One of the main developments in unconventional superconductivity in the past two decades has been the discovery that most unconventional superconductors form phase diagrams that also contain other strongly correlated states. Many systems of interest are therefore close to more than one instability, and tuning between the resultant ordered phases is the subject of intense research¹. In recent years, uniaxial pressure applied using piezoelectric-based devices has been shown to be a particularly versatile new method of tuning^2,3, leading to experiments that have advanced our understanding of the fascinating unconventional superconductor Sr₂RuO₄ (refs. ^4–9). Here we map out its phase diagram using high-precision measurements of the elastocaloric effect in what we believe to be the first such study including both the normal and the superconducting states. We observe a strong entropy quench on entering the superconducting state, in excellent agreement with a model calculation for pairing at the Van Hove point, and obtain a quantitative estimate of the entropy change associated with entry to a magnetic state that is observed in proximity to the superconductivity. The phase diagram is intriguing both for its similarity to those seen in other families of unconventional superconductors and for extra features unique, so far, to Sr₂RuO₄.

High-quality single crystals are essentially needed for the investigation of the novel bulk properties of unconventional superconductors. The availability of such crystals grown by the floating-zone method has helped to unveil the unconventional superconductivity of the layered perovskite Sr₂RuO₄, which is considered as a strong candidate of a topological spin-triplet superconductor. Yet, recent progress of investigations urges further efforts to obtain ultimately high-quality crystalline samples. In this paper, we focus on the method of preparation of feed rods for the floating-zone melting and report on the improvements of the crystal growth. We present details of the improved methods used to obtain crystals with superconducting transition temperatures T_c that are consistently as high as 1.4 K, as well as the properties of these crystals.

Applying in-plane uniaxial pressure to strongly correlated low-dimensional systems has been shown to tune the electronic structure dramatically. For example, the unconventional superconductor Sr₂RuO₄ can be tuned through a single Van Hove point, resulting in strong enhancement of both T_c and H_c2. Out-of-plane (c axis) uniaxial pressure is expected to tune the quasi-two-dimensional structure even more strongly, by pushing it towards two Van Hove points simultaneously. Here, we achieve a record uniaxial stress of 3.2 GPa along the c axis of Sr₂RuO₄. H_c2 increases, as expected for increasing density of states, but unexpectedly T_c falls. As a first attempt to explain this result, we present three-dimensional calculations in the weak interaction limit. We find that within the weak-coupling framework there is no single order parameter that can account for the contrasting effects of in-plane versus c-axis uniaxial stress, which makes this new result a strong constraint on theories of the superconductivity of Sr₂RuO₄.

The interplay of electronic and structural degrees of freedom in solids is a topic of intense research. More than 60 years ago, Lifshitz discussed a counterintuitive possibility: lattice softening driven by conduction electrons at topological Fermi surface transitions. The effect that he predicted, however, was small and has not been convincingly observed. Using a piezo-based uniaxial pressure cell to tune the ultraclean metal strontium ruthenate while measuring the stress-strain relationship, we reveal a huge softening of the Young’s modulus at a Lifshitz transition of a two-dimensional Fermi surface and show that it is indeed driven entirely by the conduction electrons of the relevant energy band.

The unconventional superconductor Sr₂RuO₄ has long served as a benchmark for theories of correlated-electron materials. The determination of the superconducting pairing mechanism requires detailed experimental information on collective bosonic excitations as potential mediators of Cooper pairing. We have used Ru L₃-edge resonant inelastic x-ray scattering to obtain comprehensive maps of the electronic excitations of Sr₂RuO₄ over the entire Brillouin zone. We observe multiple branches of dispersive spin and orbital excitations associated with distinctly different energy scales. The spin and orbital dynamical response functions calculated within the dynamical mean-field theory are in excellent agreement with the experimental data. Our results highlight the Hund metal nature of Sr₂RuO₄ and provide key information for the understanding of its unconventional superconductivity.

Pyrochlore oxide ${\mathrm{Cd}}_2{\mathrm{Re}}_2{\mathrm{O}}_7$ exhibits successive structural transitions upon cooling that break its inversion symmetry. The low-temperature noncentrosymmetric metallic phases are believed to be some odd-parity multipole ordered states that are associated with a Fermi-liquid instability due to the strong spin-orbit interaction (SOI) and electronic correlation. However, their microscopic ordering pictures and the driving force of the phase transitions are still unclear. We determined the electronic structure of the lowest temperature phase of ${\mathrm{Cd}}_2{\mathrm{Re}}_2{\mathrm{O}}_7$ by combining quantum oscillation measurements with electronic structure calculations. The observed Fermi surfaces were well reproduced based on the optimized crystal structure, and we demonstrated the strong influence of the antisymmetric SOI. From the mass enhancement factor, we elucidated the strongly correlated nature of the electronic states. In addition, we visualized the microscopic picture of the $3z^2-r^2$-type metallic electric-toroidal-quadrupole (ETQ) order characterized by the Re-O bond order. These results corroborate that the metallic ETQ order is driven by a Fermi-liquid instability associated with the strong SOI and electronic correlation, as has been theoretically proposed. Our results provide the basis for exploring unconventional phenomena expected in the metallic ETQ order.