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Roton dispersion relations were firstly predicted by Landau and have been extensively explored in correlated quantum systems at low temperatures. Recently, the roton-like dispersion relations were theoretically extended to classical acoustics, which, however, have remained elusive in reality. Here, we report the experimental observation of roton-like dispersions in acoustic metamaterials with beyond-nearest-neighbour interactions at ambient temperatures. The resulting metamaterial supports multiple coexisting modes with different wavevectors and group velocities at the same frequency and broadband backward waves, analogous to the “return flow” termed by Feynman in the context of rotons. By increasing the order of long-range interaction, we observe multiple rotons on a single dispersion band, which have never appeared in Landau’s prediction or any other condensed-matter or classical-wave studies. Moreover, we have also theoretically proposed and experimentally observed multidirectional roton-like dispersion relations in a two-dimensional nonlocal acoustic metamaterial. The realization of roton-like dispersions in metamaterials could pave the way to explore novel physics and applications on quantum-inspired phenomena in classical systems.

Dirac cones (DCs) play a pivotal role in various unique phenomena ranging from massless electrons in graphene to robust surface states in topological insulators (TIs). Recent studies have theoretically revealed a full Dirac hierarchy comprising an eightfold bulk DC, a fourfold surface DC, and a twofold hinge DC, associated with a hierarchy of topological phases including first-order to third-order three-dimensional (3D) topological insulators, using the same 3D base lattice. Here, we report the first experimental observation of the Dirac hierarchy in 3D acoustic TIs. Using acoustic measurements, we unambiguously reveal that lifting of multifold DCs in each hierarchy can induce two-dimensional topological surface states with a fourfold DC in a first-order 3D TI, one-dimensional topological hinge states with a twofold DC in a second-order 3D TI, and zero-dimensional topological corner states in a third-order 3D TI. Our Letter not only expands the fundamental research scope of Dirac physics, but also opens up a new route for multidimensional robust wave manipulation.

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Chiral edge states that propagate oppositely at two parallel strip edges are a hallmark feature of Chern insulators which were first proposed in the celebrated two-dimensional (2D) Haldane model. Subsequently, counterintuitive antichiral edge states that propagate in the same direction at two parallel strip edges were discovered in a 2D modified Haldane model. Recently, chiral surface states, the 2D extension of one-dimensional (1D) chiral edge states, have also been observed in a photonic analogue of a 3D Haldane model. However, despite many recent advances in antichiral edge states and chiral surface states, antichiral surface states, the 2D extension of 1D antichiral edge states, have never been realized in any physical system. Here, we report the experimental observation of antichiral surface states by constructing a 3D modified Haldane model in a magnetic Weyl photonic crystal with two pairs of frequency-shifted Weyl points (WPs). The 3D magnetic Weyl photonic crystal consists of gyromagnetic cylinders with opposite magnetization in different triangular sublattices of a 3D honeycomb lattice. Using microwave field-mapping measurements, unique properties of antichiral surface states have been observed directly, including the antichiral robust propagation, tilted surface dispersion, a single open Fermi arc connecting two projected WPs and a single Fermi loop winding around the surface Brillouin zone (BZ). These results extend the scope of antichiral topological states and enrich the family of magnetic Weyl semimetals.

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Roton dispersion relations, displaying a pronounced “roton” minimum at finite momentum, were firstly predicted by Landau and have been extensively explored in correlated quantum systems at low temperatures. Recently, the roton-like dispersion relations were theoretically extended to classical acoustics, which, however, have remained elusive in reality. Here, we report the experimental observation of roton-like dispersions in acoustic metamaterials with beyond-nearest-neighbour interactions at ambient temperatures. The resulting metamaterial supports multiple coexisting modes with different wavevectors and group velocities at the same frequency and broadband backward waves, analogous to the “return flow” termed by Feynman in the context of rotons. Moreover, by increasing the order of long-range interaction, we observe multiple rotons on a single dispersion band, which have never appeared in Landau’s prediction or any other condensed matter study. The realization of roton-like dispersions in metamaterials could pave the way to explore novel physics and applications on quantum-inspired phenomena in classical systems.

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