Observation of edge waves in a two-dimensional Su-Schrieffer-Heeger acoustic network
Li-Yang Zheng Vassos Achilleos Olivier Richoux Georgios Theocharis Vincent Pagneux


APS/123-QED







 







LAUM, UMR-CNRS 6613, Le Mans Université, Av. O. Messiaen, 72085 Le Mans, France





















In this work, we experimentally report the acoustic realization the two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model in a simple network of airchannels. We analytically study the  steady state dynamics of the system using a set of discrete equations for the acoustic pressure, leading to the 2D SSH Hamiltonian matrix without using tight binding approximation. 

By building an acoustic network operating in audible regime, we experimentally demonstrate the existence of topological band gap. More supremely, within this band gap we observe the associated edge waves even though the system is open to free space. Our results not only experimentally demonstrate topological edge waves in a zero Berry curvature system but also provide a flexible platform for the study of topological properties of sound waves. 




                             

                              
Observation of edge waves in a two-dimensional Su-Schrieffer-Heeger acoustic network
    Vincent Pagneux
    December 30, 2023
====================================================================================

The study of topological insulators has been attracting a lot of attention in 
recent years due to their appealing property for the control of wave propagation <cit.>. Among other properties, topological insulators exhibit non-trivial topological phases, leading to the existence of robust edge states on the boundaries/interfaces <cit.>. 
Various systems exhibiting non-trivial topological phases  have been investigated <cit.>.

Previous studies have shown that Chern insulators and Z_2 topological insulators possess a non-trivial topology phase stemming from non-vanishing Berry curvatures <cit.>. On the other hand, it has been recently reported that a topological phase can also appear in systems even in the absence of Berry curvature <cit.>. 
This new interesting scheme has been found in the two-dimensional (2D) Su-Schrieffer-Heeger (SSH) model, which is a 2D extension of the 1D SSH chain, 
with alternating strengths of bonds connecting identical atoms in both x- and y-directions <cit.>.

The topological phase in the 2D SSH model can be characterized by
2D Zak phase and topological edge states are consequently predicted on the boundaries of these structures. 
Due to the  difficulty in tuning the lattice couplings, at will in the quantum world, most of the attention has been devoted to study its analogues in classical systems, including photonics <cit.>, and electrical circuits <cit.>. 
However, acoustic analogues of the 2D SSH model have not been reported so far. 
The experimental realization of the 2D SSH model in acoustics not only can provide a simple and versatile platform for the study of topological edge waves, but also opens perspectives for activities involving other novel topological phase, such as high order topological insulators <cit.>.

The analyses of topological insulators is usually performed
starting from a discrete 
model with special lattice symmetry. 
However, the majority of systems in different domains of physics are described by continuum models associated with partial differential equations. One of the most popular techniques to bridge the gap between the continuous and the discrete models is the Tight-Binding approximation (TBA) <cit.>. The original idea of TBA is to singularise discrete points in space at places where the continuous field is localised, thus it is naturally associated with resonating scatterers. This approximation technique can be rigorously applied by using Wannier functions basis, leading to the evaluation of delicate overlap integrals <cit.>. 
In practice, for the application to topological insulators where the medium is periodic, it appears that the TBA provides generic 
discrete equations or dispersion relations with coupling coefficients that can be fitted with results from numerical simulation of the continuous problem <cit.>. 
An alternative approach which we utilize below, in the spirit of quantum graph theory <cit.>, is to directly obtain
a discrete model from a continuous system with coveted hopping coefficients, by combining wave propagation properties and geometrical characteristics of the system.
One advantage of the proposed 
approach is that it is constructive in the sense that: the hopping coefficients can be prescribed at will, as opposed to the common practice in continuous systems where the hopping coefficients are a posteriori calculated (e.g. by fitting or using asymptotic methods) <cit.>.









In addition, avoiding the use of resonating scatterers, the obtained discrete model can be tuned to be valid in a broadband range of frequencies.



















In this work, we theoretically and experimentally study an acoustic 2D SSH network composed of simply connected air channels. Using the conservation of flux at the network junctions, we derive a set of discrete equations and subsequently map the acoustic system to the 2D SSH Hamiltonian. 


The validity of our theoretical model
is checked by comparison of the dispersion relation with numerical simulations.  In addition, both our theoretical model and the numerical simulations  predict the appearance of topological edge states. Then

an experimental implementation of the 2D SSH is achieved in the audible regime and it contains two different boundaries, one supporting edge waves and the other not. The propagation of topological edge waves is experimentally observed, recovering the characteristic profile of the SSH edge modes and exhibiting localization only on the prescribed boundaries of the network.


















The 2D SSH model is depicted in Fig. <ref>(a), where identical nodes are arranged in a square lattice with lattice constant 2L. 
The unit cell containing four nodes (marked as α, β, γ and δ) is indicated by a gray-dashed box in Fig. <ref>(a). The intracellular (intercellular) hoppings, i.e., couplings of nodes within (between) unit cells, are denoted by s (t) as the red (blue) bonds in Fig. <ref>(a).

