The electronic phase diagrams of many highly correlated systems, and, in particular, the cuprate high temperature superconductors, are complex, with many different phases appearing with similar (sometimes identical) ordering temperatures even as material properties, such as dopant concentration, are varied over wide ranges. This complexity is sometimes referred to as “competing orders.” However, since the relation is intimate, and can even lead to the existence of new phases of matter such as the putative “pair-density wave,” the general relation is better thought of in terms of “intertwined orders.” Some of the experiments in the cuprates which suggest that essential aspects of the physics are reflected in the intertwining of multiple orders, not just in the nature of each order by itself, are selectively analyzed. Several theoretical ideas concerning the origin and implications of this complexity are also summarized and critiqued.

This article discusses fluctuating order in a quantum disordered phase proximate to a quantum critical point, with particular emphasis on fluctuating stripe order. Optimal strategies are derived for extracting information concerning such local order from experiments, with emphasis on neutron scattering and scanning tunneling microscopy. These ideas are tested by application to two model systems—an exactly solvable one-dimensional (1D) electron gas with an impurity, and a weakly interacting 2D electron gas. Experiments on the cuprate high-temperature superconductors which can be analyzed using these strategies are extensively reviewed. The authors adduce evidence that stripe correlations are widespread in the cuprates. They compare and contrast the advantages of two limiting perspectives on the high-temperature superconductor: weak coupling, in which correlation effects are treated as a perturbation on an underlying metallic (although renormalized) Fermi-liquid state, and strong coupling, in which the magnetism is associated with well-defined localized spins, and stripes are viewed as a form of micro phase separation. The authors present quantitative indicators that the latter view better accounts for the observed stripe phenomena in the cuprates.

Previously we have presented evidence for stripe order of holes and spins in ${\mathrm{La}}_{1.6-x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ with $x=0.12\mathrm{}$. Here we show, via neutron diffraction measurements of magnetic scattering, that similar order occurs in crystals with $x=0.15\mathrm{}$ and $0.20$. Zero-field-cooled magnetization measurements show that all three compositions are also superconducting, with the superconducting transition temperature increasing as the low-temperature staggered magnetization decreases. These results directly demonstrate an intimate connection between stripe correlations and superconductivity.

Neutron-diffraction experiments on ceramic powders of $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ (nonsuperconducting, with $x=0.0\mathrm{} \mathrm{and} 0.15$ have confirmed the existence of long-range, three-dimensional, antiferromagnetic order of the Cu spins. The structure determination was aided by the use of uniaxially oriented powders. The ordering wave vector within a Cu${\mathrm{O}}_2$ plane is (½ ½) and the planes are coupled antiferromagnetically along the $c$ axis. The Néel temperature is 400±10 K for the $x=0.15\mathrm{}\pm 0.05$ sample and ≳500 K for $x=0.0\mathrm{}$.