Generalized symmetries in singularity-free nonlinear $σ$ models and their disordered phases

Abstract: We study the nonlinear $\sigma$-model in ${(d+1)}$-dimensional spacetime with connected target space $K$ and show that, at energy scales below singular field configurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $d$-representations of a discrete $d$-group $\mathbb{G}^{(d)}$ (i.e. the emergent symmetry is the dual of the invertible $d$-group $\mathbb{G}^{(d)}$ symmetry). The $d$-group $\mathbb{G}^{(d)}$ is determined such that its classifying space $B\mathbb{G}^{(d)}$ is given by the $d$-th Postnikov stage of $K$. In $(2+1)$D and for finite $\mathbb{G}^{(2)}$, this symmetry is always holo-equivalent to an invertible ${0}$-form (ordinary) symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $\sigma$-model spontaneously breaks this symmetry, and when $\mathbb{G}^{(d)}$ is finite, it is described by the deconfined phase of $\mathbb{G}^{(d)}$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $\sigma$-model in ${(3+1)}$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $\mathbb{C}P^{N-1}$ nonlinear $\sigma$-models in the large-$N$ limit.