High Tc superconductivity in manysheeted spacetime
Mark McWilliams sent me some time ago a link Scientists Detect 'Fingerprint' Of Hightemp Superconductivity Above Transition Temperature. What this fingerprint means is that Cooper pairs exists below a critical temperature T_{c1} higher than the critical temperature T_{c} for the onset of the superconductivity. The finding is surprising but nothing spectacular in a wider perspective. Also the atoms forming BoseEinstein condensates exists stably above the critical temperature for BoseEinstein condensation. The finding however suggests what the correct question might be. The reader who has decided to discover the mechanism of high T_{c} superconductivity could try either or both of the following alternatives.
 Why quantum coherence for Cooper pairs is possible only below the critical temperature T_{c} but not in the range (T_{c},T_{c1})?
 Why supra currents flow over long distances only below T_{c}?
I hasten to confess that I am an amateur in the field of high T_{c} superconductivity. After the idea about the hierarchy of Planck constants emerged for about half decade ago, I could not however resist the temptation to sketch a TGD inspired model for high T_{c} superconductors with Cooper pairs in large Planck constant phase. The results of the above article suggested that it is a high time to work out the model again.
So I had to start debunking the few years younger me. Not in the usual sense of throwing insults and using dirty rhetoric tricks but going through thoroughly the arguments of the younger colleague in the light of wisdom gained during these years. This process is not easy. I feel deep coashame while seeing this fellow to still fall in the sin of using words like 'remarkably' and even 'extremely'! And how badly structured the text of this enthusiasistic and so inpatient young fellow can be! But as a benevolent senior collegue I must tolerate these feelings. After all, these somewhat nonconventional colleagial discussions are the only manner to overcome the problems caused by the lack of the usual communications with colleagues. Sounds somewhat perverse and brings to my mind a prisoner in Stefan Zweig's novel playing chess against himself in order to avoid becoming crazy. In any case, the basic ideas survived the debunking but a lot of unnecessary ad hoc stuff had to be thrown away and the younger me had just to accept the somewhat violent modifications of his manuscript by the older colleague.
So: what did I learn in this process? What distincuishes high T_{c} cuprate superconductors from BCS type superconductors is that they are quantum critical. What is known that so called stripes ([1], [2]) containing electronic holes and carrying thus positive charge are essential for the existence of Cooper pairs whereas large enough quantum critical fluctuations of stripes are necessary for the onset of superconductivity. Magnetic fields are usually regarded as enemy of superconductivity but for spin 1 Cooper pairs magnetic field tends to stabilize the pairs. In high T_{c} superconductors the breaking of antiferromagnetic disorder induced by the formation of stripes is known to be essential for the formation of Cooper pairs.
In TGD inspired model of biosuperconductivity magnetic flux tubes are the carriers of superconducting phases consisting of dark variants of ordinary p"../articles/ characterized by a large value of Planck constant. The natural question is whether this might be the case also in the case of high T_{c} cuprate superconductors.
 If the holes at stripes to organize chains consisting of parallel spins they generate dipole magnetic field patterns with long dipole cores, maybe with the length of stripe. Suppose this happens so that stripes or at least portions of them would be 1D hole ferromagnets: admittedly somewhat esoteric creatures but making mathematical sense.
 The magnetic flux tubes (in TGD sense!) assignable to these dipole field patterns should accompany stripes and dark Cooper pairs with large Planck constant would reside at these flux tubes.
 The transversal fluctuations of the flux tubes would be present already below T_{c1} and would replace phonons as a mechanism generating the energy gap. Transversal 1D phonons induced as occillations of lattice atoms would be in question. This could explain BCS type characteristics of high T_{c} superconductivity.
 Large enough quantum fluctuations lead to reconnections of flux tubes so that the topology of the resulting network starts to quantum fluctuate. Below T_{c} the reconnection probability becomes high enough to create so long flux tubes that macroscopic supra currents can flow. This process is a special case of a phenomenon known as percolation: the wetting of sand represents a basic example of this phenomenon. Magnetic percolation provides a beautiful interpretation and perhaps also a generalization of the quantum highway metaphor discussed by Jan Zaanen. In fact, also S=0 zero Cooper pairs could become stable below T_{c} since their decay to S=1 pairs would become impossible for topological reasons.
I do not want to bore the reader more except by gluing the abstract of the chapter Superconductivity in ManySheeted Spacetime of the book "pAdic Length Scale Hypothesis and Dark Matter Hierarchy" devoted to high T_{c} superconductivity.
