Crystalline Topology: from Metamaterials to Flat-Band Superconductors

Symmetries crucially underlie the classification of topological phenomena in condensed matter physics. The first endeavors in the study of topological effects in non-interacting systems focused predominantly on the role of symmetries that act locally in space. Nevertheless, most materials, both natural as well as engineered, also possess crystalline symmetries. Recent theoretical works unveiled that these crystalline symmetries can stabilize flavors of topology that have been previously overlooked. The contribution of this dissertation to the field of crystalline topological insulators is twofold. First, with the help of mechanical metamaterials, we provide the experimental validation of two variants of topology that necessitate crystalline symmetries: higher-order and fragile topology. Second, we theoretically establish the relevance of these topological effects to stabilize superconductivity once interactions are included in bands with quenched kinetic energy. The idea of higher-order topology is rooted in the extension of the modern theory of charge polarization to higher multipole moments. A two-dimensional topological insulator with a quantized quadrupole moment is predicted to have gapped yet topological one-dimensional edge modes, which stabilize zero-dimensional in-gap corner states. While based on the concept of charge moments, we can use the same theory as an elegant tool to characterize the bands of neutral bosonic systems such as photonic or phononic crystals. Here, we present the first observation of higher-order topological bands in an elastic metamaterial. We characterize experimentally the bulk, edge, and corner physics and find the predicted gapped edge and in-gap corner states. We further corroborate our findings by comparing the mechanical properties of a topologically non-trivial system to samples in other phases. Fragile topological bands, instead, challenge our very notion of topology: while answering to the most basic definition of topology, one can trivialize these bands through the addition of trivial ones. Moreover, they lack a rigorous bulk-boundary correspondence that ensures conducting gapless edge modes in the bulk's gap of the system. Here, we fully characterize the symmetry properties of the response of an acoustic metamaterial to establish the topologically fragile nature of its bands. Additionally, we identify a protected spectral signature associated with this flavor of topology in the form of spectral flow under twisted boundary conditions. Narrow fragile topological bands also emerge when one stacks two layers of graphene on top of each other with a small relative twist angle. Superconductivity was recently observed at partial fillings of these bands. It is an intriguing question whether the topological properties of the bands play any role in the superconducting transition. In two-dimensional systems such as twisted bilayer graphene, the superfluid weight determines the critical temperature through the Berezinksii-Kosterlitz-Thouless mechanism. At the mean-field level, the superfluid weight is bounded by the quantum metric of the bands. A non-zero Chern number or fragile topology, in turn, set a lower bound for the quantum metric. Here, we show that there exist bounds also within bands where space group symmetries force the electrons to be displaced from the atomic sites, even when their Berry curvature is identically zero. Furthermore, using exact Monte Carlo simulations, we find that an attractive Hubbard interaction in flat bands results in superconductivity as predicted by our bounds, thereby extending their validity beyond the mean-field regime. These results establish the importance of crystalline topology for flat-band superconductivity