Stable Interacting (2 + 1)d Conformal Field Theories at the Boundary of a class of (3 + 1)d Symmetry Protected Topological Phases

Motivated by recent studies of symmetry protected topological (SPT) phases, we explore the possible gapless quantum disordered phases in the $(2+1)d$ nonlinear sigma model defined on the Grassmannian manifold $\frac{U(N)}{U(n)\times U(N-n)}$ with a Wess-Zumino-Witten (WZW) term at level $k$, which is the effective low energy field theory of the boundary of certain $(3+1)d$ SPT states. With $k=0$, this model has a well-controlled large-$N$ limit, $i\mathrm{.}e\mathrm{.}$ its renormalization group equations can be computed exactly with large-$N$. However, with the WZW term, the large-$N$ and large-$k$ limit alone is not sufficient for a reliable study of the nature of the quantum disordered phase. We demonstrate that through a combined large-$N$, large-$k$ and $ϵ-$generalization, a stable fixed point in the quantum disordered phase can be reliably located in the large$-N$ limit and leading order $ϵ-$expansion, which corresponds to a $(2+1)d$ strongly interacting conformal field theory.