  It has recently been established that the high temperature (high-Tc)
superconducting state coexists with short-range charge-density-wave order and
quenched disorder arising from dopants and strain. This complex, multiscale
phase separation invites the development of theories of high temperature
superconductivity that include complexity. The nature of the spatial interplay
between charge and dopant order that provides a basis for nanoscale phase
separation remains a key open question, because experiments have yet to probe
the unknown spatial distribution at both the nanoscale and mescoscale (between
atomic and macroscopic scale). Here we report micro X-ray diffraction imaging
of the spatial distribution of both the charge-density-wave puddles (domains
with only a few wavelengths) and quenched disorder in HgBa2CuO4+y, the single
layer cuprate with the highest Tc, 95 kelvin. We found that the
charge-density-wave puddles, like the steam bubbles in boiling water, have a
fat-tailed size distribution that is typical of self-organization near a
critical point. However, the quenched disorder, which arises from oxygen
interstitials, has a distribution that is contrary to the usual assumed random,
uncorrelated distribution. The interstitials-oxygen-rich domains are spatially
anti-correlated with the charge-density-wave domains, leading to a complex
emergent geometry of the spatial landscape for superconductivity.
