A close connection between antiferromagnetism and superconductivity is at the core of high-temperature superconductivity. Here, we put forward the projected BCS theory for the unification of antiferromagnetism and strongly correlated superconductivity. Specifically, it is shown that, with the
d-wave pairing symmetry, the projected BCS theory can provide excellent trial states at general doping for the exact ground states of the $t\text{\ensuremath{-}}J$ model in the square lattice, generating a unified theory of high-temperature superconductivity as a continuous function of hole concentration. A key to the success of the projected BCS theory is an accurate treatment of the strong correlation between Cooper pairs, which not only causes the breakdown of superconductivity itself at half filling but also defines the precise nature of strongly correlated superconductivity at moderate doping. Also, via the proper incorporation of particle number fluctuations, the projected BCS theory allows direct computation of the superconducting order parameter, shedding important light on the pseudogap phenomenon.
In order to go beyond the conventional BCS plus RPA approach for the description of pairing correlation, we have developed a possible canonical theory. In this theory, we can exactly get rid of the well-known difficulty of the particle-number non-conservation in the BCS approximation. We give, in this paper, its approximate form in order to clarify the relation with the conventional approach. We show that our theory in its approximate form can reproduce the same result as that obtained by the BCS plus RPA approach. The equivalence between the approach developed by Suzuki et al. and the present one is also shown, within the RPA order at least. We give some perspective for the description of higher-order non-linear effect, such as the coupling between the pairing rotation and pairing vibration, in conclusion.
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.