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

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.