arXiv:2404.16125

Classification of connected étale algebras in multiplicity-free modular fusion categories up to rank nine

Ken Kikuchi

Published Apr 24, 2024• 2 citations

Abstract

We classify connected étale algebras $A$'s in multiplicity-free modular fusion categories $\mathcal B$'s with $\text{rank}(\mathcal B)\le9$. We also identify categories $\mathcal B_A$'s of right $A$-modules. The results have physical applications in constraining renormalization group flows. As demonstration, we study massive renormalization group flows from non-unitary minimal models to predict ground state degeneracies and prove spontaneous $\mathcal B$-symmetry breaking.

DOI: 10.48550/arxiv.2404.16125

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arXiv:2404.16125

Classification of connected étale algebras in multiplicity-free modular fusion categories up to rank nine

Ken Kikuchi

Published Apr 24, 2024• 2 citations

arXiv HTML PDF

Abstract

We classify connected étale algebras

A

's in multiplicity-free modular fusion categories

\mathcal B

's with

\text{rank}(\mathcal B)\le9

. We also identify categories

\mathcal B_A

's of right

A

-modules. The results have physical applications in constraining renormalization group flows. As demonstration, we study massive renormalization group flows from non-unitary minimal models to predict ground state degeneracies and prove spontaneous

\mathcal B

-symmetry breaking.

DOI: 10.48550/arxiv.2404.16125

Related papers

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We classify connected étale algebras $A$ 's in multiplicity-free modular fusion categories (MFCs) $\mathcal{B}$ 's at rank six, namely $\operatorname{rank}(\mathcal{B})=6$ . There are eight MFCs in total and the result indicates that only $s o(5)_2$ has nontrivial connected étale algebra. We briefly mention anyon condensation as it is used to determine the category of right $A$ -modules in $s o(5)_2$ . Finally, we discuss physical applications, specifically proving spontaneous $\mathcal{B}$ -symmetry breaking (SSB) of these MFCs. The discussion also includes predicting ground state degeneracies and SSB in massive renormalization group flows from two non-unitary minimal models.We classify connected étale algebras $A$ 's in multiplicity-free modular fusion categories (MFCs) $\mathcal{B}$'s at rank six, namely $\operatorname{rank}(\mathcal{B})=6$. There are eight MFCs in total and the result indicates that only $s o(5)_2$ has nontrivial connected étale algebra. We briefly mention anyon condensation as it is used to determine the category of right $A$-modules in $s o(5)_2$. Finally, we discuss physical applications, specifically proving spontaneous $\mathcal{B}$-symmetry breaking (SSB) of these MFCs. The discussion also includes predicting ground state degeneracies and SSB in massive renormalization group flows from two non-unitary minimal models.

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arXiv:2407.21353Jul 31, 20240 citations

We classify connected étale algebras

A

's in multiplicity-free modular fusion categories

\mathcal B

's with

\text{rank}(\mathcal B)\le9

. We also identify categories

\mathcal B_A

's of right

A

\mathcal B

-symmetry breaking.

We classify connected étale algebras

A

's in multiplicity-free modular fusion categories (MFCs)

\mathcal{B}

's at rank six, namely

\operatorname{rank}(\mathcal{B})=6

. There are eight MFCs in total and the result indicates that only

s o(5)_2

has nontrivial connected étale algebra. We briefly mention anyon condensation as it is used to determine the category of right

A

-modules in

s o(5)_2

. Finally, we discuss physical applications, specifically proving spontaneous

\mathcal{B}

-symmetry breaking (SSB) of these MFCs. The discussion also includes predicting ground state degeneracies and SSB in massive renormalization group flows from two non-unitary minimal models.

We classify connected étale algebras in (possibly non-unitary) modular fusion categories

\mathcal B

's with

\text{rank}(\mathcal B)\le5

. We also comment on Lagrangian algebra, anyon condensation, and physical applications. Concretely, we prove certain spontaneous

\mathcal B

-symmetry breaking and predict ground state degeneracies in massive renormalization group flows from non-unitary minimal models.

We develop a systematic method to classify connected étale algebras

A

's in (possibly degenerate) pre-modular category

\mathcal B

. In particular, we find the category of

A

-modules,

\mathcal B_A

, have ranks bounded from above by

\lfloor\text{FPdim}(\mathcal B)\rfloor

. For demonstration, we classify connected étale algebras in some

\mathcal B

's, which appear in physics. Physically, the results constrain (or fix) ground state degeneracies of (certain)

\mathcal B

-symmetric gapped phases. We study massive deformations of rational conformal field theories such as minimal models and Wess-Zumino-Witten models. In most of our examples, the classification suggests the symmetries

\mathcal B

's are spontaneously broken.

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.

Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models

\mathcal{M}(kq + I, q) \to\mathcal{M}(kq-I, q)

induced by

ϕ_{(1,2k+1)}

. They vastly generalize the previously proposed ones

k=I=1

by Zamolodchikov,

k=1, I>1

by Ahn and Lässig, and

k=2

by Dorey et al. All the other

\mathbb{Z}_2

preserving renormalization group flows sporadically known in the literature (e.g.

\mathcal{M}(10,3) \to \mathcal{M}(8,3)

studied by Klebanov et al) fall into our proposal (e.g.

k=3, I=1

). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the

SU(2)_{q-2}

fusion ring.