The acoustic realization of the 2D SSH model consists of a network structure which is shown in Fig. <ref>(b). As it can be seen, the network is composed by two types of rigid square blocks with widths L_1, L_2, and height H as shown in the inset of Fig. <ref>(b). By placing the two blocks centered in a square lattice substrate with a lattice constant 2L, two types of airborne channels of width w_1 and w_2 are fabricated. Then, covering the top of the structure with an additional plate, these channels form rigidly closed waveguides for the acoustic waves and the 2D SSH acoustic network is constructed. A single unit cell of the network is marked by a gray-dashed box in Fig. <ref>(b), and the four junctions between the air-channels correspond to the nodes of the unit cell in Fig. <ref>(a). 
The sound pressures at the each junction is coupled with those on its neighboring junctions through the channels of  alternating widths w_1 and w_2. Thus, this coupling between neighboring junctions can be easily tuned solely by the widths  w_1,2  which, as we will show below, play the same role as the intercellular and intracellular hoppings in the 2D SSH model, which we call s and  t respectively.


Our theoretical treatment is based on the following fact: as long as the widths of the air channels are much smaller than the channel length, i.e., w_1,2≪ L, sound wave propagation between junctions can be well described assuming monomode propagation<cit.>. Further on, to derive our discrete model we label
the center of each unit cell using the normalized coordinates m and n as shown in Fig. <ref>(a). 
Considering the unit cell at position (m, n) and employing




the continuity of flux at each of the four junctions [see supplemental materials (SM)], we derive the following system of discrete equations describing the sound pressure at each junction  

    ε p_α ^m,n   =    t p_β^m-1,n+s p_β^m,n+t p_γ^m,n+1+s p_γ^m,n, 
    ε p_β ^m,n   =    s p_α^m,n + t p_α^m+1,n + t p_δ^m,n+1+s p_δ^m,n, 
    ε p_γ ^m,n   =    t p_δ^m-1,n+s p_δ^m,n+s p_α^m,n+t p_α^m,n-1, 
    ε p_δ ^m,n   =    s p_β^m,n+t p_β^m+1,n +s p_γ^m,n+ t p_γ^m,n-1.


In Eqs. (<ref>) p_i with i=α, β, γ, δ is the pressure at each junction, 

t = w_1/(w_1+w_2) is the intercellular hopping coefficient, s=1-t is the intracellular coefficient, and the “energy" term ε = 2cos2π f L/c depends on both the length of each channel L and on the frequency  f while c is the speed of sound in air. To obtain the corresponding dispersion relation we seek solutions in form of Bloch waves as p_i^m,n=p_ie^i k_x m+i k_y n, where k_x, k_y are the wave vectors along the x- and y-directions in the first Brillouin zone (BZ), as shown in the inset of Fig. <ref>(a). By substituting the wave solution into Eqs. (<ref>),
we derive the following eigenvalue equation


    [           0  s+t e^ik_x s+t e^-ik_y           0; s+t e^-ik_x           0           0 s+t e^-ik_y;  s+t e^ik_y           0           0  s+t e^ik_x;           0  s+t e^ik_y s+t e^-ik_x           0 ]Ψ
    =
    εΨ,

where Ψ=[p_α; p_β; p_γ; p_δ] is the basis consisting of the pressures of the junctions marked in Fig. <ref>(d). 
It can be seen that, Eq. (<ref>)) has exactly the same form as the 2D SSH Hamiltonian introduced in Refs. <cit.> which confirms that the proposed network is an acoustic realization of the 2D SSH model. Thus, here we directly  bridge the interesting topological properties of the model to the acoustic realm.

At this point we emphasize the fact that the hopping coefficients in Eq. (<ref>) are directly given by the width of the channels in great contrast to TBA where these coefficients are derived as overlapping integrals of wavefunctions. Moreover, another remarkable advantage of our proposed methodology is that a large variety of discrete systems with desired coupling coefficients can be exactly mapped to an acoustic network.





We obtain the dispersion relation by solving the eigenvalue problem of Eq. (<ref>).  Note that according to our modeling there is only one  free hopping coefficient t since s=1-t. Using the values of  t=0.714 and L=0.125 m which correspond to the experimental setup, in Figure <ref>(a) we show the dispersion curves obtained by solving Eq. (<ref>). The band structure is characterized by four propagating branches (gray curves) and two full gaps around 550 Hz and 900 Hz. To verify the theoretical results we implement numerical calculations using a finite elements method for a network with  t=0.714, L=0.125 m and a channel width w_1/L=0.02 (not the same as in the experiments) and results are shown in Fig. <ref>(b). In this case, by comparing Figs. <ref>(a) and (b) we observe that the dispersion relations are almost identical confirming that the wave propagation is very well described by the discrete network model.
Furthermore, we also simulate a network which has the same channel width as our experimental setup i.e. w_1/L=0.2 and the corresponding dispersion curve is shown in Fig. <ref>(c). In this case, since our main assumption that w_1,2≪ L is not well satisfied, the theoretical model and numerical calculation exhibit a shift of between the two dispersion curves. However, in the low frequency regime and around the first band gap that we will focus our study, the discrete network model in Eq. (<ref>) describes the physical system quite accurately. 











