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Quantum Current and Holographic Categorical Symmetry
by Tian Lan, Jing-Ren Zhou
Submission summary
| Authors (as registered SciPost users): | Tian Lan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2305.12917v2 (pdf) |
| Date submitted: | 2023-06-16 04:58 |
| Submitted by: | Lan, Tian |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| 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 condensed is also specified. 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, condensed 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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Report
The manuscript provides an important progress towards a physical understanding of the “holographic categorical symmetry,” which relates a global symmetry in n-dimensions to a topological order with a gapped boundary in (n+1)-dimensions. In particular, the authors show that the mathematical data describing the topological order in (n+1)-dimensions can also be understood as describing “quantum currents” of the physical system in n-dimensions. The concept of “quantum currents” appears to be novel and insightful. The emphasis is on 1-dimensional models, with a possible generalization to higher dimensions discussed at the end.
The authors also discuss a way of constructing commuting projector Hamiltonian lattice models with a categorical symmetry, which are at the fixed point of the lattice renormalization group flow which is rigorously defined in the manuscript. The concept of “condensed quantum currents” is introduced, which holds the same mathematical information as a gapped boundary of a topological order in (n+1)-dimensions.
Overall, the manuscript is very well-written, and contains interesting results together with many explicit examples to guide the readers through various mathematically abstract concepts. The referee thus recommends publication of the manuscript in SciPost.
Requested changes
Below are minor suggestions:
- It appears that the notation for the objects in the Drinfeld Center in Eq. (2.24) follows that in Appendix A, rather than the notation of Definition 2.10. It will be helpful if Appendix A is explicitly referred to around Eq. (2.24).
- A lattice version of Noether current had been previously discussed in the literature, for instance in https://arxiv.org/pdf/2201.01327.pdf. It might be helpful if the relation to the current work (if any) is briefly mentioned.
- In the construction of fixed point models based on Frobenius algebra objects, it might be interesting to briefly mention how one may realize different symmetry protected topological phases within the general framework, when the symmetry is not spontaneously broken.
Below please find some typos:
- Below (2.19) “orthogonal complement *of* ker U in V”
- Below (4.34) “we can *similarly* extract”
- Definition 4.13 “a quantum current (Q,\beta) is a collection *of* symmetric operators”
- Theorem 5.12 “Given *a* Frobenius algebra”
- Top of page 44, “… at fixed-point is no *longer* C”
Author: Tian Lan on 2023-11-16 [id 4120]
(in reply to Report 1 on 2023-11-07)We thank the referee very much for the positive assessment.
We will revise the manuscript according to the suggestions: 1. The realization of 1+1D symmetry protected topological phases is already mentioned in the current version in section 6.2, using the Frobenius algebra corresponding to $(H\subset G, \omega_2 \in H^2(H,U(1))$ constructed in Example E.10. Here $H$ is the unbroken subgroup and $\omega_2$ describes the SPT invariant. For symmetry unbroken case one just takes $H=G$. We will make this part more explicit in the revision. 2. We will add the following paragraph in the introduction discussing the relation between our quantized current and the previous works: We like to comment on the difference between our formulation and the traditional notion of current in quantum mechanics or current operator in quantum field theory. The current carried by a charged quantum particle is traditionally defined as the charge times the probability current. Based on such notion, one can only conclude that the expectation value of the charge is locally conserved. Similarly, a current operator in quantum field theory is such an operator that its expectation value (correlation function) satisfies a local conservation condition. We consider these traditional notions of current as only "semi-classical", in that (1) the local conservation is only satisfied on average, at a macroscopic or statistical level; (2) they can not be used to deal with the discrete or quantized charge transport in a single quantum mechanical process; and (3) they usually require continuous space-time and continuous symmetry. There are recent works [2201.01327] extending the "semi-classical" current to lattice systems. Our formulation, on the contrary, is truly quantum: it can apply to discrete space and discrete symmetry, and can be used to analyse quantized charge transport exactly instead of on average. 3. We also plan to add clarifications to some arguments and remarks.