In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg MCSLG theory to describe the continuous topological quantum phase transition TQPT. In particular, we consider the TQPT etween a spin-polarized phase a state without topological order and a Z2 topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the Z2 topological order is broken down by the condensation of Z2 charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to Z2 topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.

In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2A or Z2B projective symmetry group. We find that the Z2A translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translationsymmetry protected topological orders. We introduced four Z2 topological indices k=0,1 at k=(0,0), (0,π),(π,0), and(π,π) to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-byeven, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2A topological orders for a system on three-dimensional cubic lattice.

The crossover from thermal hopping to quantum tunneling is studied. We show that the decay rate-with dissipation can accurately be determined near the crossover temperature. Besides considering the Wentzel-Kramers-Brillouin (WKB) exponent, we also calculate contribution of the fluctuation modes around the saddle point and give an extended account of a previous study of crossover region. We deal with two dangerous fluctuation modes whose contribution can't be calculated by the steepest descent method and show that higher order couplings between the two dangerous modes need to be taken into considerations. At last the crossover from thermal hopping to quantum tunneling in the molecular agnet Mn12Ac is studied.

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.

In this paper, the degenerate ground states of Z2 topological order on a plane with holes (the so-called surface codes) are used as the protected code subspace to build a topological quantum computer by tuning their quantum tunneling e ect. Using a designer Hamiltonian-the Kitaev toric-code model as an example, we study quantum tunneling e ects of the surface codes and obtain its e ective theory. Finally, we show how to do topological quantum computation including the initialization, the unitary transformation and the measurement.