Superconductivity dome rises from damped phonons

I received a link to an interesting popular article "Superconductivity Dome Rises from Damped Phonons" (see this). The article tells about findings Setty et al (see this) about BCS superconductivity near ferro-electic phase transition.

The system studied is a conventional superconductor near its critical temperature also in the vicinity of the ferroelectric phase transition. It is known that the critical temperature for superconductivity has a dome-like peak in this region. The origin of this peak has remained poorly understood and an explanation for the dome has been proposed in the article.

1. In the BCS model for the conventional low temperature superconductivity phonons bind electrons to Cooper pairs. If the phonons are damped for some reason, Tc is expected to decrease. In ferroelectric superconductors near critical temperature for the transition to ferro-electricity the situation is opposite to this. What could happen?
2. Electron-photon scattering as an analog of Compton scattering is what matters. The scattering of the phonon is Stokes or anti-Stokes depending on whether the scattered phonon gains or loses energy. In anti-Stokes scattering phonon gives energy for the electron, which disfavors the formation of pairs whereas Stokes scattering favors the formation of Cooper pairs.
3. Near the phase transition to ferro-electricity phonon damping occurs. This means that the phonon life-time gets shorter. In ordinary materials this would lead to a reduction of critical temperature but in ferroelectrets the critical temperature has a dome-like peak around the critical criticality for the transition to ferro-eletricity. Ferroelectric transitions involve a non-linear phonon-electron coupling. This anharmonic coupling implies that scattering now also involves final states with 2 phonons. This implies that anti-Stokes scattering is suppressed more than Stokes scattering. The proposal is that this raises the critical temperature and causes the dome-like structure.

1. Soft modes have a long wavelength, which approaches zero at the ferroelectric critical temperature T_c,f. Since soft modes generate long range correlations and induce a polarization of the ferroelectret, their wavelengths are much longer than the lattice constant.

2. Soft modes corresponds to photon energy not larger than 10^-4 eV, which is near the gap energy E_gap of superconductor with critical temperature Tc=10^-4 eV. This corresponds to photon wave length about λ_γ=1.24× 10^-2 m and phonon wavelength λ_ph around 2.56 × 10^-7 m for c_s= 6× 10³ m/s, that is c_s=2× 10^-5 c. The wavelength is much longer than atomic size in accordance with the generation of long range correlations. Interestingly, in the TGD framework, λ_ph corresponds to p-adic length scale L(167)= 2^(167-151)/2× L(151), L(151)= 10 nm.
3. Could soft phonons associated with the ferroelectric transition with energies below this 10^-4 eV compete with thermal excitations by reducing the energies of electrons via the Stokes scattering and in this manner raise the critical temperature?

   This suggests that the coupling of electrons to soft ferroelectric phonons with frequencies below 10^-10 Hz facilitates the formation of Cooper pairs so that their thermal decay is compensated and T_c increases.

1. In the TGD framework, Cooper pairs are dark in the sense that they have h_eff> h and reside at the magnetic flux tubes. The creation of Cooper pairs requires an increase of h_eff. Phonon or photon exchange could transform an ordinary electron pair to a dark pair, which is Galois singlet so that (using p-adic mass scale as a unit) it has 4-momentum with integer-valued components and expressible as sum of algebraic integer valued momenta of dark quarks.
2. This is not enough: there must be a mechanism reducing the value of of the dark electron pair so that it cannot decay back to the ordinary electrons. The decay can be prevented by Fermi statistics in the presence of a Fermi sphere. This is possible if the state can decay to a Galois singlet dark electron pair with energy so small that decay products would belong inside the Fermi sphere.

   This requires an emission of a dark photon or a dark photon pair which is necessarily a Galois singlet transforming to photons or phonons (in ferroelectrets there is a strong coupling between photons and phonons). The reduction of energy would correspond to the gap energy E_gap.

3. For ordinary superconductors with Tc measured in Kelvins, the gap energy is E_gap≃ 10^-4 eV. Could the exchange of phonons with energy in the energy range of soft phonons give rise to the dark states, which decay to Cooper pairs stable below Tc?

   For high Tc superconductors the gap energy is considerably stronger: for T= 100 K the gap energy is about E_gap≃ 10^-2 eV and by factor 100 larger than for T= 1 K. For photons, one would have have λ_γ ≃ 1.24 × 10^-4 m not far from the p-adic length scale L₁₇₉≃ 1.6× 10^-4 m. This corresponds to the size of a large neuron which is an important length scale in biology.

4. One can ask whether the high Tc superconductivity in biomatter could involve this kind of mechanism. At physiological temperatures one would have E_gap≃ 3× 10^-2 eV and this is not far from the cell membrane potential. Living matter is full of ferroelectrets meaning that photons and photons are strongly coupled. Therefore also in living matter, soft phonons near the criticality of ferroelectrets could compete with the thermal excitations to raise the critical temperature Tc.

   Magnetic flux tubes play a key role in the TGD based model of living matter and they can become electric with a simple deformation and generate the long range correlations via the oscillations of the flux tube length giving rise to the space-time correlates of sound waves.

A word of criticism relates to the notion of phonon in the TGD framework.

1. At the level of H, flux tubes correspond strings: at the point of the string world sheet the normal space of X⁴ ⊂ M⁸ characterized by a point of CP₂ is not unique and is characterized by points of a geodesic sphere of CP₂. The boundaries of a string at the mass shell H³ of M⁴⊂ M⁸ should characterized the phonon as an oscillation of the distance of the ends in H.
2. At the level of M⁸ everything is described in terms of momenta belonging to 3-D mass shells define by roots of polynomial defining the 4-surface. M⁸-H duality can be represented as the deformation of M⁴ containing the real projections of the mass shells and representable as an element of local CP₂. It is however far from clear what the counterpart of the flux tube picture for photons could be.

   In M⁸ there is no time and it would seem that the emission of phonon must correspond to momenta at positive and negative energy mass shells differing by the energy of phonon. The H image of X⁴ under M⁸-H duality give rise to the flux tube picture description but what does this description correspond at the level of M⁸?

3. X⁴⊂ M⁸? X⁴ should connect the two opposite mass shells of M⁸. Do the 8-momenta of X⁴ have any reasonable physical interpretation? As long as one does not have excellent reasons for the existence of X⁴, also M⁸-H duality can be challenged. One possibility is that M⁸ picture is enough in the sense that the deformations of M⁴ ⊂ M⁸ can be regarded as local CP₂ elements and allow an interpretation in terms of the space-time picture with M⁴ space-time coordinates related to M⁴ momenta essentially by inversion (see this). This would conform with the Uncertainty Principle.