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Physica C Superconductivity

Exact commutation relations for the Cooper pair operators and the problem of two interacting Cooper’s pairs

Ilya G. Kaplan, Oracio Navarro, J.A. Sánchez

Published Dec 21, 2004• 8 citations
Publisher
DOI
10.1016/j.physc.2004.11.025

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Abstract The Cooper pair has the total spin S = 0. So, in accordance with the Pauli principle, the wave functions describing the Cooper pair system should have the boson permutation symmetry, but the pairon operators (Cooper's pair operators) do not obey the boson commutation relations. The pairon operators may not be considered neither as the Bose operators, nor as the Fermi operators. In this work, we analyze the statistical properties and the commutation relations for the pairon operator and reveal that they correspond to the modified parafermi statistics of rank p = 1. Two different expressions for the Cooper pair number operator are presented. We demonstrate that the calculations with a Hamiltonian expressed via pairon operators is more convenient using the commutation properties of these operators without presenting them as a product of fermion operators. This allows to study problems in which the interactions between Cooper's pair are also included. The problem of two interacting Cooper's pairs is discussed. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Although BCS pairs of fermions are known to obey neither Bose–Einstein (BE) commutation relations nor BE statistics, we show how Cooper pairs (CPs), whether the simple original ones or the CPs recently generalized in a many-body Bethe–Salpeter approach, being clearly distinct from BCS pairs at least obey BE statistics. Hence, contrary to widespread popular belief, CPs can undergo BE condensation to account for superconductivity if charged, as well as for neutral-atom fermion superfluidity where CPs, but uncharged, are also expected to form.
Although BCS pairs of fermions are known not to obey Bose-Einstein (BE) commutation relations nor BE statistics, we show how Cooper pairs (CPs), whether the simple original ones or the CPs recently generalized in a many-body Bethe-Salpeter approach, being clearly distinct from BCS pairs at least obey BE statistics. Hence, contrary to widespread popular belief, CPs can undergo BE condensation to account for superconductivity if charged, as well as for neutral-atom fermion superfluidity where CPs, but uncharged, are also expected to form. Keywords: Cooper pairs; Bose-Einstein condensation; superconductors; superfluids 1