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 between atomic and macroscopic scale. Recently scanning micro-X-ray diffraction (S mu XRD) 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 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.