Continuous quantum phase transitions beyond Landau's paradigm in a   large-N spin model
  We study a large-N generalization of $J_1$-$J_2$ Heisenberg model on square lattice -- an $Sp(2N)$ spin model. The possible quantum spin liquid phases of the $Sp(2N)$ model are studied using the SU(2) projective construction. We find several spin liquid states at least in large $N$ limit, which include SU(2) $\pi$-flux state, SU(2) chiral spin state and $Z_2$ spin liquid states. All those spin liquid states have non-trivial quantum orders. We show how projective symmetry group, which characterizes quantum order, protects the stability of even gapless spin liquids. We also study the continuous quantum phase transition from the SU(2) $\pi$-flux state to the SU(2) chiral spin liquid state, and from the SU(2) $\pi$-flux state to the $Z_2$ spin liquid state. We show that those phase transitions are beyond the paradigm of Landau symmetry-breaking theory. The first phase transition, although a $Z_2$ symmetry breaking transition, contains critical exponents that are different from those obtained from the Ginzburg-Landau theory of Ising universality class. The second transition does not even involve a change of symmetry and has no symmetry breaking order parameter. 
Ying Ran and Xiao-gang Wen