-nearest-neighbor non-reciprocal coupling
According to the coupled-mode theory, when the wave is incident to site 10 and detected at site 20, the dynamic equation can be described as, ( ) Hz, when the source is located at the site 6, 8, and 10, respectively. The measured results are normalized by the amplitude at the source location (indicated by little arrows).

Supplementary Note 14: Tight-binding model for the coupled acoustic resonators
Due to the versatile tunability and ease of fabrication, the acoustic system is widely adapted for realizing tight-binding models. Here, we show an example to briefly introduce the connection between the designed acoustic structure and the tight-binding model. As depicted in Fig. S13(a), a single acoustic resonator has a dipole mode with eigenfrequency ω0 = 1700 Hz. When two resonators are connected by narrow waveguides, the coupling between two dipole modes will lead to eigenfrequency splitting. The resulting symmetric (at 1685 Hz) and antisymmetric (at 1728 Hz) modes are shown in Fig. S13(b). Such a two-resonator system can be described by a two-level with n being the position of the site. For each set of the first spatial moment, we calculate the averaged standard deviation σm1. The lattice size is N =100, and each calculation is averaged over 1000 realizations. As shown in Fig. S14(a) and Fig. S15(a), the averaged standard deviation starts from 0 and converges at the large disorder strength, indicating the transition from the NHSE to the Anderson localization. We also show the eigenstates at different disorder strengths [see Figs. S14(b)-14(d) and S15(b)-15(d)]. One can see that at small disorder strength, the NHSE survives 3 .