Observation of an acoustic topological Euler insulator with meronic waves.

Topological band theory has conventionally been concerned with the topology of bands around a single gap. Only recently non-Abelian topologies that thrive on involving multiple gaps were studied, unveiling a new horizon in topological physics beyond the conventional paradigm. Here, we report on the first experimental realization of a topological Euler insulator phase with unique meronic characterization in an acoustic metamaterial. We demonstrate that this topological phase has several nontrivial features: First, the system cannot be described by conventional topological band theory, but has a nontrivial Euler class that captures the unconventional geometry of the Bloch bands in the Brillouin zone. Second, we uncover in theory and probe in experiments a meronic configuration of the bulk Bloch states for the first time. Third, using a detailed symmetry analysis, we show that the topological Euler insulator evolves from a non-Abelian topological semimetal phase via. the annihilation of Dirac points in pairs in one of the band gaps. With these nontrivial properties, we establish concretely an unconventional bulk-edge correspondence which is confirmed by directly measuring the edge states via. pump-probe techniques. Our work thus unveils a nontrivial topological Euler insulator phase with a unique meronic pattern and paves the way as a platform for non-Abelian topological phenomena.

Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems.

While a significant fraction of topological materials has been characterized using symmetry requirements1-4, the past two years have witnessed the rise of novel multi-gap dependent topological states5-9, the properties of which go beyond these approaches and are yet to be fully explored. Although already of active interest at equilibrium10-15, we show that the combination of out-of-equilibrium processes and multi-gap topological insights galvanize a new direction within topological phases of matter. We show that periodic driving can induce anomalous multi-gap topological properties that have no static counterpart. In particular, we identify Floquet-induced non-Abelian braiding, which in turn leads to a phase characterized by an anomalous Euler class, being the prime example of a multi-gap topological invariant. Most strikingly, we also retrieve the first example of an 'anomalous Dirac string phase'. This gapped out-of-equilibrium phase features an unconventional Dirac string configuration that physically manifests itself via anomalous edge states on the boundary. Our results not only provide a stepping stone for the exploration of intrinsically dynamical and experimentally viable multi-gap topological phases, but also demonstrate periodic driving as a powerful way to observe these non-Abelian braiding processes notably in quantum simulators.

Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates.

Topological phases of matter have revolutionised the fundamental understanding of band theory and hold great promise for next-generation technologies such as low-power electronics or quantum computers. Single-gap topologies have been extensively explored, and a large number of materials have been theoretically proposed and experimentally observed. These ideas have recently been extended to multi-gap topologies with band nodes that carry non-Abelian charges, characterised by invariants that arise by the momentum space braiding of such nodes. However, the constraints placed by the Fermi-Dirac distribution to electronic systems have so far prevented the experimental observation of multi-gap topologies in real materials. Here, we show that multi-gap topologies and the accompanying phase transitions driven by braiding processes can be readily observed in the bosonic phonon spectra of known monolayer silicates. The associated braiding process can be controlled by means of an electric field and epitaxial strain, and involves, for the first time, more than three bands. Finally, we propose that the band inversion processes at the Γ point can be tracked by following the evolution of the Raman spectrum, providing a clear signature for the experimental verification of the band inversion accompanied by the braiding process.

Topological Euler Class as a Dynamical Observable in Optical Lattices.

The last years have witnessed rapid progress in the topological characterization of out-of-equilibrium systems. We report on robust signatures of a new type of topology-the Euler class-in such a dynamical setting. The enigmatic invariant (ξ) falls outside conventional symmetry-eigenvalue indicated phases and, in simplest incarnation, is described by triples of bands that comprise a gapless pair featuring 2ξ stable band nodes, and a gapped band. These nodes host non-Abelian charges and can be further undone by converting their charge upon intricate braiding mechanisms, revealing that Euler class is a fragile topology. We theoretically demonstrate that quenching with nontrivial Euler Hamiltonian results in stable monopole-antimonopole pairs, which in turn induce a linking of momentum-time trajectories under the first Hopf map, making the invariant experimentally observable. Detailing explicit tomography protocols in a variety of cold-atom setups, our results provide a basis for exploring new topologies and their interplay with crystalline symmetries in optical lattices beyond paradigmatic Chern insulators.

Data Mining for Three-Dimensional Organic Dirac Materials: Focus on Space Group 19.

We combined the group theory and data mining approach within the Organic Materials Database that leads to the prediction of stable Dirac-point nodes within the electronic band structure of three-dimensional organic crystals. We find a particular space group P212121 (#19) that is conducive to the Dirac nodes formation. We prove that nodes are a consequence of the orthorhombic crystal structure. Within the electronic band structure, two different kinds of nodes can be distinguished: 8-fold degenerate Dirac nodes protected by the crystalline symmetry and 4-fold degenerate Dirac nodes protected by band topology. Mining the Organic Materials Database, we present band structure calculations and symmetry analysis for 6 previously synthesized organic materials. In all these materials, the Dirac nodes are well separated within the energy and located near the Fermi surface, which opens up a possibility for their direct experimental observation.