Mean-field theory of spin-liquid states with finite energy gap and topological orders

The mean-field theory of a T- and P-symmetric spin-liquid state is developed. The quasiparticle excitations in the spin-liquid state are shown to be spin-1/2 neutral fermions (the spinons) and charge e spinless bosons (the holons). The spin-liquid state is shown to be characterized by a nontrivial topological order. Although our discussions are based on the mean-field theory, the concept of the topological order and the associated universal properties (e.g., the quantum number of the quasiparticles) are expected to be valid beyond the mean-field theory. We also discuss the dynamical stability of the mean-field theory.