Topological phases and quantum criticality from $SU(2)$ Chern-Simons-matter theories

Motivated by recent numerical studies where various $SU(2)$ Chern-Simons-matter theories emerge, we analytically study topological phases and quantum criticality in two-dimensional systems described by such theories. First, we classify $SU(2)_k$ topological orders in all lattice spin systems with a $p4\times SO(3)$ symmetry, where $k$ is an arbitrary nonzero integer. We find that for each odd $k$, the topological order can emerge in systems with an arbitrary Lieb-Schultz-Mattis (LSM) anomaly, and the symmetry cannot permute anyons. If the system has a nontrivial (respectively, trivial) LSM anomaly, then there is exactly one (respectively, nine) symmetry-enriched topological (SET) phases. On the other hand, $SU(2)_k$ topological order with any even $k$ can only emerge in systems with a trivial LSM anomaly. If $k\notin\{6, 10, 14, \cdots\}$, the symmetry cannot permute anyons, and there are 16 SET phases. If $k\in\{6, 10, 14, \cdots\}$, there are 4 different ways how the symmetry can permute anyons, and there are 64 SET phases. Next, we analyze the $SU(2)_k$ Chern-Simons theories coupled to $N_f$ flavors of gapless matter fields that can be either bosonic or fermionic. For both types of theories, we consider a joint large-$N_f$ and large-$k$ limit with $N_f/k$ fixed, and compute the scaling dimensions of the bilinear operators of the bosons or fermions to the order of $1/N_f$. These results sharpen our understanding of these emergent exotic topological phases and quantum criticality, and provide useful guidance to explore them further.