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Abstract
In this contribution, we pay tribute to the scientific achievements of Professor Rodney J. Bartlett on account of his seminal contributions to the many-body electron correlation problem. We are here concerned with strongly correlated situations as met in the theory of superconductivity. In condensed matter physics, one often makes use of the famous Bardeen–Cooper–Schrieffer (BCS) formulation, while quantum chemists often instigate an approach that originates in Yang's concept of off-diagonal long-range order (ODLRO), and the Coleman–Sasaki extreme state. Our aim is to demonstrate that both approaches are essentially equivalent by deriving the BCS gap equation from the assumption of the presence of ODLRO.
Note
The diagonal tail terms in our model are in fact small, see e.g. Leggett [16], and can be neglected. Furthermore, they do not occur in the BCS theory. Note that the tail is indirectly included since the trace of the total second-order density matrix equals the number of pairings, which correctly yields both the kinetic energy and the interaction contributions. The statistical reasoning in this paper should refer to the thermalisation, which will be discussed elsewhere. Note also that Equations (8) and (15) should, to be exact, contain the tail contributions, before they are neglected as in Equation (19). While the BCS mechanism is commensurate with condensation, ODLRO yields the condensation of N(N-1)/2 pairings into N/2 physical pairs.