A mutual-Chern-Simons Lagrangian is derived as a minimal field theory description of the phasestring model for doped antiferromagnets. Such an e ective Lagrangian is shown to retain the full symmetries of parity, time-reversal, and global SU (2) spin rotation, in contrast to conventional Chern-Simons theories where first two symmetries are usually broken. Two ordered phases, i.e., antiferromagnetic and superconducting states, are found at low temperatures as characterized by "dual" Meissner e ects and dual flux quantization conditions due to the mutual-Chern-Simons gauge structure. A "dual" confinement in charge/spin degrees of freedom occurs such that no true spincharge separation is present in these ordered phases, but the spin-charge separation/deconfinement serves as a driving force in the unconventional phase transitions of these ordered states to disordered states.

We study several di erent Z2 topological ordered states in frustrated spin systems. The e ective theories for those di erent Z2 topological orders all have the same form-a Z2 gauge theory which can also be written as a mutual U(1)×U(1) Chern-Simons theory. However, we find that the di erent Z2 topological orders are reflected in di erent projective realizations of lattice symmetry in the same e ective mutual Chern-Simons theory. This result is obtained by comparing the groundstate degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasi-particles calculated from the slave-particle theory and from the e ective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.

The existence of conserved spin Hall currents is shown in a strongly correlated system without involving spin-orbit coupling. The spin Hall conductivity is determined by intrinsic bulk properties, which remains finite even when the charge resistivity diverges in strong magnetic fields at zero temperature. The state in the latter limit corresponds to a spin Hall insulator. Such a system is a doped Mott insulator described by the phase string theory, and the spin Hall e ect is predicted to coexist with the Nernst e ect to characterize the intrinsic properties of the low-temperature pseudogap phase.