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Since the discovery of high-temperature superconductivity in 1986 by Bednorzand Mueller, great efforts have been devoted to finding out how and why itworks. From the d-wave symmetry of the order parameter, the importance ofantiferromagnetic fluctuations, and the presence of a mysterious pseudogapphase close to the Mott state, one can conclude that high-Tc superconductorsare clearly distinguishable from the well-understood BCS superconductors. Thed-wave superconducting state can be understood through a Gutzwiller-typeprojected BCS wave-function. In this review article, we revisit the Hubbardmodel at half-filling and focus on the emergence of exotic superconductivitywith d-wave symmetry in the vicinity of the Mott state, starting from laddersystems and then studying the dimensional crossovers to higher dimensions. Thisallows to confirm that short-range antiferromagnetic fluctuations can mediatesuperconductivity with d-wave symmetry. Ladders are also nice prototype systemsallowing to demonstrate the truncation of the Fermi surface and the emergenceof a Resonating Valence Bond (RVB) state with preformed pairs in the vicinityof the Mott state. In two dimensions, a similar scenario emerges fromrenormalization group arguments. We also discuss theoretical predictions forthe d-wave superconducting phase as well as the pseudogap phase, and addressthe crossover to the overdoped regime. Finally, cold atomic systems withtunable parameters also provide a complementary insight into this outstandingproblem.

Superconductivity close to the Mott state: From condensed-matter systems to superfluidity in optical lattices