Experimental demonstration of acoustic semimetal with topologically charged nodal surface

Model Hamiltonian of the charged nodal surface

In a 3D system, the total Berry flux through any closed surface is quantized and defines the first Chern number of the closed surface. When the Chern number is nonzero, there must be band degeneracy that carries the charge of Berry flux for a system with time reversal symmetry. Weyl points (4) are one class of such band degeneracy points with nonzero Berry charges. A simple Hamiltonian of the Weyl point with charge +1 can be written asĤ=qxσx+qyσy+qzσz(1)where q_x, q_y, and q_z are three wave vector components measured from the Weyl point, and σ_x, σ_y, and σ_z are Pauli matrices. Figure 1A illustrates the Berry flux density emitted by the Weyl point, with a total Berry flux of 2π.

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However, Weyl points are not the only kind of topologically charged geometric objects in momentum space. Here, we introduce a nodal surface that also carries a nonzero charge. A simple Hamiltonian of the charged nodal surface (with charge +1) is given byĤ=qz(qxσx+qyσy)+qzσz(2)

The eigenvalues of the effective Hamiltonian, E±=±qz1+qρ2 (with qρ=qx2+qy2), are degenerate at q_z = 0 and change linearly with q_z when away from the nodal surface, indicating the presence of a nodal surface (25, 26). The subscripts “+” and “−” correspond to the upper and lower bands, respectively. Analytically, the Hamiltonian in Eq. 2 gives rise to the Berry flux distribution of the nodal surfaceB±=∓12(1+qρ2)3/2 ẑ(3)which emits a Berry flux of 2π in total. As illustrated in Fig. 1B, now the Berry flux concentrates around q_ρ = 0 and decays quickly away from it. In contrast, the Berry flux of a Weyl point originates from a singular point (the Weyl point) (Fig. 1A), even though they both have the same charge of Berry flux. Nodal surfaces can also have other integer charges of Berry flux, i.e., they belong to ℤ classification, as discussed in detail in section S1. Note that for a general form of the charged nodal surface, the Berry flux is no longer limited to the z direction as in Eq. 3.

Tight-binding model

The Hamiltonian of Eq. 2 can be embedded in a simple tight-binding model. The realizations of nodal surfaces protected by different symmetries have been theoretically discussed in very recent works (24–26). None of these works, however, showed a nodal surface that carries a nonzero ℤ charge of Berry flux. Here, we consider a system exhibiting both time reversal symmetry T and a twofold screw symmetry along the z direction C˜2z. We define a compound anti-unitary symmetry operator G2z≡TC˜2z, which, in real space, acts asG2z:(x,y,z,t) ↦ (−x,−y,z+h/2,−t)(4)

Here, h represents the lattice constant along the z direction. It is easy to check that G2z2=−1 for a Bloch wave function on the k_z = π/h plane with arbitrary k_x and k_y, where k_x, k_y, and k_z are the wave vectors along the x, y, and z directions, respectively. Therefore, k_z = π/h forms a nodal surface due to a Kramers degeneracy on this surface (26). We emphasize here that nodal surface being protected by G_2z is a symmetry argument only and does not depend on the detailed form of the Hamiltonian. To further construct a charged nodal surface, one has to break either the inversion symmetry or time reversal symmetry or both to achieve a nonzero Berry flux (28). Therefore, the idea to construct a charged nodal surface is to consider a system with G_2z symmetry and properly break inversion or time reversal symmetry to get the nodal surface charged. In this work, we implement inversion symmetry breaking.

Figure 1C shows a simple tight-binding model with G_2z symmetry. Two sublattices (red and blue spheres) are located on the z = 0 and z = h/2 planes. When projected to the x-y plane, the two sublattices are respectively projected to the A and B sites of a hexagonal lattice with a lattice constant a. There is a coupling t_c (cyan bond) between neighboring A (or B) sites and a coupling t₀ (black bond) between neighboring A and B sites. Both t_c and t₀ are real, as required by time reversal symmetry. In addition, this system exhibits G_2z symmetry and breaks inversion symmetry. According to the above analysis, this tight-binding model should have a nodal surface at k_z = π/h. This is confirmed by the calculated band structure (see section S2).

In addition to the nodal surface, we also note the existence of Weyl points at the K and K′ points of the bulk Brillouin zone. The charges of these two Weyl points are identical due to time reversal symmetry. Since the total charges inside the Brillouin zone must vanish (29), the charges of Weyl points must be compensated by the charge of the nodal surface, as these points and the nodal surface are the only band-degenerate features in the Brillouin zone. Hence, we conclude that the nodal surface here should also be charged. This has been directly verified by calculating the Chern number (see section S2). The charge distribution of this tight-binding model with t_c = −0.1 and t₀ = −0.6 is shown schematically in Fig. 1D, where the orange spheres and the cyan surface label the Weyl points and nodal surface with charges +1 and −2, respectively. As mentioned above, the Berry flux of a nodal surface actually concentrates around some special points in the momentum space (Fig. 1B). In this tight-binding model, the Berry flux of the nodal surface can be well captured by two charge −1 nodal surface Hamiltonians located at H ≡ (4π/3a,0, π/h) and H^′ ≡ (−4π/3a,0, π/h), respectively. To see this point, we expand the Hamiltonian near H and H^′ with respect to q_x, q_y, and q_z, and keeping to the lowest order, the Hamiltonian becomesĤ=3tcI+32[aht0qz (±qxσx−qyσy)∓ 6htcqzσz](5)where (q_x, q_y, q_z) represent the wave vector measured from H or H^′, I is the 2 × 2 identity matrix, and the upper and lower signs represent the Hamiltonian near H and H^′ points, respectively. Equations 5 and 2 share a similar form up to some constants and a unit matrix. We, hence, conclude that the total charge of the nodal surface is 2 sgn (t_c), which is independent of t₀. It is worth pointing out that for a more general system beyond the above tight-binding model, G_2z symmetry only protects the presence of a nodal surface at k_zh = π and does not guarantee the existence of nonzero charge. In section S3, we demonstrate a topological phase transition, where a nodal surface changes its charge by emitting or absorbing an integer number of Weyl points, through controlling the strength of the next nearest layer couplings. We also note that the nodal surface is stably charged before this topological transition, provided that G_2z symmetry is preserved.

