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Symmetry indicators allow to identify topological crystalline insulators using symmetry-group analysis of their Bloch states. An extension of this approach to superconducting systems requires defining an appropriate atomic limit for superconductors. Here, we introduce such a notion of atomic limit and derive a symmetry indicator for inversion-symmetric superconductors. This indicator allows for a refined topological classification including higher-order phases for systems in the superconducting phase with and without time-reversal symmetry. We further elucidate their bulk-boundary correspondence using Dirac surface theories. This indicator is well suited for a search of topological superconductors using first-principles calculations.

Magic-angle twisted bilayer graphene has received a lot of interest due to its flat bands with potentially nontrivial topology that lead to intricate correlated phases. A spectrum with flat bands, however, does not require a twist between multiple sheets of two-dimensional materials, but can be realized with an appropriate periodic potential. Here, we propose the imposition of a tailored potential onto a single graphene layer through local perturbations that could be created via lithography or adatom manipulation, which also results in an energy spectrum featuring flat bands. First-principle calculations for an appropriate adatom decoration of graphene indeed show the presence of flat bands and a symmetry-indicator analysis further reveals the bands' topological nature. This nontrivial topology manifests itself in corner-localized states with a filling anomaly.

Inversion symmetry is a key symmetry in unconventional superconductors, and even its local breaking can have profound implications. Non-centrosymmetric superconductors is a class of materials, that break inversion symmetry locally, while preserving it globally. For such systems, there is a competition on a microscopic level between the spin-orbit coupling associated with the local lack of inversion and hybridizing terms that “restore” inversion. Investigating a layered system with alternating mirror-symmetry breaking, we study this competition considering the spin response of different superconducting order parameters for the case of strong spin-orbit coupling. We also identify several regimes with possible topological superconducting phases using a symmetry-indicator analysis.

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter.

Yu-Shiba-Rusinov states are subgap bound states of a magnetic impurity in a superconducting host. Usually magnetic impurities in such systems are modelled as classical spins. However, a wide selection of magnetic adatoms and substrates show experimental signatures, which are not adequately captured by such approach and require quantum treatment of impurity spin. In this work we take into account the quantum nature of the impurity spin in a single-site approximation to study a system of magnetic impurity in a spin-split superconductor, i.e., a superconductor in proximity to a ferromagnetic insulator. We investigate the spectral properties of the single-particle excitations and compare the results of this quantum approach to the classical approach, which conventionally predicts fully polarized YSR excitations even in the absence of exchange and external magnetic field.