The SU(2) slave-boson mean-field theory for the ${\mathrm{tt}}^{\prime }J$ model is analyzed. The role of next-nearest-neighbor hopping $t^{\prime }$ on the phase diagram is studied. We find a pseudogap phase in hole-doped (HD) materials (where $t^{\prime }<0).\mathrm{}$ The pseudogap phase is a U(1) spin liquid (the staggered-flux phase) with a U(1) gauge interaction and no fractionalization. This agrees with experiments on HD samples. The same calculation also indicates that a positive $t^{\prime }$ favors a $Z_2$ state with true spin-charge separation. The $Z_2$ state that exists when $t^{\prime }\gtrsim 0.5J$ can be a candidate for the pseudogap phase of electron-doped (ED) cuprates (if such a phase exists). The experimental situation in ED materials is also addressed.

• Received 27 February 2003

DOI:https://doi.org/10.1103/PhysRevB.68.024501