Acoustic semimetal with a charged nodal surface

The tight-binding model can be implemented directly in a physical structure, as we have shown in (30). We note that the charged nodal surface is not limited to the tight-binding model. To simplify the experimental demonstration, below we show that a suitably designed 3D phononic crystal can host a charged nodal surface. Figure 2A shows the unit cell geometry of this phononic crystal, in which the solid ingredient (colored) is designed fully connected for the convenience of practical sample fabrication. The phononic crystal is arranged by a hexagonal lattice in the x-y plane and has a spatial symmetry C˜2z as required. In particular, the slanted cyan cylinders are used to break the inversion symmetry. Figure 2B presents the bulk band structure of the crystal along several high symmetry directions of the reciprocal space (see Fig. 1D). We focus on the lowest two bands. As expected, double degeneracy emerges in the whole momentum surface k_z = π/h, of which the degenerate bands split linearly (e.g., see Γ-A direction). The degenerate bands are very flat, which spans from 9.75 to 9.79 kHz. Similar to the tight-binding model, in addition to the nodal surface, a node point of frequency 9.43 kHz is observed at K (K′) point, around which the dispersion curves are linear in all directions. Our further calculations confirm that the charge distributions of the Weyl points and the nodal surface are the same as the above tight-binding model (as labeled in Fig. 1D). The lowest two bands are isolated from higher ones, which facilitate the experimental detection of the topologically nontrivial surface states. To further confirm the topological nature of the nodal surface, we consider a phononic crystal slab of finite thickness along the y direction and calculate the in-plane dispersions for the slab surfaces XZ₁ and XZ₂ imposed with rigid boundary conditions. The detailed surface truncations are specified in Fig. 2C (right panel). As exemplified by the momentum k_z = 0.5π/h, the in-plane dispersion of the slab (Fig. 2C, left panel) exhibits one gapless one-way edge mode for each of the XZ₁ and XZ₂ surfaces. As there is no other band degeneracy except for the Weyl point and the nodal surface, the existence of topological edge states justifies the nontrivial topology of the nodal surface.

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Experimental demonstrations

The topologically nontrivial surface states are confirmed in our airborne sound experiments. Figure 3A shows the experimental setup. The phononic crystal slab has been fabricated precisely by 3D printing technique. Additional hard cover plates are integrated on the XZ₁ and XZ₂ surfaces to emulate the rigid boundary conditions involved in our full-wave simulations. Note that the XZ₁ and XZ₂ surfaces cannot be related by the spatial symmetry and thus offer two comparative surface dispersions measured from a single sample. To excite and detect the pressure distribution on a given surface, a rectangular lattice of circular holes has been perforated through the cover plate. A point-like sound source is positioned at the bottom left corner of the sample surface (see Materials and Methods), which excite mostly the states propagating toward the top and right. This treatment gives a relatively long distance of sound propagation and relaxes the finite size effect to some extent. During our measurement, all the holes on the substrate are sealed except for the one reserved for the sound probe. By Fourier transforming the measured surface field, we can map out the in-plane dispersion at any desired frequency. Figure 3B shows the dispersions for the XZ₁ surface, measured at the frequencies of Weyl point (9.43 kHz, left panel) and nodal surface (9.75 kHz, right panel). In addition to the projected bulk bands (gray regions), the measured surface arcs (bright color), on which the surface acoustic states carry specific propagation directions (labeled by arrows), agree well with those simulation results (white curves) for both frequencies. The band broadening comes from the finite size effect. Similar agreements are also observed in Fig. 3C for the XZ₂ surface. Note that this new surface allows the presence of some additional topologically trivial surface states (black curves) near the surface Brillouin zone boundary k_z = π/h. Comparing Fig. 3B with Fig. 3C, both the topological surface arcs start from the projection of Weyl point at 9.43 kHz but end by apparently different momenta at the nodal surface frequency 9.75 kHz. This points out a major difference from Weyl semimetals, wherein the Fermi arcs are always pinned at the projection of the Weyl points independent of the boundary condition. In contrast, the Fermi arcs in our system connect the projections of the Weyl points and the nodal surface. The end point of the Fermi arc on the nodal surface can shift depending on the boundary condition. Last, in Fig. 3D, we present the measured frequency dispersions for k_z = 0.5π/h in the bulk gap, which reproduce well the numerical data for both surfaces again and demonstrate the one-way property of these chiral edge states (10, 14).

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