Observation of topological phases without crystalline counterparts

Abstract

Topological phases have been extensively studied primarily in crystalline systems with translational symmetry. Recent theoretical studies, however, have demonstrated the existence of topological phases in quasicrystals that are absent in crystals. Despite numerous experimental observations of topological phases in various crystalline systems, observing these phases without crystalline counterparts remains challenging due to very complex models. Here, we design a practically realizable tight-binding model with nearest-neighbor hopping on the Ammann-Beenker quasicrystalline lattice. This model respects eight-fold rotational and chiral symmetries, resulting in a higher-order topological phase with eight zero-energy corner modes that have no crystalline counterparts. We experimentally explore the topological phase in an acoustic quasicrystal. Surprisingly, we also discover symmetry-protected zero-energy modes near the center of the quasicrystal in a topologically trivial phase, a phenomenon not seen in crystals. We further experimentally observe these modes in a topologically trivial acoustic quasicrystal. Our work represents the first experimental observation of topological phases in quasicrystals without crystalline counterparts, paving the way for the study of exotic topological physics in quasicrystals.

Figures (18)

• Figure 1: Quasicrystalline structure gives rise to topological phases without crystalline counterparts. (a) Illustration of the tight-binding model in Eq. (\ref{['H_TB']}) on an octagonal quasicrystal. Each vertex of the quasicrystal hosts a cell, depicted as brown-filled polygons, containing either eight, six, or four lattice sites. Connections between neighboring sites indicate the presence of hopping between these sites. The bonds shown as red lines denote sign-reversal hoppings. The red-filled circles near the corners represent the spatial distribution of the local DOS at zero energy, with the relative size of the circles indicating its magnitude. (b) Energy spectra of the tight-binding Hamiltonian in Eq. (\ref{['H_TB']}) on the quasicrystal with respect to the intra-cell hopping strength $t_0$. The system exhibits a topological regime with eight corner modes for $0<t_0<0.67$ and a trivial regime with two zero-energy modes for $t_0>4.15$. Between these regions, the bulk spectrum becomes gapless. Inset: the energy spectrum at $t_0=0.3$ with respect to the state index, highlighting the presence of eight zero-energy states. (c) Topological invariants $\chi_p$ as a function of $t_0$ with $p=0,\dots,4$. In the topological regime, $\chi_p=\pm N_b/8$, where $N_b$ is the number of boundary sites. For the quasicrystalline structure shown in (a), $N_b=40$.
• Figure 2: Observation of the topological phase in an acoustic quasicrystal. (a) Photograph of the fabricated acoustic quasicrystal sample. (b) Spatial distribution of the measured acoustic pressure field, summed over the 1832--1842 Hz frequency range. (c) Fourier spectrum of (b) in the $k_x$--$k_y$ space, which reveals the quasicrystalline feature of the states. (d) Measured spatial distribution of the acoustic pressure at the frequency of $1866.2$ Hz, showing strong peaks at eight corners, consistent with the theoretical prediction. Inset: simulated eigenfrequencies of the acoustic quasicrystal versus the state index. (e) Measured response spectra as a function of frequency detected at corner $D_1$ (red line), bulk $D_2$ (blue line), and edge $D_3$ (black line) given a source $S$ as highlighted in (a). The dashed line describes the simulated result for corner modes considering an acoustic loss $\alpha=0.008$, which closely matches the measured result (red line).
• Figure 3: Observation of the zero-energy modes in a topologically trivial acoustic quasicrystal. (a) Photograph of the fabricated acoustic quasicrystal sample in the trivial regime (see Supplemental Material Sec. 5 for construction details). (b) Simulated eigenfrequencies of the trivial quasicrystal versus the state index, displaying two degenerate states at the frequency of 1874.5 Hz within the energy gap. (c) Measured spatial distribution of the acoustic pressure at the frequency of 1874.5 Hz, showing strong peaks at the central cell, which agrees well with the theoretical results. (d) Response spectra as a function of frequency measured at a central position $D_1$ (red line) and a position away from the central cell $D_2$ (blue line) given a source $S$ as highlighted in (a). The measurement results agree well with the simulated one at the central cell by including an acoustic loss $\alpha=0.008$ (dashed line).
• Figure 4: Construction of the tight-binding model on an octagonal quasicrystalline lattice. (a) An AB tiling octagonal quasicrystalline lattice. (b)-(d) We first place a cell consisting of eight sites at each vertex [see (b)] followed by introducing two edges connecting these sites [see (c)]. In (d), all sites lacking inter-cell bonds are removed and the remaining sites within each cell are connected. (e) Illustration of the obtained tight-binding model. Here, we set $R_c=0.2$.
• Figure 5: Modification of certain hoppings opens the bulk energy gap. (a) A cell of seven sites is changed to a cell of four sites. (b) All grey polygons are transformed into even-sided shapes by shifting the blue inter-cell bonds. (c) The sign of an intra-cell hopping within each four-site cell is reversed (red line), and the magnitude of an intra-cell hopping in all eight-site cells---except the eight-site cell at the center---is increased (grey line). (d) The sign of an inter-cell hopping within a slender rhombuse structure is also reversed.