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A review of the phases of copper oxides (especially the ‘strange metal’), discussing their high-temperature superconductivity and their various forms of quantum matter, and the implications for fundamental theory. High- T c superconductors reviewed Since the discovery of high temperature copper oxide superconductors nearly thirty years ago, much has been learned about the nature of the superconducting state and the novel forms of quantum matter involved. This Review concentrates on the related issues that have not been resolved, notably the complexity of the phase diagram for the copper oxides, and the simplicity and insensitivity to material details of the 'normal' state at elevated temperatures. The discovery of high-temperature superconductivity in the copper oxides in 1986 triggered a huge amount of innovative scientific inquiry. In the almost three decades since, much has been learned about the novel forms of quantum matter that are exhibited in these strongly correlated electron systems. A qualitative understanding of the nature of the superconducting state itself has been achieved. However, unresolved issues include the astonishing complexity of the phase diagram, the unprecedented prominence of various forms of collective fluctuations, and the simplicity and insensitivity to material details of the ‘normal’ state at elevated temperatures.

The search continues for nickel oxide-based materials with electronic properties similar to cuprate high-temperature superconductors 1 – 10 . The recent discovery of superconductivity in the doped infinite-layer nickelate NdNiO 2 (refs. 11 , 12 ) has strengthened these efforts. Here, we use X-ray spectroscopy and density functional theory to show that the electronic structure of LaNiO 2 and NdNiO 2 , while similar to the cuprates, includes significant distinctions. Unlike cuprates, the rare-earth spacer layer in the infinite-layer nickelate supports a weakly interacting three-dimensional 5 d metallic state, which hybridizes with a quasi-two-dimensional, strongly correlated state with 3 d x 2 − y 2 symmetry in the NiO 2 layers. Thus, the infinite-layer nickelate can be regarded as a sibling of the rare-earth intermetallics 13 – 15 , which are well known for heavy fermion behaviour, where the NiO 2 correlated layers play an analogous role to the 4 f states in rare-earth heavy fermion compounds. This Kondo- or Anderson-lattice-like ‘oxide-intermetallic’ replaces the Mott insulator as the reference state from which superconductivity emerges upon doping. X-ray spectroscopy and density functional theory are used to show that the electronic structure of the parent compound of superconducting infinite-layer nickelates, while similar to the copper-based high-temperature superconductors, has significant differences.

A highlight of Fermi-liquid phenomenology, as explored in neutral 3 He, is the observation that in the collisionless regime shear stress propagates as if one is dealing with the transverse phonon of a solid. The existence of this “transverse zero sound” requires that the quasiparticle mass enhancement exceeds a critical value. Could such a propagating shear stress also exist in strongly correlated electron systems? Despite some noticeable differences with the neutral case in the Galilean continuum, we arrive at the verdict that transverse zero sound should be generic for mass enhancement higher than 3. We present an experimental setup that should be exquisitely sensitive in this regard: the transmission of terahertz radiation through a thin slab of heavy-fermion material will be strongly enhanced at low temperature and accompanied by giant oscillations, which reflect the interference between light itself and the “material photon” being the actual manifestation of transverse zero sound in the charged Fermi liquid.

Copper oxide high-TC superconductors possess a number of exotic orders that coexist with or are proximal to superconductivity. Quantum fluctuations associated with these orders may account for the unusual characteristics of the normal state, and possibly affect the superconductivity1–4. Yet, spectroscopic evidence for such quantum fluctuations remains elusive. Here, we use resonant inelastic X-ray scattering to reveal spectroscopic evidence of fluctuations associated with a charge order5–14 in nearly optimally doped Bi2Sr2CaCu2O8+δ. In the superconducting state, while the quasielastic charge order signal decreases with temperature, the interplay between charge order fluctuations and bond-stretching phonons in the form of a Fano-like interference increases, an observation that is incompatible with expectations for competing orders. Invoking general principles, we argue that this behaviour reflects the properties of a dissipative system near an order–disorder quantum critical point, where the dissipation varies with the opening of the pseudogap and superconducting gap at low temperatures, leading to the proliferation of quantum critical fluctuations, which melt charge order.X-ray scattering experiments show that the quantum fluctuations associated with charge order take a form that is incompatible with the idea of competition between charge order and superconductivity.

Strange metals possess highly unconventional electrical properties, such as a linear-in-temperature resistivity 1 – 6 , an inverse Hall angle that varies as temperature squared 7 – 9 and a linear-in-field magnetoresistance 10 – 13 . Identifying the origin of these collective anomalies has proved fundamentally challenging, even in materials such as the hole-doped cuprates that possess a simple bandstructure. The prevailing consensus is that strange metallicity in the cuprates is tied to a quantum critical point at a doping p * inside the superconducting dome 14 , 15 . Here we study the high-field in-plane magnetoresistance of two superconducting cuprate families at doping levels beyond p *. At all dopings, the magnetoresistance exhibits quadrature scaling and becomes linear at high values of the ratio of the field and the temperature, indicating that the strange-metal regime extends well beyond p *. Moreover, the magnitude of the magnetoresistance is found to be much larger than predicted by conventional theory and is insensitive to both impurity scattering and magnetic field orientation. These observations, coupled with analysis of the zero-field and Hall resistivities, suggest that despite having a single band, the cuprate strange-metal region hosts two charge sectors, one containing coherent quasiparticles, the other scale-invariant ‘Planckian’ dissipators. Measurements of high-field magnetotransport in overdoped cuprates indicate that the strange-metal regime exists beyond the critical doping, and that it has both coherent and incoherent contributions.

