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The Berry phase associated with energy bands in crystals can lead to quantized quantities, such as the quantization of electric dipole polarization in an insulator, known as a one-dimensional (1D) topological insulator (TI) phase. Recent theories have generalized such quantization from dipole to higher multipole moments, giving rise to the discovery of multipole TIs, which exhibit a cascade hierarchy of multipole topology at boundaries of boundaries: A quantized octupole moment in the three-dimensional (3D) bulk can induce quantized quadrupole moments on its two-dimensional (2D) surfaces, which then produce quantized dipole moments along 1D hinges. The model of 2D quadrupole TI has been realized in various classical structures, exhibiting zero-dimensional (0D) in-gap corner states. Here we report on the realization of a quantized octupole TI on the platform of a 3D acoustic metamaterial. By direct acoustic measurement, we observe 0D corner states, 1D hinge states, 2D surface states, and 3D bulk states, as a consequence of the topological hierarchy from octupole moment to quadrupole and dipole moment. The critical conditions of forming a nontrivial octupole moment are further demonstrated by comparing with another two samples possessing a trivial octupole moment. Our work thus establishes the multipole topology and its full cascade hierarchy in 3D geometries.
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Electrical transport in three dimensional topological insulators(TIs) occurs through spin-momentum locked topological surface states that enclose an insulating bulk. In the presence of a magnetic field, surface states get quantized into Landau levels giving rise to chiral edge states that are naturally spin-polarized due to spin momentum locking. It has been proposed that p-n junctions of TIs in the quantum Hall regime can manifest unique spin dependent effects, apart from forming basic building blocks for highly functional spintronic devices. Here, for the first time we study electrostatically defined n-p-n junctions of bulk insulating topological insulator BiSbTe$_{1.25}$Se$_{1.75}$ in the quantum Hall regime. We reveal the remarkable quantization of longitudinal resistance into plateaus at 3/2 and 2/3 h/e$^2$, apart from several partially developed fractional plateaus. Theoretical modeling combining the electrostatics of the dual gated TI n-p-n junction with Landauer Buttiker formalism for transport through a network of chiral edge states explains our experimental data, while revealing remarkable differences from p-n junctions of graphene and two-dimensional electron gas systems. Our work not only opens up a route towards exotic spintronic devices but also provides a test bed for investigating the unique signatures of quantum Hall effects in topological insulators.
The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topological corner states, as well as three-dimensional (3D) systems having one-dimensional (1D) topological hinge states. Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamond-lattice acoustic metamaterial. The bulk acoustic bandstructure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedron-like structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices.
Discovery of novel topological orders of condensed matters is of a significant interest in both fundamental and applied physics due to the associated quantum conductance behaviors and unique symmetry-protected backscattering-immune propagation against defects, which inspired similar fantastic effects in classical waves system, leading to the revolution of the manipulation of wave propagation. To date, however, only few theoretical models were proposed to realize acoustic topological states. Here, we theoretically and experimentally demonstrate a two dimensional acoustic topological insulators with acoustic analogue of quantum spin Hall Effect. Due to the band inversion mechanism near the double Dirac cones, acoustic one-way pseudospin dependent propagating edge states, corresponding to spin-plus and spin-minus, can be observed at the interface between two graphene-like acoustic crystals. We have also experimentally verified the associated topological immunity of such one-way edge states against the different lattice defects and disorders, which can always lead to inherent propagation loss and noise. We show that this unique acoustic topological phenomenon can offer a new promising application platform for the design of novel acoustic devices, such as one-way sound isolators, acoustic mode switchers, splitters, filters etc.
High-order topological insulators (TIs) are a family of recently-predicted topological phases of matter obeying an extended topological bulk-boundary correspondence principle. For example, a two-dimensional (2D) second-order TI does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D TI, but instead has topologically-protected zero-dimensional (0D) corner states. So far, higher-order TIs have been demonstrated only in classical mechanical and electromagnetic metamaterials exhibiting quantized quadrupole polarization. Here, we experimentally realize a second-order TI in an acoustic metamaterial. This is the first experimental realization of a new type of higher-order TI, based on a breathing Kagome lattice, that has zero quadrupole polarization but nontrivial bulk topology characterized by quantized Wannier centers (WCs). Unlike previous higher-order TI realizations, the corner states depend not only on the bulk topology but also on the corner shape; we show experimentally that they exist at acute-angled corners of the Kagome lattice, but not at obtuse-angled corners. This shape dependence allows corner states to act as topologically-protected but reconfigurable local resonances.
The influence of dephasing on the quantum spin Hall effect (QSHE) is studied. In the absence of dephasing, the longitudinal resistance in a QSHE system exhibits the quantum plateaus. We find that these quantum plateaus are robust against the normal dephasing but fragile with the spin dephasing. Thus, these quantum plateaus only survive in mesoscopic samples. Moreover, the longitudinal resistance increases linearly with the sample length but is insensitive to the sample width. These characters are in excellent agreement with the recent experimental results [science {bf 318}, 766 (2007)]. In addition, we define a new spin Hall resistance that also exhibits quantum plateaus. In particular, these plateaus are robust against any type of dephasing and therefore, survive in macroscopic samples and better reflect the topological nature of QSHE.
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