Luttinger-volume violating Fermi liquid in the pseudogap phase of the cuprate superconductors

Based on the NMR measurements on Bi2Sr2-xLaxCuO6+delta (La-Bi2201) in strong magnetic fields, we identify the nonsuperconducting pseudogap phase in the cuprates as a Luttinger-volume violating Fermi liquid (LvvFL). This state is a zero-temperature quantum liquid that does not break translational symmetry, and yet, the Fermi surface encloses a volume smaller than the large one given by the Luttinger theorem. The particle number enclosed by the small Fermi surface in the LvvFL equals the doping level p, not the total electron number n(e) = 1 - p. Both the phase string theory and the dopon theory are introduced to describe the LvvFL. For the dopon theory, we can obtain a semiquantitative agreement with the NMR experiments.