More than 30 years after the discovery of high-temperature superconductivity in copper oxides, its mechanism remains a mystery. Electron pairing mediated solely by lattice vibrations—phonons—is thought to be insufficient to account for the high transition temperatures. He et al. found a rapid and correlated increase of the superconducting gap and electron-phonon interactions as the chemical composition of their bismuth-based cuprate samples was varied across a critical doping concentration. The interplay of electron-phonon with electron-electron interactions may lead to enhanced transition temperatures. Science , this issue p. 62 Angle-resolved photoemission uncovers an interplay between various types of interaction in a cuprate superconductor. Electron-boson coupling plays a key role in superconductivity for many systems. However, in copper-based high–critical temperature ( T c ) superconductors, its relation to superconductivity remains controversial despite strong spectroscopic fingerprints. In this study, we used angle-resolved photoemission spectroscopy to find a pronounced correlation between the superconducting gap and the bosonic coupling strength near the Brillouin zone boundary in Bi 2 Sr 2 CaCu 2 O 8+δ . The bosonic coupling strength rapidly increases from the overdoped Fermi liquid regime to the optimally doped strange metal, concomitant with the quadrupled superconducting gap and the doubled gap-to- T c ratio across the pseudogap boundary. This synchronized lattice and electronic response suggests that the effects of electronic interaction and the electron-phonon coupling (EPC) reinforce each other in a positive-feedback loop upon entering the strange-metal regime, which in turn drives a stronger superconductivity.

Surprising observations in the evolution of electronic states in electron-doped iridates provide fresh insight into the melting of the Mott state and might lead to a fuller understanding of corresponding processes in copper-oxide superconductors. It is widely believed that high-temperature superconductivity in the cuprates emerges from doped Mott insulators 1 . When extra carriers are inserted into the parent state, the electrons become mobile but the strong correlations from the Mott state are thought to survive—inhomogeneous electronic order, a mysterious pseudogap and, eventually, superconductivity appear. How the insertion of dopant atoms drives this evolution is not known, nor is whether these phenomena are mere distractions specific to hole-doped cuprates or represent genuine physics of doped Mott insulators. Here we visualize the evolution of the electronic states of (Sr 1− x La x ) 2 IrO 4 , which is an effective spin-1/2 Mott insulator like the cuprates, but is chemically radically different 2 , 3 . Using spectroscopic-imaging scanning tunnelling microscopy (SI-STM), we find that for a doping concentration of x ≈ 5%, an inhomogeneous, phase-separated state emerges, with the nucleation of pseudogap puddles around clusters of dopant atoms. Within these puddles, we observe the same iconic electronic order that is seen in underdoped cuprates 1 , 4 , 5 , 6 , 7 , 8 , 9 . We investigate the genesis of this state and find evidence at low doping for deeply trapped carriers, leading to fully gapped spectra, which abruptly collapse at a threshold of x ≈ 4%. Our results clarify the melting of the Mott state, and establish phase separation and electronic order as generic features of doped Mott insulators.

We put forward here the case that the anomalous electron states found in cuprate superconductors and related systems are rooted in a deeply non-classical fermion sign structure. The collapse of Mottness, as advocated by Phillips and supported by recent dynamical cluster approximation results on the Hubbard model, sets the necessary microscopic conditions. The crucial insight is due to Weng, who demonstrated that, in the presence of Mottness, the fundamental workings of quantum statistics change, and we will elaborate on the effects of this Weng statistics with an emphasis on characterizing it further using numerical methods. The pseudo-gap physics of the underdoped regime appears as a consequence of the altered statistics and the profound question is how to connect this by a continuous quantum phase transition to the overdoped regime ruled by normal Fermi-Dirac statistics. Proof of principle follows from Ceperley's constrained path integral formalism, in which states can be explicitly constructed showing a merger of Fermi-Dirac sign structure and scale invariance of the quantum dynamics.

Phases of classical matter, such as solids and liquids, are ruled by emergence principles that are well understood. Although the same principles govern forms of quantum matter that have no secrets for physicists, such as the superfluids, having to deal with fermions and the associated Fermi sign problem shatters this analogy. This Perspective addresses the Fermion sign problem and describes experiments on metals undergoing quantum phase transitions exhibiting scale-invariant electronic behavior, a description of which is at odds with established quantum theory.

We study the coexisting smectic modulations and intra—unit-cell nematicity in the pseudogap states of underdoped Bi 2 Sr 2 CaCu 2 O 8+δ . By visualizing their spatial components separately, we identified 2π topological defects throughout the phase-fluctuating smectic states. Imaging the locations of large numbers of these topological defects simultaneously with the fluctuations in the intra—unit-cell nematicity revealed strong empirical evidence for a coupling between them. From these observations, we propose a Ginzburg-Landau functional describing this coupling and demonstrate how it can explain the coexistence of the smectic and intra—unit-cell broken symmetries and also correctly predict their interplay at the atomic scale. This theoretical perspective can lead to unraveling the complexities of the phase diagram of cuprate high-critical-temperature superconductors.