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 Z-N(1)) introduced recently, or an explicit Z(1) 1-form symmetry. We demonstrate that for many quantum phase transitions involvN ing a ZN(1) or Z-(1) symmetry, the following expectation value 〈(log OC)2〉 takes the form N 〈(log OC)2〉 ∼ − Aε P + b log P, where OC is an operator defined associated with loop C (or its interior A), which reduces to the Wilson loop operator for cases with an explicit ZN(1) 1-form symmetry. P is the perimeter of C, and the b log P term arises from the sharp corners of the loop 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 OC: SC = 〈0|OC|1〉/〈0|1〉 where |0〉 and |1〉 are many-body states with different topological nature.