The picture shows the pathways of superconducting pairs running around the CDW puddles. The puddle size distribution is a power law. Between two points (black dots) there are an infinite number of pathways not only distinguished by the number of times a path goes around a single puddle, but also distinguished by the way the path is passing though the pattern of CDW puddles. This space can be mapped into a hyperbolic space

The evolution of the Universe after the Big Bang started from a disordered state of massless particles at a very high temperature. Later, with the temperature decreasing down, these particles begin to gain a mass thanks to the spontaneous symmetry breaking of Gauge symmetries. This effect, is the base for the Higgs mechanism. The massive particles form protons and neutrons and finally the atoms.

In the classic physics we know that jiggling atoms by decreasing temperature form three major states of aggregation: gas -> liquid -> solid. The phase transition from liquid to solidis called a transition form disorder to order. The jiggling atoms in the liquid approaching zero temperature are trapped in specific spots of the space and form periodic arrays of atoms. Scientists say that the symmetry of the system is broken since it is invariant only under a small subset of translations by a lattice vector.
In the XX century we have learned that there is also a fourth state of matter: the superfluid phase. An example of this state is superfluid-helium. Cooling down liquid helium, the atoms do not like to get fixed in spots of the Euclidean space, forming a crystal, but they prefer to order their velocities and to move all together coherently in the fourth state of matter.
Also electrons packed inside metals form a liquid, called Fermi liquid, which manifests itself in the electrical current. By decreasing the temperature, electrons could form an ordered electronic crystal called -Wigner crystal- or -Peirels charge-density-wave order- in some metals. On the contrary in other metals, electrons prefer to undergo in the fourth state of matter called here superconductivity. This phase transition occurs by spontaneous symmetry breaking of gauge symmetry and with the formation of the off-diagonal long-range order. Therefore superconductivity in metals is a condensed-matter state analog of the Higgs phenomena, in which Cooper pairs of electrons spontaneously break the U(1) Gauge "symmetry" associated with light and electromagnetism.

There is another beautiful ordered state of matter: life. About life we know a lot of details, and we know that it emerged below about 400K, in our planet 5 billions of years ago, but we do not know its physical laws.
We know that also at the origin of living matter there is a disorder-to-order transition from the random coil made by a polypeptide chain to a partial ordered state where some portions of the chain remain as random coil portions and other portions form a periodic regular crystalline order (the alfa or beta chain).

  Arrested phase transitions from the disordered phase (random coil) to the inhomogeneous phase common in all biological polypeptide chains of the life world: made of short-random-coil-portions, short-alpha-helix-portions and short-beta sheet-portions.
The transition from two homogeneous phases a solid (the homogeneous iceberg) and a liquid (homogeneous ocean) to the inhomogeneous phase made of both small-solid-ice-islands coexisting with interstitial-liquid-water.

Physicists have been fascinated since many years by the possibility that the off-diagonal long-range order of electrons could get stable in the same temperature range as living matter T< 400 K. The search for new superconducting materials was focused for years on pure single crystalline metals, and disorder was considered to be detrimental for superconductivity. In perfect crystals superconductivity was considered to be impossible above 30K. In 1986 Mueller and Bednorz have shown that superconductivity at high can appear at higher temperature not in perfect metals but in ceramics. Recently the group of Emerets has reached the superconducting Tc record of 203K in pressurized sulfur hydride after a procedure of sample annealing which indicate the crystal instability.
Disorder was considered to be not detrimental but an essential ingredient in these materials to reach the exotic high temperature superconductivity only by few scientists including Antonio Bianconi, Takeshi Egami, Vladimir Krezin, Alex Mueller, James Phillip since many years.

The Nature article by Campi et al. appearing on Sept 17, 2015 sheds light on the emergent spatial order in the mesoscopic space between the atomic structure and macroscopic structure in a metal superconducting at about 100K.

While the average macroscopic crystal shows a perfect cubic lattice symmetry, probing the nanoscale and mesoscale ordering of electrons the authors have found that at – 30°C (250K) the electrons get aggregated and form puddles of the electronic crystals called charge density waves (CDW).
The main discovery is that this order is highly inhomogeneous the phase transition is arrested like in the scenario of melted iceberg in Antartica or in the biological polypeptide. The CDW order extends over small puddles like the small ice crystals in the melting iceberg or like in the short alpha helix pieces in the polypeptide chain of proteins. The statistical distribution of the electronic CDW crystal puddles has been measured by scanning micro x-ray diffraction using top world level technologies in focusing synchrotron radiation x-ray beams. The scientists discovered that the distribution of the puddles size and puddle density follows a power law distribution over more than one order of magnitude describing a complex fractal-like self-organization.

Not all electrons form electronic crystal puddles, other electrons remain in the fluid phase below 20°C (250K) and at about -175°C (95K) they undergo to the so called exotic high critical temperature superconducting order. In this topologically confined space the pairs do not move a isotropic 3D world but they run only along pathways in the interstitial space available between the CDW puddles. Different paths connecting two points are not topological equivalent as shown in the top image. Between two points (black dots) there are an infinite number of pathways not only distinguished by the number of times a path goes around a single puddle, but also distinguished by the way the path is passing though the pattern of CDW puddles. The authors argue that this emergent spatial interstitial space can be mapped into a hyperbolic geometry

The hyperbolic space is a non Euclidean geometry like geometries needed in general relativity, and recently found in network theory and in quantum gravity. These results finally open new venues in the field for the design of new superconducting metamaterials taking advances of the atomic self organization in the mesoscopic world.

Antonio Bianconi

1) Campi et al., Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor Nature 525, 359–362 (17 September 2015)

2) Erica W. Carlson Condensed-matter physics: Charge topology in superconductors Nature 525, 329–330 (17 September 2015) doi:10.1038/525329°