Abstract
Pure quantum phenomena are characterized by intrinsic recurrences in space and time. We use this intrinsic periodicity as a quantization condition to derive a heuristic description of the essential quantum phenomenology of superconductivity. The resulting description is based on fundamental quantum dynamics and geometrical considerations, rather than on microscopical characteristics of the superconducting materials. This allows us to investigate the related gauge symmetry breaking in terms of the competition between quantum recurrence and thermal noise. We also test the validity of this approach to describe the case of carbon nanotubes.
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Notes
The semiclassical quantization of the magnetic flux induced by the PBCs is similar to the quantization of the angular momentum in a cylindric Schrödinger problem. The quantum number \(n\) can also be regarded as describing the quantized angular momentum of the current. Thus, for momentum conservation, once that a current is created \(n \ge 1\) it cannot decay to the state \(n=0\). A similar effect can be observed in superfluidity of a toroidal Bose–Einstein condensate where the case with zero angular momentum is only possible by introducing a potential barrier that breaks the rotational invariance of the superflow [29].
By considering the relativistic modulation of time periodicity (relativistic Doppler effect) associated to boosts of an elementary particle, the resulting dispersion relation of the energy spectrum is \(E_{n}(\bar{\mathbf p} ) = n \bar{E} (\bar{\mathbf p} ) = n h / T(\bar{\mathbf p} ) = n \sqrt{ \bar{m} ^{2} c^{4} + {\bar{ \mathbf p }}^{2} c^{2}}\). The energy bands of a material are the analogous of the quantum levels of the energy spectrum of a second quantized field. The vacuum energy \(n h / 2 T(\bar{\mathbf p} )\) corresponds to half twist, anti-PBCs. Interaction between the ECCs and the atom in the lattice can result in deformations of this perfectly harmonic band structure.
For a more accurate evaluation we must consider the reduced speed of light in the material.
As in particular lattice configurations of Armchair CNs [21], the local extremal point in the dispersion relation of the time periodicity determining the mass spectrum of the ECC may occur with small residual values of the axial momentum.
It is possible to show in several different ways that the PBCs of the matter field induces a periodicity to the related gauge field (modulo gauge transformation), as for instance by considering the effect of the PBCs of the gauge connection (15) or by using the similar formalism of field theory at finite temperature (Euclidean periodicity), as shown in [30]. A detailed description of this aspect is given in [6].
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Dolce, D., Perali, A. The role of quantum recurrence in superconductivity, carbon nanotubes and related gauge symmetry breaking. Found Phys 44, 905–922 (2014). https://doi.org/10.1007/s10701-014-9816-y
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DOI: https://doi.org/10.1007/s10701-014-9816-y