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arXiv:2308.09670

Classification of modular data up to rank 12

Siu‐Hung Ng, Eric C. Rowell, Xiao-Gang Wen

Published Aug 18, 2023• 3 citations
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Abstract

We use the computer algebra system GAP to classify modular data up to rank 12. This extends the previously obtained classification of modular data up to rank 6. Our classification includes all the modular data from modular tensor categories up to rank 12, with a few possible exceptions at rank 12 and levels $5,7$ and $14$. Those exceptions are eliminated up to a certain bound by an extensive finite search in place of required infinite search. Our list contains a few potential unitary modular data which are not known to correspond to any unitary modular tensor categories (such as those from Kac-Moody algebra, twisted quantum doubles of finite group, as well as their Abelian anyon condensations). It remains to be shown if those potential modular data can be realized by modular tensor categories or not. We provide some evidence that all may be constructed from centers of near-group categories or gauging group symmetries of known modular tensor categories, with the exception of a total of five cases at rank 11 (with $D^2 =1964.590$) and 12 (with $D^2 =3926.660$). The classification of modular data corresponds to a classification of modular tensor categories (up to modular isotopes which are not expected to be present at low ranks). The classification of modular tensor categories leads to a classification of gapped quantum phases of matter in 2-dimensional space for bosonic lattice systems with no symmetry, as well as a classification of generalized symmetries in 1-dimensional space.

DOI
10.48550/arxiv.2308.09670

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We use the computer algebra system GAP to classify modular data up to rank 12. This extends the previously obtained classification of modular data up to rank 6. Our classification includes all the modular data from modular tensor categories up to rank 12, with a few possible exceptions at rank 12 and levels 5,75,75,7 and 141414. Those exceptions are eliminated up to a certain bound by an extensive finite search in place of required infinite search. Our list contains a few potential unitary modular data which are not known to correspond to any unitary modular tensor categories (such as those from Kac-Moody algebra, twisted quantum doubles of finite group, as well as their Abelian anyon condensations). It remains to be shown if those potential modular data can be realized by modular tensor categories or not. We provide some evidence that all may be constructed from centers of near-group categories or gauging group symmetries of known modular tensor categories, with the exception of a total of five cases at rank 11 (with D2=1964.590D^2 =1964.590D2=1964.590) and 12 (with D2=3926.660D^2 =3926.660D2=3926.660). The classification of modular data corresponds to a classification of modular tensor categories (up to modular isotopes which are not expected to be present at low ranks). The classification of modular tensor categories leads to a classification of gapped quantum phases of matter in 2-dimensional space for bosonic lattice systems with no symmetry, as well as a classification of generalized symmetries in 1-dimensional space.
Classification of Rank 6 Modular Categories with Galois Group ⟨(012)(345)⟩\langle (012)(345)\rangle⟨(012)(345)⟩
Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low rank, applied to the specific case of a rank 6 category with Galois group ⟨(012)(345)⟩\langle(012)(345)\rangle⟨(012)(345)⟩, and show that certain symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We show that all the rank 6 MTC's with this Galois group have modular data conjugate to either the product of the semion category with (A1,5)12(A_1, 5)_{\frac{1}{2}}(A1​,5)21​​ or a certain modular subcategory of C(so5,9,ejπi/9)\mathcal{C}(\mathfrak{so}_5, 9, e^{jπi/9})C(so5​,9,ejπi/9) with gcd(18,j)=1(18, j) = 1(18,j)=1.
We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular S -matrix S and − S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 non-trivial prime UMTCs are given in Sect. 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Sect. 6.
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.
We classify connected étale algebras in (possibly non-unitary) modular fusion categories B\mathcal BB's with rank(B)≤5\text{rank}(\mathcal B)\le5rank(B)≤5. We also comment on Lagrangian algebra, anyon condensation, and physical applications. Concretely, we prove certain spontaneous B\mathcal BB-symmetry breaking and predict ground state degeneracies in massive renormalization group flows from non-unitary minimal models.
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.
Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach [L. Bhardwaj et al., SciPost Phys. 19, 056 (2025); L. Bhardwaj et al., arXiv: 2502.20440]. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider “all-boson type” fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups G G (possibly with an ’t Hooft anomaly) via generalized gauging—that is, by stacking with an H H -symmetric TFT and gauging a subgroup H H . The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.