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1 Acoustic toological insulator and robust one-wa sound transort Cheng e 1,, Xu Ni 1, ao Ge 1, Xiao-Chen Sun 1, Yan-Bin Chen 1, Ming-ui Lu 1,, Xiao-Ping Liu 1,, and Yan-Feng Chen 1, 1 National Laborator of Solid State Microstructures& Deartment of Materials Science and Engineering, Nanjing Universit, Nanjing, Jiangsu 10093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing Universit, Nanjing, Jiangsu 10093, China Corresondence and request for materials should be addressed to M.. Lu (luminghui@nju.edu.cn), X. P. Liu (xliu@nju.edu.cn) and Y. F. Chen (fchen@nju.edu.cn). This PDF file includes: Sulementar Text : Theoretical model Page -4 Figs. S1 to S Page 5-14 Sulementar Text : Band inversion for hotonic cases based on accidentall degenerated double Dirac cone & Fig. S Page Sulementar Text : Double Dirac cone based on zone folding mechanism & Fig. S Page Full Reference List Page 19 NATURE PYSICS 1

2 Theoretical model ere, k P method is adoted to calculate toological order (sin Chern number) of our hononic grahene, which has been successfull alied in designing dielectric based hotonic toological insulators 1. Based on the C 6v smmetr of our hononic grahene, the basis set used in our analsis is chosen to be [,, d, d ]. Then, a 4 4 amiltonian near the Brillouin zone (BZ) x x x center (Γ oint) can be described as ( k) ( k), (S1) 0 where 0 onl has diagonal terms { M, M, M, M} (M is relative to the middle ga of and d states) corresonding to two two-fold degenerate states with eigenvalues,,, d, d x. x x Awa from Γ oint, the eigenvalue is{ E, E, E, E } { M Nk, M Nk, M Nk, M Nk }. d d ( k) reresents the erturbation term with k awa from Γ oint, which can be exressed as ( k) xx x xdx xdx x dx dx dxx dx dxdx dxdx dx x dx dx dx dx dx, (S) In Eq. (S), we can get the interaction of states according to odd or even arit of satial inversion relative to x or axis of the unit cell as xx x x k P d d kp kp d d kp E Dk Fk x x x x x x x x x k P x x E Ed kp d d kp kp d d kp E Fk Dk x x x x k P x E Ed kp d d kp kp d d kp E E x x x x x x d Gkxk. (S3) Similarl, we can also get NATURE PYSICS

3 dxdx dxdx dx dx dxdx d x x x x d x x x x d x d x E Dk Fk d x E Fk Dk d x SUPPLEMENTARY INFORMATION k Pd k Pd Gk k x x x x Ak Ak kpd Ak k Pd Ak x. (S4) It should be noticed that the arameter A in Eq. (S4) is ure imaginar. Then, the new basis can be hbridized as [ x i, dx idx, x i, dx idx ]/ with wave vectors 1 k k ik. After omitting high order erturbation terms, the amiltonian can be rewritten as x ( k) ( DF N) ( DF N) Ak M k M k Ak * ( DF N) M k Ak Ak ( DF N) M k * (S5) Eq. (S5) can be further simlified as c, (S6) with (k) = A( k ˆ k ˆ ) ( M Bk ) ˆ, where ˆ is Pauli matrices and B ( D F N)/. x x z This reresentation of our sstem amiltonian described as Eq. (S6) is similar to the amiltonian in BZ model of quantum sin hall effect (QSE) in a two-dimensional electron toological insulator. This new basis can then reresent acoustic sin+ and sin-, enabling the acoustic counterart of QSE. In addition, in order to describe the toolog order of our sstem, we can also calculate the sin Chern numbers via 3 1 Cs [sgn( B ) sgn( M )]. (S7) It is worth noting that Cs 0 if BM<0, corresonding to an ordinar hase, and Cs sgn( B) if BM>0, corresonding to a toological hase. B decreasing the filling factor of our hononic grahene, the sgn( BM ) in our model can continuousl var from negative to ositive, indicating NATURE PYSICS 3

4 the occurrence of band inversion and further leading to toological transition. The exact transition oint corresonds to the case when the four energ bands touch forming a double Dirac cone ( E Ed ). Thus, the sin Chern number near the double Dirac cone can be easil determined as C s =-1, +1, -1, +1 from lower to uer four bands 1,4. 4 NATURE PYSICS

5 SUPPLEMENTARY INFORMATION Fig. S1. Band structure of grahene-like hononic grahene with different filling factors: r/a=0.45, and 0.3. Center to center distance of the nearest neighboring rods is set to be a=1cm. Insets show the hononic states at the BZ center corresonding to different arities. NATURE PYSICS 5