One essential property of the 2D SSH model is the appearance of topological edge states when the system acquires a non-trivial topological phase which is achieved by tuning the hopping t. In the acoustic network, this transition takes place at the critical case when w_1=w_2 (t=0.5). When w_1>w_2 (t>0.5), the network is in the topological non-trivial phase, and can exhibit topological edge waves. Thus our experimental
setup with  t=0.714 is designed to fall in the non-trivial phase.
To predict the presence of edge modes theoretically,  we calculate the dispersion relation
of a supercell containing 8 unit cells with open ends (assuming zero pressure field), see SM. 
The resulting dispersion curves are depicted in Fig. <ref>(d).
It can be seen that inside each band gap, there is a degenerate edge wave branch marked with a black line, (see SM).  The dispersion curves  for the supercell obtained by numerical simulations using zero pressure at the boundaries, are shown in Fig. <ref>(e), and <ref>(f) corresponding to the same channel widths as panels (b) and (c) respectively.
Note that from now on we will use zero pressure boundary conditions in all simulations.
Both simulations confirm the appearance of an edge wave branch inside the band gaps
verifying our theoretical prediction.




Let us now  turn to the experimental realization of the SSH network:  its total size is 4 m × 2 m  
and it is constructed using building blocks of size L_1=0.1 m, and L_2=0.13 m, which leads to the air channels of width w_1=0.025 m, and w_2=0.01 m as shown in Fig. <ref>(a).
To emphasise the importance of the edge configuration, 
 two types of edges are simultaneously investigated:
 one supporting edge waves [in green in  Fig. <ref>(a)] and the other one with alternative blocks that does not support  edge waves [in red in  Fig. <ref>(a)].


We focus on the first band gap around ∼ 550 Hz as marked by the orange area in Figs. <ref>(d)-(f).  
We experimentally identify this bandgap and the results are shown in Fig. <ref>(b) where the measured pressure amplitude at each junction of line x=24 [Fig. <ref>(a)] as a function of frequency is plotted. For this experiment, the source is placed at location  A (see Fig. <ref>(a)). Experiments are compared with numerical results shown in Fig. <ref>(c), and both confirm the existence of a band gap in the frequency range from ∼ 540 to ∼ 620 Hz. A footprint of the edge waves also appears as bright spots inside the bandgap in Figs. <ref>(b)-(c) located at y=0 (green edge).















To better characterize the edge waves of the acoustic 2D SSH network we perform additional experiments using a source close
to the green edge [position B in Fig. <ref>(a)]. 




The normalised edge wave profiles measured at three different lines [x=8 (pink), x=18 (orange), and x=24 (blue) as indicated in Fig. <ref>(a)] of the network are presented in Fig. <ref>(a). The edge wave, which is excited using a source at  570 Hz, 
 is revealed as the acoustic field is localised on the green edge (y=0) and is decaying into the bulk. All the experimental profiles are found to be invariant along the x-axis and they exhibit the typical pattern provided by SSH models with sub-lattice symmetry <cit.>. Note that in our airborne experiment it is impossible to achieve an exact zero pressure boundary condition due to a slight leakage into free space.  However,  as shown in Fig. <ref>(b), exact zero pressure boundary conditions used in numerical simulations lead to the same profiles confirming the robustness of the system with respect to boundary conditions.





 
 




A characteristic field distribution measured at all the junctions located at x ≥ 24[Fig. <ref>(a)] is shown 
in Fig. <ref>(c) for a frequency of 582 Hz. The corresponding numerical results are also shown in Fig. <ref>(d), visualising 
the field distribution within the whole network, in good agreement with the experiment.
 We clearly observe that the acoustic field is only localised at the green edges [bottom 
and left of Fig. <ref>(c)-(d)] while it vanishes in the other two edges. This is a direct consequence of the particular design of the device which combines two different types of edges: the red edges see the bulk as trivial and the green ones as topological. 








In conclusion, by applying a 1D approximation in each connection of an acoustic network, we exactly mapped a continuous system to 
the recently proposed 2D SSH model. The latter although with zero curvature is known to support topological edge waves which we observed in this work using an airborne experimental setup. These results show that, despite the fact that the system is open to the free space, the edges are able to support localised waves.
This work provides an acoustic demonstration of topological edge waves in a very simple system that is the 2D SSH model and paves the way for the experimental study of other novel topological phase in 
acoustic systems, such as higher-order topological modes. 







This work has been funded by the APAMAS, Sine City LMac, and the Acoustic Hub projects.

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