Universal Features of Higher-Form Symmetries at Phase Transitions

Xiao-Chuan Wu; Chao-Ming Jian; Cenke Xu

doi:10.21468/SciPostPhys.11.2.033

paper

Universal Features of Higher-Form Symmetries at Phase Transitions

arXiv:2101.10342 · doi:10.21468/SciPostPhys.11.2.033

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 $Z_{N}^{(1)}$ 1-form symmetry. We demonstrate that for many quantum phase transitions involving a $Z_{N}^{(1)}$ or $Z ~_{N}^{(1)}$ symmetry, the following expectation value $⟨ (lo g O_{C})^{2} ⟩$ takes the form $⟨ (lo g O_{C})^{2} ⟩ \sim - \frac{A}{ε} P + b lo g P$ , where $O_{C}$ is an operator defined associated with loop $C$ (or its interior $A$ ), which reduces to the Wilson loop operator for cases with an explicit $Z_{N}^{(1)}$ 1-form symmetry. $P$ is the perimeter of $C$ , and the $b lo g P$ 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 $O_{C}$ : $S_{C} = ⟨ 0∣ O_{C} ∣1 ⟩ / ⟨ 0∣1 ⟩$ where $∣0 ⟩$ and $∣1 ⟩$ are many-body states with different topological nature.

arXiv comment9 pages, 2 figures

Universal Features of Higher-Form Symmetries at Phase Transitions

Abstract

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