Colloquium: Order and quantum phase transitions in the cuprate superconductors

It is now widely accepted that the cuprate superconductors are characterized by a long-range order similar to that present in the Bardeen-Cooper-Schrieffer (BCS) theory, that associated with the condensation of Cooper pairs. The author argues that many physical properties of the cuprates require interplay with additional order parameters associated with a proximate Mott insulator. A classification of Mott insulators in two dimensions is proposed. Experimental evidence so far shows that the class appropriate to the cuprates has collinear spin correlations, bond order, and confinement of neutral, spin $S=1/2\mathrm{}$ excitations. Proximity to second-order quantum phase transitions associated with these orders, and with the pairing order of BCS, has led to systematic predictions for many physical properties. In this context the author reviews the results of recent neutron scattering, fluxoid detection, nuclear magnetic resonance, and scanning tunneling microscopy experiments.