6 Numericall calculated Berr curvature and sin Chern numbers We have shown non-zero sin Chern numbers via k P method in our revious version. Although the rigorous BZ model concerns two totall degenerate electron sins, it is still alicable in our acoustic case with two seudosins. The band inversion haens at the center of Brillouin zone and the non-zero Berr curvature can thus be exected to aear near oint, where the two lower (or uer) disersions have a ver slight difference. We also can numericall calculate Berr curvature b rojecting the Bloch state of the two disersion bands into two acoustic seudosin states via the similar k P method as we indicated in theoretical model section of the sulementar materials. For examle, the numericall calculated Berr curvature for the two lower bulk bands of TPC (r/a=0.3) are shown in Fig. S. The corresonding sin Chern numbers can be obtained b integration of Berr curvature (BC) over the first Brillouin zone as 1 Cs BCd k. We get C s =-1 and +1 for two lower BZ bands, which match the results obtained via our theoretical model 3. Fig. S. The Berr curvature of two lower bands of TPC (r/a=0.3) corresonding to the second and the third bands in Fig. S1. 6 NATURE PYSICS

7 SUPPLEMENTARY INFORMATION Fig. S3. Filling factor induced band inversion. Center to center distance of the nearest neighboring rods is set to be a=1cm. The double Dirac cone aears at r=0.398 cm. Red and blue dots reresent the eigen frequencies of the double degenerate and d states at BZ center, resectivel. Dashed lines guide for the ee. NATURE PYSICS 7

8 Fig. S4. Measured transmission sectra of 1-laer TPC and OPC along Γ-K direction. An obvious band ga can be found in our frequenc range of interest (from kz to 0.53 kz). Center to center distance of the nearest neighboring rods is set to be a=1cm. The transmission for P13 of Fig. 3D and P4 of Fig. 3E in main text clearl resemble the tical transmission roert in the resence of bulk band ga. If desired, a larger transmission extinction in the band ga could be obtained b using more hononic laers. 8 NATURE PYSICS

9 SUPPLEMENTARY INFORMATION Fig. S5. Projected band structure of an ordinar band-ga-guiding waveguide b removing one row of rods (r/a=0.45). Four toological trivial waveguide and edge modes can be observed in the bulk band ga. NATURE PYSICS 9

10 Fig. S6. Contrast ratio for toological waveguide (uer anel) and ordinar band-ga-guiding waveguide (lower anel). Contrast ratio is defined as (T d -T o )/(T d +T o ), where T d and T o are measured transmission with defect and without defect. Red, blue and green lines corresond to the case with a cavit, a disorder and a bend in the waveguide. In the toological waveguide (uer anel), desite a ver small variation ossible due to the intrinsic loss and measurement noise, the exerimentall derived contrast ratio is ver close to zero eseciall for the frequenc range near the center of the band ga (dashed line guides for the ee), indicating the backscattering-immune and robust roagating roert. In the ordinar waveguide (lower anel), the contrast ratio, however, fluctuates from -1 to 1, suggesting defect-sensitive transmission. Shadow regions reresent the bulk band ga. 10 NATURE PYSICS

11 SUPPLEMENTARY INFORMATION Fig. S7. Measured transmission sectra of the toological waveguide with various forms of cavities. Black line is the transmission without an defects. Red, blue, green, can and magenta lines reresent the transmission b removing 1,, 3, 4 and 6 rods, resectivel. Although near the band edge the transmission is slightl decreased with large cavit (ossibl due to the combination of longer detour distance and intrinsic loss), the robust transmission revails in these scenarios. NATURE PYSICS 11

12 Fig. S8. Projected band structure of grahene-like hononic crstals for underwater sound. The same structure as that aeared in main text (r/a=0.3 TPC and r/a=0.45 OPC) is used excet for the change of the background from air to water (densit is 1000 kg/m 3 and seed is 1490 m/s). In this case, a double Dirac cone will aear near r/a= The toological henomenon will take lace around 90 kz. 1 NATURE PYSICS

13 SUPPLEMENTARY INFORMATION Fig. S9. Zoom-in acoustic edge states of Fig. 1c in main text. There exists a tin ga between two edge states at k =0 with a bandga of kz. The relative bandga is less than 0.5%, which is too small to be noticed in our current exeriments (Figs. d, e. 3c and 3d) where an increment 0.01 kz. NATURE PYSICS 13

14 Fig. S10. Reduced ga b adjusting the boundar condition. The tin ga can be further reduced b changing the filling factor of a row of clinders to be r/a=0.1. The relative bandga is less than 0.0%. 14 NATURE PYSICS

15 SUPPLEMENTARY INFORMATION Band inversion for hotonic cases based on accidentall degenerated double Dirac cone In rincile, our wa of realizing a double Dirac cone, via accidentall degenerac resulted from an aroriate filling factor of a simle honecomb lattice, could be generalized to design hotonic double Dirac cone. owever, the major difficulties lie in the limited range of otical index and imedance contrast between the background materials and the constituent materials, which is usuall less than 10 for ure dielectric materials based hotonic crstals. Such a small contrast is not enough for realizing accidental double Dirac cone using our mechanism. owever, in acoustic, the index and imedance contrast can var wildl because the large densit as well velocit difference can commonl exist between two acoustic materials. For examle, acoustic refractive index contrast between stainless steel and air [343/6010 vs. 1] is larger than 17 and acoustic imedance (densitvelocit) contrast between them [ vs ] is larger than We find that such a large index and imedance contrast is the ke in obtaining an accidental double Dirac cone. For examle, if we were to design a hotonic crstal with exotic materials arameters such as b 7800 and b 1/ sqrt(6010) for background and r 1 and r 1 for air holes as shown in Fig. S11, a double Dirac cone and a band inversion rocess for TM olarization as a result of different filling factors could be realized. We also note that in different configurations, e.g., in a triangle lattice, accidentall double Dirac cone in ure dielectric hotonic crstals ( 1) ma be realized with ring-shaed constituents (b changing two geometr arameters, i.e., the ring s inner radius and outer radius) 6. NATURE PYSICS 15

