This review presents recent developments in the understanding of how impurities influence the electronic states in the bulk properties of superconductors. The focus is on quasilocalized states in the vicinity of impurity sites in conventional and unconventional superconductors and the goal is to provide a unified framework for their description. The nonmagnetic impurity resonances in unconventional superconductors are directly related to the Yu-Shiba-Rusinov states around magnetic impurities in conventional $s$-wave systems. The physics behind these states, including the quantum phase transition between screened and unscreened impurities, are reviewed and recent work on $d$-wave superconductors is emphasized. The bound states are seen in scanning-tunneling spectroscopy measurements on high-$T_c$ cuprates, which are described in detail. This paper discusses very recent progress in our understanding of states coupled to impurity sites, which have their own dynamics. Also reviewed are inelastic electron-tunneling spectroscopy features that could be seen by scanning-tunneling microscopy in real space and their Fourier-transformed images and impurity resonances in the presence of an order competing with superconductivity. The last part of the review is devoted to the influence of local deviations of the impurity concentration from its average value on the density of states in $s$-wave superconductors. Discussed is how these fluctuations affect the density of states and it is shown that $s$-wave superconductors are, strictly speaking, gapless in the presence of an arbitrarily small concentration of magnetic impurities.

Establishing the appropriate theoretical framework for unconventional superconductivity in the iron-based materials requires correct understanding of both the electron correlation strength and the role of Fermi surfaces. This fundamental issue becomes especially relevant with the discovery of the iron chalcogenide superconductors. Here, we use angle-resolved photoemission spectroscopy to measure three representative iron chalcogenides, FeTe_0.56Se_0.44, monolayer FeSe grown on SrTiO₃ and K_0.76Fe_1.72Se₂. We show that these superconductors are all strongly correlated, with an orbital-selective strong renormalization in the d_xy bands despite having drastically different Fermi surface topologies. Furthermore, raising temperature brings all three compounds from a metallic state to a phase where the d_xy orbital loses all spectral weight while other orbitals remain itinerant. These observations establish that iron chalcogenides display universal orbital-selective strong correlations that are insensitive to the Fermi surface topology, and are close to an orbital-selective Mott phase, hence placing strong constraints for theoretical understanding of iron-based superconductors.

Two major themes in the physics of condensed matter are quantum critical phenomena and unconventional superconductivity. These usually occur in the context of competing interactions in systems of strongly correlated electrons. All this interesting physics comes together in the behavior of the recently discovered iron pnictide compounds that have generated enormous interest because of their moderately high-temperature superconductivity. The ubiquity of antiferromagnetic ordering in their phase diagrams naturally raises the question of the relevance of magnetic quantum criticality, but the answer remains uncertain both theoretically and experimentally. Here, we show that the undoped iron pnictides feature a unique type of magnetic quantum critical point, which results from a competition between electronic localization and itinerancy. Our theory provides a mechanism to understand the experimentally observed variation of the ordered moment among the undoped iron pnictides. We suggest P substitution for As in the undoped iron pnictides as a means to access this example of magnetic quantum criticality in an unmasked fashion. Our findings point to the iron pnictides as a much-needed setting for quantum criticality, one that offers a unique set of control parameters.

Motivated by the search for design principles of rare-earth-free strong magnets, we present a study of electronic structure and magnetic properties of the ferromagnetic metal ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$ within the local-density approximation (LDA) of the density-functional theory, and its combination with dynamical mean-field theory (DMFT). To compare these calculations, we measure magnetic and thermodynamic properties as well as x-ray magnetic circular dichroism and the photoemission spectrum of single-crystal ${\mathrm{Fe}}_3{\mathrm{GeTe}}_2$. We find that the experimentally determined Sommerfeld coefficient is enhanced by an order of magnitude with respect to the LDA value. This enhancement can be partially explained by LDA+DMFT. In addition, the inclusion of dynamical electronic correlation effects provides the experimentally observed magnetic moments, and the spectral density is in better agreement with photoemission data. These results establish the importance of electronic correlations in this ferromagnet.

Bad metal properties have motivated a description of the parent iron pnictides as correlated metals on the verge of Mott localization. What has been unclear is whether interactions can push these and related compounds to the Mott-insulating side of the phase diagram. Here we consider the iron oxychalcogenides La2O2Fe2O(Se,S)2, which contain an Fe square lattice with an expanded unit cell. We show theoretically that they contain enhanced correlation effects through band narrowing compared to LaOFeAs, and we provide experimental evidence that they are Mott insulators with moderate charge gaps. We also discuss the magnetic properties in terms of a Heisenberg model with frustrating J1-J2-J2$'${} exchange interactions on a ``doubled'' checkerboard lattice.