1eand 2esuperconductivity in spin liquids and spin crystals

The recent, unexpected discoveries of superconductors with reported transition temperatures as high as 123 K have renewed the search for mechanisms with higher transition temperatures. Among the resulting proposals is one by Anderson in which an odd-electron insulator in a resonating valence-bond or quantum-spin-liquid state becomes superconducting via pair condensation when sufficiently doped. However, Kivelson, Rokhsar, and Sethra have suggested that doping such a state introduces singly charged boson defects which form a superconducting state via Bose-Einstein condensation. We resolve this apparent contradiction by pointing out the possible existence of a remarkable transition between superconductivity with an elementary charge of the supercurrent of e to superconductivity with an elementary charge of 2e with increased doping. Moreover, both kinds of superconductivity occur, even in the spin-crystal or N\'eel state.