  The topological quantum spin liquids (SL) and the nature of quantum phase
transitions between them have attracted intensive attentions for the past
twenty years. The extended kagome spin-1/2 antiferromagnet emerges as the
primary candidate for hosting both time reversal symmetry (TRS) preserving and
TRS breaking SLs based on density matrix renormalization group simulations. To
uncover the nature of the novel quantum phase transition between the SL states,
we study a minimum XY model with the nearest neighbor (NN) ($J_{xy}$), the
second and third NN couplings ($J_{2xy}=J_{3xy}=J'_{xy}$). We identify the TRS
broken chiral SL (CSL) with the turn on of a small perturbation $J'_{xy}\sim
0.06 J_{xy}$, which is fully characterized by the fractionally quantized
topological Chern number and the conformal edge spectrum as the $\nu=1/2$
fractional quantum Hall state. On the other hand, the NN XY model ($J'_{xy}=0$)
is shown to be a critical SL state adjacent to the CSL, characterized by the
gapless spin singlet excitations and also vanishing small spin triplet
excitations. The quantum phase transition from the CSL to the gapless critical
SL is driven by the collapsing of the neutral (spin singlet) excitation gap. By
following the evolution of entanglement spectrum, we find that the transition
takes place through the coupling of the edge states with opposite chiralities,
which merge into the bulk and become gapless neutral excitations. The effect of
the NN spin-$z$ coupling $J_z$ is also studied, which leads to a quantum phase
diagram with an extended regime for the gapless SL.
