Classification of Fermionic Topological Orders from Congruence Representations

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.

32 pages, 2 figures, 6 tables

References in corpus (5)

• Non-Abelian Anyons and Topological Quantum Computation
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• On Classification of Fermionic Rational Conformal Field Theories
• Reconstruction of modular data from $S{L}_2(\mathbb{Z})$ representations
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Cited by in corpus (5)

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• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders
• Gapped boundaries of fermionic topological orders and higher central charges
• Modular extension of topological orders from congruence representations
• Most two-dimensional bosonic topological orders forbid sign-problem-free quantum Monte Carlo: Nonpositive Gauss sum as an indicator