Universal features of higher-form symmetries at phase transitions

• Published in: SciPost physics Vol. 11; no. 2; p. 033
• Main Authors: Wu, Xiao-Chuan, Jian, Chao-Ming, Xu, Cenke
• Format: Journal Article
• Language: English
• Published: SciPost 01.08.2021
• ISSN: 2542-4653; 2542-4653
• DOI: 10.21468/SciPostPhys.11.2.033

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)} Z ̃ N ( 1 ) ) introduced recently, or an explicit Z_N^{(1)} Z N ( 1 ) 1-form symmetry. We demonstrate that for many quantum phase transitions involving a Z_N^{(1)} Z N ( 1 ) or \tilde{Z}_N^{(1)} Z ̃ N ( 1 ) symmetry, the following expectation value \langle \left( O_\mathcal{C}\right)^2 \rangle ⟨ ( O ) 2 ⟩ takes the form \langle \left( \log O_\mathcal{C} \right)^2 \rangle \sim - \frac{A}{\epsilon} P + b \log P ⟨ ( log O ) 2 ⟩ ∼ − A ϵ P + b log P , where O_\mathcal{C} O 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)} Z N ( 1 ) 1-form symmetry. P P is the perimeter of \mathcal{C} , and the b \log P 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 b 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. b b 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} O : S_{\mathcal{C}} = \langle 0 | O_\mathcal{C} | 1 \rangle / \langle 0 | 1 \rangle S = ⟨ 0 | O | 1 ⟩ / ⟨ 0 | 1 ⟩ where |0\rangle | 0 ⟩ and |1\rangle | 1 ⟩ are many-body states with different topological nature.