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arXiv · 2210.03681

Classification of Fermionic Topological Orders from Congruence Representations

Gil Young Cho·Hee-cheol Kim·Donghae Seo·Minyoung You
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Abstract

The fusion rules and braiding statistics of anyons in $(2+1)$D fermionic topological orders are characterized by the modular data of a super-modular category. On the other hand, the modular data of a super-modular category form a congruence representation of the $\Gamma_\theta$ subgroup of the modular group $\mathrm{SL}_2(\mathbb{Z})$. We provide a method to classify the modular data of super-modular categories by first obtaining the congruence representations of $\Gamma_\theta$ and then building candidate modular data out of those representations. We carry out this classification up to rank $10$. We obtain both unitary and non-unitary modular data, including all previously known unitary modular data, and also discover new classes of modular data of rank $10$. We also determine the central charges of all these modular data, without explicitly computing their modular extensions.

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Gil Young Cho, Hee-cheol Kim, Donghae Seo, Minyoung You. 2022-10-07. Classification of Fermionic Topological Orders from Congruence Representations. https://doi.org/10.1103/physrevb.108.115103

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