# Structured sound tube with carbon nanotube-like topological edge states | Panda Anku

### Planar acoustic graphene sheet

We realize the acoustic counterpart of a SWCNT by rolling up an equivalent plate from a honeycomb arrangement of rigid rods (lattice constant and radius are a= 2.5cm and right= 0.25a ) as shown in Fig. 1a. The physics of the Acoustic Graphene Tube (AGT) can essentially be captured by its unrolled planar structure, which is why a topological analysis can be carried out in the geometry mentioned. To construct an effective acoustic Hamiltonian, we treat the plate as a honeycomb waveguide network (see Supplementary Information for details). Thus, after solving the wave equation and applying Bloch’s theorem, we get the following acoustic eigenvalue problem

$${{{{{{mathcal{H}}}}}}}}({{{{{{{bf{k}}}}}}}}){{{{{{{ bold symbol {psi }}}}}}}=E{{{{{{{boldsymbol{psi }}}}}}}},$$

(1)

Where (E=3cos({k}_{0}L))and ({{{{{{{mathcal{H}}}}}}}}}({{{{{{{bf{k}}}}}}}})=[0,g({{{{{{{bf{k}}}}}}}});{g}^{*}({{{{{{{bf{k}}}}}}}}),0]) with the off-diagonal term (g({{{{{{bf{k}}}}}}})=mathop{sum }nolimits_{l=1}^{3}exp (-i{{{ {{ {{bf{k}}}}}}}cdot {{{{{{{{boldsymbol{delta}}}}}}}}}}_{l})). The Vectors δl Connect Site A to its nearest neighbor Site B (see Supplementary Information for details). Interestingly, Eq. (1) is mapped exactly into graphs, considering only the nearest neighbor hopping effects. through expansion G(k) near Kpoint, ie k= K + δk the reduced Hamiltonian (delta {{{{{{{{mathcal{H}}}}}}}}}_{{{{{{{{rm{D}}}}}}}}}}) achieves the standard massless Dirac formulation (delta {{{{{{{{mathcal{H}}}}}}}}}_{{{{{{{{{rm{D}}}}}}}}}(delta {{{{{{{bf{k}}}}}}}})={upsilon }_{{{{{{{{{rm{D}}}}}}}}}(delta {k}_{x}{{{{{{{{boldsymbol{sigma}}}}}}}}}_{x}+delta {k}_{y}{{{{{{{ {boldsymbol{sigma }}}}}}}} {boldsymbol{sigma }}}}}}}}}_{y}))Where σx and σj are the Pauli matrices and ({upsilon }_{{{{{{{{rm{D}}}}}}}}}=sqrt{3}widetilde{a}/2) is the Dirac speed. The effective grating period (widetilde{a}) represents the lattice constant of the effective waveguide network. With the relation (widetilde{a} ; approx ; 0.931a)Fig. 1b shows good agreement with numerical calculations of the dispersion relation from the effective Hamiltonian.

The two achiral chair (AC) and zigzag (ZZ) topologies exhibit strikingly contrasting properties. To elucidate this experimentally, we construct a structured sheet, ie an acoustic crystal in a finite configuration, as shown in Fig. 1g. Figure 1c shows the calculated (black circles) and measured (colored contour) gapless AC dispersion relation, whose Zak phase (inset) predicts a trivial topology. To adjust the leaf interface, we define a new set of grid vectors as C and R With a1 × a2 =C × Ras shown in Figure 2a. Accordingly, the Bloch wave vector is k = kCc1+kRc2WithkC,R [0, 1), where c1 and c2 are the reciprocal lattice vectors. To calculate the band structures of the structured sheet, we use a ribbon-shaped unit cell as shown in Fig. 1d and f, where the Floquet periodic conditions are imposed at the lateral boundaries. In this case, the structure can be considered as a one-dimensional sonic crystal with a period of A. Accordingly, the wave vector along the edge direction is defined as k. The relation between k and kC can be expressed as k = 2πkC/A with kC [0, 1), where (A=a^{prime}=sqrt{3}a) for the AC edge and A = a for the ZZ edge. The corresponding degenerate eigenmode profiles (marked by a d in Fig. 1c) exhibit delocalized sound throughout the bulk. The measured dispersion relation was obtained through a Fourier transformation of the detected acoustic pressure fields (Methods). In contrast to the AC case, the ZZ interface hosts a non-zero quantized Zak phase that is accompanied by a topological non-trivial edge state as both experimental and numerical data show in Fig. 1e. Referring to this, Fig. 1f illustrates the computed surface excitations that are localized along the rigid boundary, which have also been confirmed with the finite-sheet computations where the ZZ interface adjacently boarders a rigid wall (Fig. 1g). The green line in this figure, marks the path along which the decaying intensity has been spectrally measured in Fig. 1h. Interestingly, the topologically protected edge state that has been launched by a point source at frequency f = 7.75 kHz, is not obtained by a prototypical breaking of the time-reversal, mirror, or inversion symmetry. Instead, the topology of the ZZ interface leads to a non-zero Zak phase whose continuous but finite width in momentum space, passes through the K or (K^{prime}) points of the one-dimensional (1D) BZ in an otherwise pristine lattice.

