Nonperturbative effects of Topological Theta-term

TL;DR

This paper analyzes the nonperturbative effects of a topological $\Theta$-term on (2+1)-dimensional principal chiral models (PCMs) defined on simple compact Lie groups and focuses on $\Theta=\pi$. It proves that the disordered phase at $\Theta=\pi$ is either gapless and conformal or gapped with a two-fold ground-state degeneracy, using nonperturbative boundary/global arguments and RG flow considerations. For $G=SU(2)$, the bulk-boundary correspondence maps the disordered phase to a (1+1)-dimensional boundary theory described by an $O(4)$ NLSM with a $\Theta$-term, where $\Theta=2\pi k$ yields $SU(2)_k$ WZW CFTs, implying protected gapless boundary states. The results generalize to other simple Lie groups and have direct implications for edge states of (3+1)D SPT phases, providing a framework to analyze boundaries via PCM with $\Theta=\pi$.

Abstract

We study the effects of a topological Theta-term on 2+1 dimensional principal chiral models (PCM), which are nonlinear sigma models defined on Lie group manifolds. We find that when Theta = pi, the nature of the disordered phase of the principal chiral model is strongly affected by the topological term: it is either a gapless conformal field theory, or it is gapped and two-fold degenerate. The result of our paper can be used to analyze the boundary states of three dimensional symmetry protected topological phases.

Figures (3)

• Figure 1: $(a)$. We compactify the space of model Eq. \ref{['pc']} and Eq. \ref{['theta']} to a two dimensional cylinder. When $\Theta = 2\pi$ there are gapless boundary states localized at the two boundaries. $(b)$. The first possibility when we tune $\Theta$ from $2\pi$ to $0$, the bulk gap closes at $\Theta = \pi$. $(c)$. The second possibility, the two states at $\Theta = 0$ and $\Theta = 2\pi$ have level crossing at $\Theta = \pi$.
• Figure 2: The two possible RG flows for the coupling constants $g$ and $\Theta$ of the (2+1)-d PCM on a compact Lie group $G$ with Theta term, Eq. \ref{['pc']} (and Eq. \ref{['theta']} for $G=SU(2)$). There is always an ordered phase with small $g$. For $G=SU(2)$, the phase transition is in the conventional three-dimensional O(4) Wilson-Fisher universality class when $\Theta = 0$ and $2\pi$, while the fixed points at $\Theta=\pi$ are presumably in the different universality classes.
• Figure 3: When the O(4) symmetry of Eq. \ref{['theta']} is broken down to O(3)$\times Z_2$, the Skyrmion number is quantized on space $S^2$. We map Eq. \ref{['Philag']} to a one dimensional tight-binding model. Hopping between nearest neighbor sites corresponds to changing the Skyrmion number by 1. The $\Theta$-term grants two types of monopoles a factor $\exp(i\Theta/2)$ and $\exp(- i\Theta/2)$ respectively, which forbids nearest neighbor hopping when $\Theta = \pi$ due to destructive interference between these two types of monopoles.