# Canonical quantization of nonlinear sigma models with theta term, with applications to symmetry-protected topological phases

## TL;DR

This paper rigorously quantizes nonlinear sigma models with theta term on arbitrary manifolds, establishing their suitability for describing symmetry-protected topological phases of bosons in various dimensions.

## Contribution

It provides a canonical quantization framework for NLSMs with theta term on curved manifolds, confirming their role in modeling SPT phases without topological order.

## Key findings

- Unique ground state at theta = 2πk with a finite energy gap
- Ground state wave functional independent of metric, given by Wess-Zumino term
- Models exhibit symmetry-preserving ground states without topological degeneracy

## Abstract

We canonically quantize $O(D+2)$ nonlinear sigma models (NLSMs) with theta term on arbitrary smooth, closed, connected, oriented $D$-dimensional spatial manifolds $\mathcal{M}$, with the goal of proving the suitability of these models for describing symmetry-protected topological (SPT) phases of bosons in $D$ spatial dimensions. We show that in the disordered phase of the NLSM, and when the coefficient $\theta$ of the theta term is an integer multiple of $2\pi$, the theory on $\mathcal{M}$ has a unique ground state and a finite energy gap to all excitations. We also construct the ground state wave functional of the NLSM in this parameter regime, and we show that it is independent of the metric on $\mathcal{M}$ and given by the exponential of a Wess-Zumino term for the NLSM field, in agreement with previous results on flat space. Our results show that the NLSM in the disordered phase and at $\theta=2\pi k$, $k\in\mathbb{Z}$, has a symmetry-preserving ground state but no topological order (i.e., no topology-dependent ground state degeneracy), making it an ideal model for describing SPT phases of bosons. Thus, our work places previous results on SPT phases derived using NLSMs on solid theoretical ground. To canonically quantize the NLSM on $\mathcal{M}$ we use Dirac's method for the quantization of systems with second class constraints, suitably modified to account for the curvature of space. In a series of four appendices we provide the technical background needed to follow the discussion in the main sections of the paper.