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Extending the planar theory of anyons to quantum wire networks
by Tomasz Maciazek, Mia Conlon, Gert Vercleyen, J. K. Slingerland
Submission summary
| Authors (as registered SciPost users): | Mia Conlon |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2301.06590v2 (pdf) |
| Date submitted: | 2024-10-10 12:33 |
| Submitted by: | Conlon, Mia |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
The braiding of the worldlines of particles restricted to move on a network (graph) is governed by the graph braid group, which can be strikingly different from the standard braid group known from two-dimensional physics. It has been recently shown that imposing the compatibility of graph braiding with anyon fusion for anyons exchanging at a single wire junction leads to new types of anyon models with the braiding exchange operators stemming from solutions of certain generalised hexagon equations. In this work, we establish these graph-braided anyon fusion models for general wire networks. We show that the character of braiding strongly depends on the graph-theoretic connectivity of the given network. In particular, we prove that triconnected networks yield the same braiding exchange operators as the planar anyon models. In contrast, modular biconnected networks support independent braiding exchange operators in different modules. Consequently, such modular networks may lead to more efficient topological quantum computer circuits. Finally, we conjecture that the graph-braided anyon fusion models will possess the (generalised) coherence property where certain polygon equations determine the braiding exchange operators for an arbitrary number of anyons. We also extensively study solutions to these polygon equations for chosen low-rank multiplicity-free fusion rings, including the Ising theory, quantum double of Z2, and Tambara-Yamagami models. We find numerous solutions that do not appear in the planar theory of anyons.
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- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
List of changes
* In Appendix A - changed the word anyon to particle to hopefully make the distinction between discussing the graph braid group and the anyons on a graph.
* We included the fact that we study multiplicity free anyon models in the Abstract, Introduction and Conclusion.
* In Section 2, included a sentence about the Haagreup theory as an example of non commutative fusion.
* Mentioned that we are restricting to unitary anyon models in the Introduction.
* Included a sentence in Section 2, where we discuss the pentagon equation that we consider unitary solutions of the pentagon equation.
* Included the algebraic presentation of the graph braid group for each graph studied in the paper.
* We have rewritten Section 3 to explicitly include the presentation for the generators for the graph braid group on a trijunction.
* We have included an additional figure ( Figure 19.), which displays that the G-symbol does not preserve the topological charge of a x b.
* We have included a summary of main results section.
* We have uploaded a new Figure 5.
* Removed the phrasing ``composite anyon'' from the paper.
* We have included a sentence in Section 2 that we define an anyon model as a ribbon fusion category not a modular fusion category as we do not study the analogue of modularity on a graph.
* Author name changed from A.Conlon to M.Conlon.
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Dear Editor,
The authors used the questions/comments/suggestions of both referees to improve their manuscript. I am satisfied by the answers and changes to the manuscript concerning my first referee report. I therefore recommend the manuscript for publication.
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