Universal Features of Higher-Form Symmetries at Phase Transitions

We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept categorical symmetry (labelled as $tilde{Z}_N^{(1)}$) introduced recently, or an explicit $Z_N^{(1)}$ 1-form symmetry. We demonstrate that for many quantum phase transitions involving a $Z_N^{(1)}$ or $tilde{Z}_N^{(1)}$ symmetry, the following expectation value $ langle left( log O_mathcal{C} right)^2 rangle$ takes the form $langle left( log O_mathcal{C} right)^2 rangle sim - frac{A}{epsilon} P+ b log P $, where $O_mathcal{C}$ is an operator defined associated with loop $mathcal{C}$ (or its interior $mathcal{A}$), which reduces to the Wilson loop operator for cases with an explicit $Z_N^{(1)}$ 1-form symmetry. $P$ is the perimeter of $mathcal{C}$, and the $b log P$ term arises from the sharp corners of the loop $mathcal{C}$, which is consistent with recent numerics on a particular example. $b$ is a universal microscopic-independent number, which in (2+1)d is related to the universal conductivity at the quantum phase transition. $b$ can be computed exactly for certain transitions using the dualities between (2+1)d conformal field theories developed in recent years. We also compute the strange correlator of $O_mathcal{C}$: $S_{mathcal{C}} = langle 0 | O_mathcal{C} | 1 rangle / langle 0 | 1 rangle$ where $|0rangle$ and $|1rangle$ are many-body states with different topological nature.