More than 30 years have passed since the discovery of high-T_c superconducting copper oxides. While the mechanism of the high-T_c superconductivity still remains elusive, the electronic phase diagram is a key ingredient to understand it. It has been believed that parent compounds of cuprates were universally antiferromagnetic Mott insulators and that superconductivity would develop upon doping either holes or electrons in the insulators (“doped Mott-insulator scenario”). However, our recent discovery of superconductivity in the parent and heavily-underdoped compounds with the Nd₂CuO₄(T′) structure urged a serious reassessment to the above scenario. We start this review with our experimental results for the T′ cuprates and their implications. The key material issue is to remove excess oxygen ions residing in the interstitial sites without introducing oxygen vacancies in the CuO₂ planes, the playground of high-T_c superconductivity, by elaborate synthesis procedures. There has been significant progress also in the electronic-structure calculation techniques. Unlike 30 years ago, it is now possible to predict the coordination-dependent different ground states between T- and T′-La₂CuO₄: Cu is octahedral and square-planar coordinated in the T and T′ structures, respectively. High-T_c superconductivity remains confined, at least at present, only to copper oxides. Reassessment of the electronic state by the state-of-the-art calculation methods may unveil unique features of copper oxides not shared by other transition-metal oxides.

The Coulomb interaction among massless Dirac electrons in graphene is unscreened around the isotropic band-crossing points, causing a logarithmic velocity renormalization and a Dirac cone reshaping. In less symmetric materials possessing anisotropic Dirac cones with tilted axes, the Coulomb interaction can provide still more exotic phenomena which have not been experimentally unveiled yet. Using site-selective nuclear magnetic resonance, we find a non-uniform cone reshaping accompanied by a bandwidth reduction and an emergent ferrimagnetism in tilted Dirac cones that appear on the verge of charge ordering in an organic compound. Our theoretical analyses based on the renormalization-group approach and the Hubbard model show that these observations are the direct consequences of the long-range and short-range parts of the Coulomb interaction, respectively. The cone reshaping and the bandwidth renormalization, as well as the novel magnetism revealed here, can be ubiquitous and vital for many Dirac cone materials in general.

The null energy condition can be violated stably in general scalar-tensor theories of gravity, which gives rise to the possibilities of healthy non-singular cosmologies. However, it has been reported that in many cases cosmological solutions are plagued with instabilities or have some pathologies somewhere in the whole history of the universe. We show that non-singular models (with flat spatial sections) in general suffer either from gradient instabilities or some kind of pathology in the tensor sector. This implies that one must go beyond the Horndeski theory to implement healthy non-singular cosmologies.