  The concept of intertwined orders has been introduced to describe the
cooperative relationship between antiferromagnetic spin correlations and
electron (or hole) pair correlations that develop in copper-oxide
superconductors. This contrasts with systems in which, for example,
charge-density-wave (CDW) order competes for Fermi surface area with
superconductivity. La$_{2-x}$Ba$_x$CuO$_4$ with $x=0.125$ provides an example
in which the ordering of spin stripes coincides with the onset of
two-dimensional superconducting correlations. The apparent frustration of the
interlayer Josephson coupling has motivated the concept of the
pair-density-wave superconductor, a state that theoretical calculations show to
be energetically competitive with the uniform $d$-wave superconductor. Even at
$x=0.095$, where there is robust superconductivity below 32~K in zero field,
the coexistence of strong, low-energy, incommensurate spin excitations implies
a spatially modulated and intertwined pair wave function. Recent observations
of CDW order in YBa$_2$Cu$_3$O$_{6+x}$ and other cuprate families have raised
interesting questions regarding the general role of charge modulations and the
relation to superconductivity. While there are differences in the doping
dependence of the modulation wave vectors in YBa$_2$Cu$_3$O$_{6+x}$ and
La$_{2-x}$Ba$_x$CuO$_4$, the maximum ordering strength is peaked at the hole
concentration of 1/8 in both cases. There are also possible connections with
the quantum oscillations that have been detected about the same hole
concentration but at high magnetic fields. Resolving these relationships
remains a research challenge.
