Generalized symmetries in singularity-free nonlinear \(\sigma\) models and their disordered phases

• Published in: arXiv.org 2024-11
• Main Authors: Pace, Salvatore D, Zhu, Chenchang, Beaudry, Agnès, Xiao-Gang, Wen
• Format: Article
• Language: English
• Subjects: Classification; Electrodynamics; Equivalence; Gauge theory; Group theory; Singularities; Symmetry
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2310.08554

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.