Quantum Current and Holographic Categorical Symmetry
- URL: http://arxiv.org/abs/2305.12917v3
- Date: Sat, 23 Dec 2023 05:34:07 GMT
- Title: Quantum Current and Holographic Categorical Symmetry
- Authors: Tian Lan and Jing-Ren Zhou
- Abstract summary: A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance.
The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
- Score: 62.07387569558919
- License: http://creativecommons.org/licenses/by/4.0/
- 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 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.
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