16 Fig. S11. Photonic band structure of TM olarization with different filling factors: r/a=0.41, and 0.37 (honecomb lattice). Insets show the electric field at the BZ center, revealing different arities. The ermittivit and ermeabilit of background are b 7800 and b 1/ sqrt(6010), resectivel. The rods are air holes with r 1 and 1. r 16 NATURE PYSICS

17 SUPPLEMENTARY INFORMATION Double Dirac cone based on zone folding mechanism. In our model, the hononic accidental double Dirac cone is constructed b engineering the nearest neighbor couling via tuning the filling factor in a honecomb lattice with C 6v smmetr. According to grou theor, the C 6v smmetr ossesses two two-dimensional irreducible reresentations, which can then be leveraged to construct a double Dirac cone b forming a four-fold accidental degenerac. This accidental nature of such a kind of degenerac means that it can be easil lifted b again tuning the filling factor, leading to a band inversion rocess. As a comarison, the double Dirac cone realized in comosite honecomb lattice is originated from the energ band folding mechanism 1. The schematic for such a folding mechanism is illustrated in Fig. S1a b examining two choices of unit cell and their associated Brillouin zone (BZ). The first is a hexagonal conventional (non-rimitive) unit cell (marked as I ) and the second is a rhombic rimitive unit cell (marked as II ). Note that this hexagonal unit cell is three times larger than the rhombic rimitive one (left anel of Fig. S1a). Consequentl, the first BZ of the rhombic rimitive unit cell is three times larger than that of the hexagonal one (right anel of Fig. S1a). The I M I of the rhombic rimitive unit cell is equivalent to I M I, KII M II and K ' II M (red lines) of the hexagonal unit cell. Therefore, the double Dirac cone of hexagonal unit cell I is formed b folding two two-fold degenerate oints at K II and ' K II (of rhombic rimitive unit cell) to the center of BZ. For examle, Fig. S1b and Fig. S1c show the energ bands calculated with the hexagonal unit cell and the rhombic rimitive unit cell with arameter r=1/3a, and rod The six energ bands along I M I direction of the rhombic rimitive unit cell in Fig. S1b come from the three-folded energ bands along II M I K II M II direction of the hexagonal unit cell in Fig. S1c. In this case, the degenerac at K ( K ) resulted from b the II ' II C 6v smmetr of the rhombic rimitive unit cell gives rise to a double Dirac cone b zone folding. Since this double Dirac cone is NATURE PYSICS 17

18 determined b the lattice smmetr of the rhombic rimitive unit cell, its degenerac cannot be broken b just changing the clindrical radius or ermittivit. For examle, Fig. S1d shows that the double Dirac cone still exists when the radius is changed to r=0.4a. An effective wa to lift the degenerac is to change the lattice smmetr so as to make the hexagonal conventional (non-rimitive) unit cell become the rimitive unit cell 1. Fig. S1. Double Dirac cone based on zone folding mechanism. a, the schematic of BZ folding mechanism. The shadow regions reresent the equivalent areas after folding. Photonic band structure of TM olarization using, b, a hexagonal conventional (non-rimitive) unit cell, c, a rhombic rimitive unit cell in a honecomb lattice with r/a=1/3, and rod d, band structure with r/a=0.4 using the hexagonal unit cell. Insets show the electric field at the BZ center. 18 NATURE PYSICS

19 SUPPLEMENTARY INFORMATION References: 1. Wu, L.-. & u, X. Scheme for achieving a toological hotonic crstal b using dielectric material. Phs. Rev. Lett. 114, 3901 (015).. Bernevig, B. A., ughes, T. L. & Zhang, S.-C. Quantum sin all effect and toological hase transition in gte quantum wells. Science 314, (006). 3. Li,. et al. Stabilization of the Quantum Sin all Effect b Designed Removal of Time-Reversal Smmetr of Edge States. Phs. Rev. Lett. 110, 6680 (013). 4. Mousavi, S.., Khanikaev, A. B. & Wang, Z. Toologicall rotected elastic waves in hononic metamaterials. Nat. Commun. 6, 868 (015). 5. Chen, Z.-G. et al. Accidental degenerac of double Dirac cones in a hononic crstal. Scientific Reorts 4, 4613 (014). 6. Li, Y. & Mei, J. Double Dirac cones in two-dimensional dielectric hotonic crstals. Otics exress 3, (015). NATURE PYSICS 19