In this chapter a model for high T_{c} superconductivity as quantum critical phenomenon is developed. The relies on the notions of quantum criticality, dynamical quantized Planck constant requiring a generalization of the 8D imbedding space to a book like structure, and manysheeted spacetime. In particular, the notion of magnetic flux tube as a carrier of supra current of central concept.
With a sufficient amount of twisting and weaving these basic ideas one ends up to concrete model for high T_{c} superconductors as quantum critical superconductors consistent with the qualitative facts that I am personally aware. The following minimal model looks the most realistic option found hitherto.
 The general idea is that magnetic flux tubes are carriers of supra currents. In antiferromagnetic phases these flux tube structures form small closed loops so that the system behaves as an insulator. Some mechanism leading to a formation of long flux tubes must exist. Doping creates holes located around stripes, which become positively charged and attract electrons to the flux tubes.
 The higher critical temperature T_{c1} corresponds to a formation local configurations of parallel spins assigned to the holes of stripes giving rise to a local dipole fields with size scale of the order of the length of the stripe. Conducting electrons form Cooper pairs at the magnetic flux tube structures associated with these dipole fields. The elongated structure of the dipoles favors angular momentum L=2 for the pairs. The presence of magnetic field favors Cooper pairs with spin S=1.
 Stripes can be seen as 1D metals with delocalized electrons. The interaction responsible for the energy gap corresponds to the transversal oscillations of the magnetic flux tubes inducing oscillations of the nuclei of the stripe. These transverse phonons have spin and their exchange is a good candidate for the interaction giving rise to a mass gap. This could explain the BCS type aspects of high T_{c} superconductivity.
 Above T_{c} supra currents are possible only in the length scale of the flux tubes of the dipoles which is of the order of stripe length. The reconnections between neighboring flux tube structures induced by the transverse fluctuations give rise to longer flux tubes structures making possible finite conductivity. These occur with certain temperature dependent probability p(T,L) depending on temperature and distance L between the stripes. By criticality p(T,L) depends on the dimensionless variable x=TL/hbar only: p=p(x). At critical temperature T_{c} transverse fluctuations have large amplitude and makes p(x_{c}) so large that very long flux tubes are created and supra currents can run. The phenomenon is completely analogous to percolation.
 The critical temperature T_{c} = x_{c}hbar/L is predicted to be proportional to hbar and inversely proportional to L (, which is indeed to be the case). If flux tubes correspond to a large value of hbar, one can understand the high value of T_{c}. Both Cooper pairs and magnetic flux tube structures represent dark matter in TGD sense.
 The model allows to interpret the characteristic spectral lines in terms of the excitation energy of the transversal fluctuations and gap energy of the Cooper pair. The observed 50 meV threshold for the onset of photon absorption suggests that below T_{c} also S=0 Cooper pairs are possible and have gap energy about 9 meV whereas S=1 Cooper pairs would have gap energy about 27 meV. The flux tube model indeed predicts that S=0 Cooper pairs become stable below T_{c} since they cannot anymore transform to S=1 pairs. Their presence could explain the BCS type aspects of high T_{c} superconductivity. The estimate for hbar/hbar_{0} = r from critical temperature T_{c1} is about r=3 contrary to the original expectations inspired by the model of of living system as a superconductor suggesting much higher value. An unexpected prediction is that coherence length is actually r times longer than the coherence length predicted by conventional theory so that type I superconductor could be in question with stripes serving as duals for the defects of type I superconductor in nearly critical magnetic field replaced now by ferromagnetic phase.
 TGD predicts preferred values for r=hbar/hbar_{0} and the applications to biosystems favor powers of r=2^{11}. r=2^{11} predicts that electron Compton length is of order atomic size scale. Biosuperconductivity could involve electrons with r=2^{22} having size characterized by the thickness of the lipid layer of cell membrane.
At qualitative level the model explains various strange features of high T_{c} superconductors. One can understand the high value of T_{c} and ambivalent character of high T_{c} super conductors, the existence of pseudogap and scalings laws for observables above T_{c}, the role of stripes and doping and the existence of a critical doping, etc...
References
[1] V. J. Emery, S. A. Kivelson, and J. M. Tranquada (1999), Stripe phases in hightemperature superconductors , Perspective, Vol. 96, Issue 16, 88148817, August 3.
[2] J. Zaanen (2006), Superconductivity: Quantum Stripe Search, Nature vol 440, 27 April.
[3] Jan Zaanen (2007), Watching Rush Hour in the World of Electrons. Science vol 315. p. 372.
For details see the chapter BioSystems as SuperConductors: Part I.
