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

High Temperature superconductivity in a hyperbolic geometry of complex matter from nanoscale to mesoscopic scale