Mottness and spin liquidity in a doped organic superconductor $\kappa$-(BEDT-TTF)$_4$Hg$_{2.89}$Br$_8$

It has been more than 40 years since superconductivity was discovered in organic conductors, and the way scientists view organic superconductors has changed over time. At first, the fact that organic conductors exhibit superconductivity was a novelty in itself, and subsequently it was shown that behind the superconductivity is the physics of electron correlation, which has been a focus in condensed matter physics at large. Amid the remarkable development of correlation physics, the unique characteristics of organic conductors, e.g., a variety of lattice geometries and the highly compressible feature, led to the elucidation of fundamental principles and the finding of new phenomena, such as bandwidth-controlled Mott transitions and possible quantum spin liquids. However, most organic superconductors have commensurate band fillings, such as a half or a quarter, whereas inorganic superconductors, such as high-$T_{\rm c}$ cuprates and iron-based superconductors, have often been investigated under the variation of their band fillings. Thus, the physical linkage between organic and inorganic superconductors has remained unresolved. In this review article, we focus on the layered nonstoichiometric superconductor, $\kappa$-(BEDT-TTF)$_4$Hg$_{2.89}$Br$_8$, which is exceptional among organic conductors in that the nonstoichiometry serves as doping to a half-filled band. Moreover, the strong correlation of electrons and a geometrically frustrated triangular lattice make this system exhibit the unique phenomena involved in Mottness, spin liquidity, and superconductivity, which are key concepts of correlated electron physics. This review will summarize what we learned from the pressure study of $\kappa$-(BEDT-TTF)$_4$Hg$_{2.89}$Br$_8$ and how they relate to the extensively studied issues in inorganic materials.