  While it was known that High Temperature Superconductivity appears in
cuprates showing complex multiscale phase separation due to inhomogeneous
charge density wave (CDW) order, the spatial distribution of CDW domains
remained an open question for a long time, because of the lack of experimental
probes able to visualize their spatial distribution at mesoscale, between
atomic and macroscopic scale. Recently scanning micro X-ray Diffraction (SmXRD)
revealed CDW crystalline electronic puddles with a complex fat-tailed spatial
distribution of their size. In this work we have determined and mapped the
anisotropy of the Charge Density Waves (CDW) puddles in HgBa2CuO4+y (Hg1201)
single crystal. We discuss the emergence of high temperature superconductivity
in the interstitial space with hyperbolic geometry, that opens a new paradigm
for quantum coherence at high temperature where negative dielectric function
and interference between different pathways can help to raise the critical
temperature.
