Motivated by the recent work of QED$_3$-Chern-Simons quantum critical points of fractional Chern insulators (Phys. Rev. X textbf{8}, 031015, (2018)), we study its non-Abelian generalizations, namely QCD$_3$-Chern-Simons quantum phase transitions of f
ractional Chern insulators. These phase transitions are described by Dirac fermions interacting with non-Abelian Chern-Simons gauge fields ($U(N)$, $SU(N)$, $USp(N)$, etc.). Utilizing the level-rank duality of Chern-Simons gauge theory and non-Abelian parton constructions, we discuss two types of QCD$_3$ quantum phase transitions. The first type happens between two Abelian states in different Jain sequences, as opposed to the QED3 transitions between Abelian states in the same Jain sequence. A good example is the transition between $sigma^{xy}=1/3$ state and $sigma^{xy}=-1$ state, which has $N_f=2$ Dirac fermions interacting with a $U(2)$ Chern-Simons gauge field. The second type is naturally involving non-Abelian states. For the sake of experimental feasibility, we focus on transitions of Pfaffian-like states, including the Moore-Read Pfaffian, anti-Pfaffian, particle-hole Pfaffian, etc. These quantum phase transitions could be realized in experimental systems such as fractional Chern insulators in graphene heterostructures.
It has been proposed that the deconfined criticality in $(2+1)d$ -- the quantum phase transition between a Neel anti-ferromagnet and a valence-bond-solid (VBS) -- may actually be pseudo-critical, in the sense that it is a weakly first-order transitio
n with a generically long correlation length. The underlying field theory of the transition would be a slightly complex (non-unitary) fixed point as a result of fixed points annihilation. This proposal was motivated by existing numerical results from large scale Monte-Carlo simulations as well as conformal bootstrap. However, an actual theory of such complex fixed point, incorporating key features of the transition such as the emergent $SO(5)$ symmetry, is so far absent. Here we propose a Wess-Zumino-Witten (WZW) nonlinear sigma model with level $k=1$, defined in $2+epsilon$ dimensions, with target space $S^{3+epsilon}$ and global symmetry $SO(4+epsilon)$. This gives a formal interpolation between the deconfined criticality at $d=3$ and the $SU(2)_1$ WZW theory at $d=2$ describing the spin-$1/2$ Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level $1/k$. We show that at leading order, there is a fixed point annihilation at $d^*approx2.77$, with complex fixed points above this dimension including the physical $d=3$ case. The pseudo-critical properties such as correlation length, scaling dimensions and the drifts of scaling dimensions as the system size increases, calculated crudely to leading order, are qualitatively consistent with existing numerics.
