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Quantum Current and Holographic Categorical Symmetry

by Tian Lan, Jing-Ren Zhou

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Submission summary

Authors (as registered SciPost users): Tian Lan
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-02-05
Date submitted: 2023-12-27 03:45
Submitted by: Lan, Tian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


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 one-dimensional lattice systems and analyse the fixed-point models. It is proved that in the fixed-point models, superconducting quantum currents form a Lagrangian algebra in $Z_1(\mathcal{C})$ and the boundary-bulk 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)

Reports on this Submission

Anonymous Report 1 on 2024-1-14 (Invited Report)


The authors have addressed all the questions raised in my previous report. I would thus recommend the paper for publication in its current form.

  • validity: -
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