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arXiv:2312.00868 · Physical review. B./Physical review. B

Modular extension of topological orders from congruence representations

Donghae Seo, Minyoung You, Gil Young Cho, Hee‐Cheol Kim

Published Apr 2, 2024• 3 citations
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Abstract

We present an efficient method to compute the modular extension of both fermionic topological orders and ${\mathbb{Z}}_{2}$-symmetric bosonic topological orders in two spatial dimensions, from congruence representations of ${\mathrm{SL}}_{2}(\mathbb{Z})$ and its subgroups. To demonstrate the validity of our approach, we provide explicit calculations for topological orders with ranks up to 10 for the fermionic cases and up to 6 for the bosonic cases. Along the way, we clarify the relation between fermionic rational conformal field theories, which live on the boundary of the corresponding fermionic topological orders, and modular extensions. We show that the ${\mathrm{SL}}_{2}(\mathbb{Z})$ representation of the Ramond-Ramond sector can be determined from the Nuveu-Schwarz-Nuveu-Schwarz sector using the modular extensions.

DOI
10.1103/physrevb.109.155105

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We present an efficient method to compute the modular extension of both fermionic topological orders and Z2{\mathbb{Z}}_{2}Z2​-symmetric bosonic topological orders in two spatial dimensions, from congruence representations of SL2(Z){\mathrm{SL}}_{2}(\mathbb{Z})SL2​(Z) and its subgroups. To demonstrate the validity of our approach, we provide explicit calculations for topological orders with ranks up to 10 for the fermionic cases and up to 6 for the bosonic cases. Along the way, we clarify the relation between fermionic rational conformal field theories, which live on the boundary of the corresponding fermionic topological orders, and modular extensions. We show that the SL2(Z){\mathrm{SL}}_{2}(\mathbb{Z})SL2​(Z) representation of the Ramond-Ramond sector can be determined from the Nuveu-Schwarz-Nuveu-Schwarz sector using the modular extensions.
Two-dimensional topologically ordered states such as fractional quantum Hall fluids host anyonic excitations, which are relevant for realizing fault-tolerant topological quantum computers. Classification and characterization of topological orders have been intensely pursued in both the condensed matter and mathematics literature. These topological orders can be bosonic or fermionic depending on whether the system hosts fundamental fermionic excitations or not. In particular, emergent topological orders in usual solid state systems are fermionic topological orders because the electron is a fermion. Recently, bosonic topological orders have been extensively completely classified up to rank 6 using representation theory. Inspired by their method, we provide in this paper a systematic method to classify the fermionic topological orders by explicitly building their modular data, which encodes the self and mutual statistics between anyons. Our construction of the modular data relies on the fact that the modular data of a fermionic topological order forms a projective representation of the ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\theta}}$ subgroup of the modular group SL2(Z){\mathrm{SL}}_{2}(\mathbb{Z})SL2​(Z). We carry out the classification up to rank 10 and obtain both unitary and nonunitary modular data. This includes all previously known unitary modular data, and also two new classes of modular data of rank 10. We also determine the chiral central charges (mod 12\frac{1}{2}21​) via a novel method, which does not require the explicit computation of modular extensions.
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We classify possible topological quantum field theories in 2+1 dimensions with up to 22 anyons in the bosonic case and up to 26 in the fermionic case, through the modular data of modular and supermodular tensor categories. We construct such data via congruence representations of SL(2,Z)\mathrm{SL}(2,\mathbb{Z})SL(2,Z) using a numerical optimization algorithm that scales to high rank. We chart the relations between the resulting theories, including anyon condensations and Chern-Simons realizations. We find examples of modular data that do not appear to arise from any Chern-Simons theory by any sequence of condensations of anyons or gaugings of ordinary global symmetries. These examples give qualitatively new topological quantum field theories, which go beyond any notion of gauge theory. For examples in this class with 11 and 12 anyon types, we solve the pentagon and hexagon equations explicitly, showing that these modular data are realized by full unitary topological quantum field theories. We also introduce https://www.AnyonAtlas.org, an online resource for exploring the data.
2+12+12+1D bosonic topological orders can be characterized by the S,TS,TS,T matrices that encode the statistics of topological excitations. In particular, the S,TS,TS,T matrices can be used to systematically obtain the gapped boundaries of bosonic topological orders. Such an approach, however, does not naively apply to fermionic topological orders (FTOs). In this work, we propose a systematic approach to obtain the gapped boundaries of 2+12+12+1D abelian FTOs. The main trick is to construct a bosonic extension in which the fermionic excitation is "condensed" to form the associated FTOs. Here we choose the parent bosonic topological order to be the Z2\mathbb{Z}_2Z2​ topological order, which indeed has a fermionic excitation. Such a construction allows us to find an explicit correspondence between abelian FTOs (described by odd KKK-matrix KFK_FKF​) and the "fermion-" condensed Z2\mathbb{Z}_2Z2​ topological orders (described by even KKK-matrix KBK_BKB​). This provides a systematic algorithm to obtain the modular covariant boundary partition functions as well as the boundary topological excitations of abelian FTOs. For example, the ν=1−1mν=1-\frac{1}{m}ν=1−m1​ Laughlin's states have exactly one type of gapped boundary when mmm is a square, whose boundary excitations form a Z2×Zm\mathbb{Z}_{2}\times\mathbb{Z}_{\sqrt{m}}Z2​×Zm​​ fusion ring. Our approach can be easily generalized to obtain gapped and gapless boundaries of non-abelian fermionic topological orders.
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