  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.
