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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.

and macroscopic world. In particular, Scanning micro X ray diffraction (SµXRD) using focused beams on the micro-scale is able to probe the k-space order in different spatial locations of the material. This allows to unveil the complex maps showing the multiscale structure in materials. In this way, ultrastructure and phase separation have been visualized in complex and heterogeneous materials for applications in different field from biomedicine [4][5][6][7] to material science [8][9][10][11]. In High Temperature Superconductors (HTS) the spatial imaging of ubiquitous nanoscale phase separation due to quenched disorder of dopants has been obtained both in cuprates [12][13][14] and iron based superconductors [15][16][17]. Recently, the interest of scientific community has been focused on the competition of short range charge density wave (CDW) order and superconductivity [18][19][20][21]. Scanning micro X-ray diffraction has been used for imaging the spatial distribution of short-range charge-density-wave puddles in La 2 CuO 4+y [20] and recently in HgBa 2 CuO 4+y [21] the single-layer cuprate with the highest T c , 95 kelvin [21]. In this work we investigate the anisotropy due to the different size of CDW puddles in the CuO 2 plane and out of plane directions.
The Hg1201 crystal was grown at ETH [22]. Crystal structure has P4/mmm symmetry with lattice parameters a=b=0.3875(5) nm and c=0.9508(2) nm at T=100K. We have identified the CDW order by single crystal X-ray diffraction using beam size of 200x200 µm 2 and photon energy of 17 KeV on the XRD1 beamline at ELETTRA synchrotron, Trieste [23]. Only selected reflections show clear CDW satellites, in agreement with Croft [18]. We focused on the CDW satellite q CDW =(0.23, 0, 0.17) around the (1 0 8) Bragg reflection, due to stripe-like modulation propagating along the tetragonal a(b)-axis with an average of only 3-4 oscillations per puddle [20]. The CDW order appears below the onset temperature T CDW =240 K [20]. Figure 1a shows the CDW satellite peak in a diffraction pattern measured at T=100K. By inspection of the profiles of the CDW peak, along the h(k) and l directions, shown in Figure 1b and 1c respectively, we clearly observe how the peak width is larger along the l direction.
Here we determine the variation of CDW puddles size both along the in plane direction and along the out of plane direction, point by point by means of Scanning micro X-ray diffraction (SµXRD) measurements. The sample was kept at T=100K where the system shows the maximum CDW order [20]. SµXRD measurements were performed on the ID13 beamline at ESRF, Grenoble, France.  It was already established, using XANES spectroscopy [27][28][29], that the cuprate superconductors show a heterogeneous copper oxide plane due to polaronic charge density wave formation [30,31] The use of scanning micro x-ray diffraction [20] has provided compelling evidence for the emergence of the hyperbolic space in the mesoscopic space with hyperbolic geometry favoring superconductivity in cuprates superconductors.
The hyperbolic space has recently been found to play a key role in different fields looking for the emergence of order in complex systems ranging from network theory to quantum gravity [32,33]. The role of a spatial hyperbolic geometry in disordered media is known to be of high relevance in the design of new metamaterials showing negative dielectric constant [34][35][36][37], as well as in the manipulation of complex light in optical metamaterials [38][39][40][41][42][43][44][45]. The hyperbolic space has recently been shown to influence critical phenomena [46]. The search for complex materials with zero or negative dielectric constant as been one of the road maps proposed for the amplification of the superconducting critical temperature to reach room temperature superconductivity [47][48][49][50][51][52][53]. The present experiments show a hyperbolic space in the mesoscale of cuprates but it could be present also in pressurized sulfur hydride superconductors [54,55], iron based superconductors [56] and diborides [57] which all show local lattice fluctuations.
Therefore further investigations are needed to investigate the substantial role of non Euclidean space in the mesoscale range in high temperature superconductors [49][50][51][52].