High Tc superconductivity in many-sheeted space-time
Mark McWilliams sent me some time ago a link Scientists Detect 'Fingerprint' Of High-temp Superconductivity Above Transition Temperature. What this fingerprint means is that Cooper pairs exists below a critical temperature Tc1 higher than the critical temperature Tc for the onset of the super-conductivity. The finding is surprising but nothing spectacular in a wider perspective. Also the atoms forming Bose-Einstein condensates exists stably above the critical temperature for Bose-Einstein condensation. The finding however suggests what the correct question might be. The reader who has decided to discover the mechanism of high Tc superconductivity could try either or both of the following alternatives.
- Why quantum coherence for Cooper pairs is possible only below the critical temperature Tc but not in the range (Tc,Tc1)?
- Why supra currents flow over long distances only below Tc?
I hasten to confess that I am an amateur in the field of high Tc 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 Tc 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 co-ashame 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 non-conventional 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 Tc cuprate superconductors from BCS type superconductors is that they are quantum critical. What is known that so called stripes (, ) 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 super-conductivity. Magnetic fields are usually regarded as enemy of super-conductivity but for spin 1 Cooper pairs magnetic field tends to stabilize the pairs. In high Tc 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 bio-superconductivity magnetic flux tubes are the carriers of super-conducting 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 Tc cuprate super-conductors.
- 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 1-D 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 Tc1 and would replace phonons as a mechanism generating the energy gap. Transversal 1-D phonons induced as occillations of lattice atoms would be in question. This could explain BCS type characteristics of high Tc superconductivity.
- Large enough quantum fluctuations lead to reconnections of flux tubes so that the topology of the resulting network starts to quantum fluctuate. Below Tc 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 Tc 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 Super-conductivity in Many-Sheeted Space-time of the book "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" devoted to high Tc superconductivity.
In this chapter a model for high Tc super-conductivity as quantum critical phenomenon is developed. The relies on the notions of quantum criticality, dynamical quantized Planck constant requiring a generalization of the 8-D imbedding space to a book like structure, and many-sheeted space-time. 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 Tc 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 anti-ferromagnetic 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 Tc1 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 1-D 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 Tc super-conductivity.
- Above Tc 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 Tc transverse fluctuations have large amplitude and makes p(xc) so large that very long flux tubes are created and supra currents can run. The phenomenon is completely analogous to percolation.
- The critical temperature Tc = xchbar/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 Tc. 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 Tc 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 Tc since they cannot anymore transform to S=1 pairs. Their presence could explain the BCS type aspects of high Tc super-conductivity. The estimate for hbar/hbar0 = r from critical temperature Tc1 is about r=3 contrary to the original expectations inspired by the model of of living system as a super-conductor 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 super-conductor could be in question with stripes serving as duals for the defects of type I super-conductor in nearly critical magnetic field replaced now by ferromagnetic phase.
- TGD predicts preferred values for r=hbar/hbar0 and the applications to bio-systems favor powers of r=211. r=211 predicts that electron Compton length is of order atomic size scale. Bio-superconductivity could involve electrons with r=222 having size characterized by the thickness of the lipid layer of cell membrane.
At qualitative level the model explains various strange features of high Tc superconductors. One can understand the high value of Tc and ambivalent character of high Tc super conductors, the existence of pseudogap and scalings laws for observables above Tc, the role of stripes and doping and the existence of a critical doping, etc...
 V. J. Emery, S. A. Kivelson, and J. M. Tranquada (1999), Stripe phases in high-temperature superconductors , Perspective, Vol. 96, Issue 16, 8814-8817, August 3.
 J. Zaanen (2006), Superconductivity: Quantum Stripe Search, Nature vol 440, 27 April.
 Jan Zaanen (2007), Watching Rush Hour in the World of Electrons. Science vol 315. p. 372.
For details see the chapter Bio-Systems as Super-Conductors: Part I.