Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems

Abstract: The usual condensed matter lattice theories do not include dynamical electromagnetic (EM) field and do not have higher symmetries naturally (unless we engineer fine-tuned toy models to realize higher symmetries). However, for gapped systems, (anomalous) higher symmetries can emerge from the usual condensed matter theories at low energies (usually in a spontaneously broken form). We pointed out that spontaneously broken emergent higher symmetries are nothing but a kind of topological orders. The emergent (anomalous) higher symmetries can be used to constrain possible phase transitions and possible phases induced by certain types of excitations in topological orders. (Anomalous) higher symmetry can also emerge in gapless systems if the gapless excitations contain gapless gauge fields. In particular, EM condensed matter systems that include the dynamical EM field have an anomalous $U(1)$ 1-symmetry if we ignore the magnetic monopoles. So EM condensed matter systems can realize some physical phenomena of anomalous higher symmetry. In particular, any gapped liquid phases of EM condensed matter systems (induced by arbitrary electric charge fluctuations and condensations) must have non-trivial bosonic topological orders.