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Quantum Current and Holographic Categorical Symmetry
by Tian Lan, JingRen Zhou
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Submission summary
Authors (as registered SciPost users):  Tian Lan 
Submission information  

Preprint Link:  https://arxiv.org/abs/2305.12917v3 (pdf) 
Date accepted:  20240205 
Date submitted:  20231227 03:45 
Submitted by:  Lan, Tian 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We establish the formulation for quantum current. Given a symmetry group $G$, let $\mathcal{C}:=\mathrm{Rep} G$ be its representation category. Physically, symmetry charges are objects of $\mathcal{C}$ and symmetric operators are morphisms in $\mathcal{C}$. The addition of charges is given by the tensor product of representations. For any symmetric operator $O$ crossing two subsystems, the exact symmetry charge transported by $O$ can be extracted. The quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. A quantum current exactly corresponds to an object in the Drinfeld center $Z_1(\mathcal{C})$. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension. To express the local conservation, the internal hom must be used to compute the charge difference, and the framework of enriched category is inevitable. To illustrate these ideas, we develop a rigorous scheme of renormalization in onedimensional lattice systems and analyse the fixedpoint models. It is proved that in the fixedpoint models, superconducting quantum currents form a Lagrangian algebra in $Z_1(\mathcal{C})$ and the boundarybulk correspondence is verified in the enriched setting. Overall, the quantum current provides a natural physical interpretation to the holographic categorical symmetry.
List of changes
1. The terminology that the quantum current is "condensed" is changed to "superconducting". A superconducting quantum current can transport charges over a long distance without costing any energy. This terminology“superconducting” fits better with the idea of quantum current than condensation.
2. A paragraph and several sentences are add in the introduction commenting on the relation to the traditional notion
of current in quantum mechanics, and other recent related works.
3. A sentence is added in Section 6.2 on the physical meaning of $(H,\omega_2)$. Table 1 is revised with a new entry on how to realize phases with nontrivial $\omega_2$.
4. A new remark is added under Convention 1, explaining the case when $\mathcal C$ is a generic fusion category and the relation to weak Hopf algebras.
5. More physical explanations are added around the definition of quantum current.
6. Other minor modifications and improvements.
Published as SciPost Phys. 16, 053 (2024)