The diffusive thermal conductivity tensor of the A 1 -phase of superfluid 3 He at low temperatures and melting pressure are calculated beyond the s-p approximation, by using the Boltzmann equation approach. The interaction between normal-normal, normal-Bogoliubov and Bogoliubov-Bogoliubov quasiparticles in the collision integrals are considered for important scattering processes such as binary process. At low temperatures, we show that the scattering between Bogoliubov and normal quasiparticles in binary processes plays an important role in the A 1 -phase, and Bogoliubov-Bogoliubov interaction is ignorable.We show that the two normal and superfluid components take part in elements of the diffusive thermal conductivity tensor differently. We obtain the result that the elements of the diffusive thermal conductivities, K xx , K yy and K zz , are proportional to T −1 , and also that the superfluid components of the diffusive thermal conductivity tensor, K xx↑ and K zz↑ , are proportional to T 3 and T , respectively.
In this paper, we investigate quantum correlation of an interacting Fermi system, which is a nodal superconductor (d-wave superconductor) at zero temperature, via quantum entanglement of two electron spins forming Cooper pairs (Werner state), tripartite and quantum discord. The energy gap depends on the angle between the electron momentum and the nodal axis; and at zero temperature we use an approximation in which the energy gap is considered as the linear function of the angle. After calculating single-electron Green's functions, the two-electron space-spin density matrix, which has X-state form, is obtained. The dependence of quantum correlation to the relative distance of electrons spins of Cooper pair and energy gap is investigated. One of the results is, for d-wave case, concurrence (as a measure of entanglement), quantum discord and tripartite are sensitive to the change of magnitude of gap. Another result is both concurrence and discord oscillate. Then, we consider three-dimensional rectangular nanosuperconducting grain in the weak coupling frame. The nano-size effect is entered via gap fluctuation. The dependence of quantum correlation to length of superconductor and lower bound of robustness of tripartite entanglement are determined. Moreover, we show that quantum correlation of d-wave nano-size superconducting grain strongly depends on length of grain (in contrast to swave case). In general, it is found that the length of grain lower, the effect of nano-size on quantum correlation higher. Quantum tripartite for nano-scale d-wave superconductor is better than for bulk d-wave superconductor. However, we find out both bulk and nano-size s-wave superconductors have the same tripartite. Furthermore, entanglement length and quantum correlation length are investigated and it is shown that there is a length of superconductor in which discord becomes zero. Also, for a given fixed length of superconductor, both a peak in discord and a peak in concurrence occur simultaneously.
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