### Rolled-up structured sheet

The topology of the rolled-up AGT is determined by a chiral vector Ch = NC = na1 + ma2 and a translation vector T as shown in Fig. 2a. The greatest common divisor of the chiral index (n, m), denoted by N = gcd(n, m), represents the periodic numbers of the units along the circumference. Based on the six-fold symmetry of the underlying lattice, distinct tube geometries can be characterized by the integer pairs ((hat{n},hat{m})N) with (0le hat{m}le hat{n}). To illustrate the basic configurations under study, in Fig. 2b we depict the unit cells of two tubes, where the left panel shows the (1, 0)14-AGT with the ZZ edge and the right panel shows the (1, 1)14-AGT with the AC edge. Further, our numerical computations also show that the way the tubes are rolled up is not the only deciding ingredient to engineer a complete band gap. Its spectral width, as shown in Fig. 2c, appears to take discrete jumps with the number of circumferential unit cells. Specifically, for each unit number (N=3d,din {mathbb{N}}), the (1, 0)N-AGT remains gapless (red dots), whereas a band gap for the (1, 1)N-AGT never shows (blue dots) (see Supplementary Information for details). To give proof of this rule, we fabricate four AGTs with ZZ and AC edge terminations and measure their bulk dispersion. Two (1, 0)N-AGTs are constructed whose well-agreeing numerical and experimental band diagrams are shown in Fig. 2d and e. According to the rule in Fig. 2c, the measured (1, 0)14-AGT and (1, 0)15-AGT bands display clearly how a single circumferential increment of the units, leads to an acoustic semiconductor- and metal-like behavior, respectively. In other words, the (1, 0)14-AGT topology entails a complete band gap, whereas the (1, 0)15-AGT always remains gapless. In contrast, as shown in Fig. 2f and g, gapless dispersion relations have been observed for both AC tubes in accordance with the (1, 1)N-AGT predictions discussed in Fig. 2c.

### Observation of tubular edge states

In the tubular geometry, the topological picture changes in comparison to the sheet, in that only a discrete number of states fall into the nontrivial range of the band gap. As discussed earlier (Fig. 1), the nontrivial topological properties of the acoustic graphene sheet with a ZZ interface are guaranteed by the non-zero Zak phase in the range of 1/3 < kC < 2/3, where the wavenumber kC can take arbitrary values from 0 to 1. However, for the rolled-up AGT, the wavenumber is discretized to N values, induced by the inherent periodicity. Hence, the number of the topological edge states can be determined by

$${N}_{{{{{{{{rm{edge}}}}}}}}}=mathop{sum }limits_{mu=0}^{N-1}|upsilon ({k}_{{{{{{{{rm{C}}}}}}}}})|,,{{{{{{{rm{with}}}}}}}},{k}_{{{{{{{{rm{C}}}}}}}}}=frac{mu }{N},{{{{{{{rm{and}}}}}}}},mu in [0,; 1,cdots ,,; N-1],$$

(2)

Where υ( kC) is the winding number (see Supplementary Information for details). A table with possible numbers of topological boundary states in (1, 0) N-AGTs, see Supplementary Information. Figure 3a shows a 3D printed finite (1, 0)14 AGT with 60 ring units (Fig. 2b) stacked along the tube axis and capped by a rigid cap. According to theory, five topological edge states are to be expected within the band gap (see additional information). Sound from a loudspeaker is directed into the structured tube in close proximity to the tube termination to excite the states of interest, while condenser microphones are deployed at the designated locations to measure the local pressure fields (see Methods section). First we calculate the natural frequencies of the AGT as shown in Fig. 3b. According to Eq. (2) five topological edge states (red dots) are found occupying the band gap from 7.66 to 7.79 kHz in a spectrum of bulk states (grey circles). We then experimentally measure the spectral response of the detected acoustic intensity for those states represented by colored peaks in Fig. 3c. Amid low-intensity waves traversing the entire tubular geometry (grey), a prominent (red) peak is observed in the spectrum, centered at 7.73 kHz, originating from the adjacent five edge states (I–V). In particular, the degenerate edge states I and II, whose eigenmodes, as shown in FIG. 3d, have an acoustic edge limitation with considerable penetration lengths. In contrast, the last three states (III–V), of which the first two are degenerate, show a much stronger localization at the end, where the sound is just able to decay over two unit rings. Finally, to visualize the full circumference of the topological tube, we scan the pressure field along the tube axis (green dashed line in Fig. 3a) in a frequency window from 6 to 9 kHz as shown in Fig. 3e. The bulk pressure hotspots resemble standing wave formations, but within the bandgap, the adjacent five states around 7.73 kHz show their edge localization due to pressure damping along the tube. In fact, as discussed in Figure 3d, the first two degenerate states (I and II) are on f1= 7.688 kHz show weak edge confinement in contrast to the highly concentrated edge states (III–V).f2= 7.791 kHz, as shown by the measured inset of Fig. 3e. In addition, we further show that the AGT edge states can be remotely excited from a distant source despite being mixed with broadband white noise (see Supplementary Information).