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 properties of a class of topological quantum phase transition (TQPT) are analyzed based on a model proposed by Haldane. We study the effect of finite temperature on this phase transition. We have found that finite temperature would drive this TQPT to be a crossover, while it is stable against the weak short range interaction. When the interaction is strong enough, however, this TQPT is unstable and other states would emerge. Then we investigate the effect of the on-site energy in the original haldane model. The critical di erence between our TQPT and the topological phase transition in conventional quantum Hall system is discussed. Finally, we discuss the potential application of our analysis to a topological phase transition proposed in a realistic system.

In this paper, the quantum phase transition between superfluid state and Mott-insulator state is studied based on an extended Bose-Hubbard model with two-and three-body on-site interactions. By employing the mean-field approximation we find the extension of the insulating 'lobes' and the existence of a fixed point in three dimensional phase space. We investigate the link between experimental parameters and theoretical variables. The possibility to obverse our results through some experimental effects in optically trapped Bose-Einstein Condensates (BEC) is also discussed.

In this paper, we propose a method to understand the nature for the quantum disorder phase of the two-dimensional (2D) high spin frustrated model. The ground state and excitation properties of a fully frustrated 2D spin-1 model are studied based on a model whose groundstate can be found exactly. By analogy to the pseudo-potential approach in the fractional quantum Hall effect, we conclude that the ground states of the fully frustrated spin-1 model are doubly degenerate valance bond solid (VBS) states along the horizontal or vertical direction of the square lattice. We also find that the ground state could be characterized by a nonzero string order, which rarely happened in the 2D case. The method that we used reveals the connection between the fractional quantum hall e ect and the frustrated 2D antiferromagnetism system. The VBS states capture the main character of the disordered phase in the 2D spin-1 frustrated system, and can be verified by a numerical method.

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.