Universal features of higher-form symmetries at phase transitions

Abstract

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)}Z̃N(1)) introduced recently, or an explicit Z_N^{(1)}ZN(1) 1-form symmetry. We demonstrate that for many quantum phase transitions involving a Z_N^{(1)}ZN(1) or \tilde{Z}_N^{(1)}Z̃N(1) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle⟨(O𝒞)2⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P⟨(logO𝒞)2⟩∼−AϵP+blogP, where O_\mathcal{C}O𝒞 is an operator defined associated with loop \mathcal{C}𝒞 (or its interior \mathcal{A}𝒜), which reduces to the Wilson loop operator for cases with an explicit Z_N^{(1)}ZN(1) 1-form symmetry. PP is the perimeter of \mathcal{C}𝒞, and the b \log PblogP term arises from the sharp corners of the loop \mathcal{C}𝒞, which is consistent with recent numerics on a particular example. bb is a universal microscopic-independent number, which in (2+1)d(2+1)d is related to the universal conductivity at the quantum phase transition. bb can be computed exactly for certain transitions using the dualities between (2+1)d(2+1)d conformal field theories developed in recent years. We also compute the "strange correlator" of O_\mathcal{C}O𝒞: S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangleS𝒞=⟨0|O𝒞|1⟩/⟨0|1⟩ where |0\rangle|0⟩ and |1\rangle|1⟩ are many-body states with different topological nature.