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

TL;DR

This paper investigates the boundary phases of 3D SPT states using a (2+1)d nonlinear sigma model with a WZW term, identifying a stable conformal fixed point through combined large-N, large-k, and epsilon-expansion methods.

Contribution

It introduces a novel approach combining large-N, large-k, and epsilon-expansion to locate stable fixed points in boundary theories of 3D SPT phases.

Findings

Identified a stable fixed point in the quantum disordered phase.

Demonstrated the effectiveness of combined large-N, large-k, and epsilon-expansion methods.

Connected boundary conformal field theories to topological phases.

Abstract

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, $i.e.$ 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 $\epsilon-$generalization, a stable fixed point in the quantum disordered phase can be reliably located in the large$-N$ limit and leading order $\epsilon-$expansion, which corresponds to a $(2+1)d$ strongly interacting conformal field theory.