Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2311.00746

Ground state degeneracy and module category

Ken Kikuchi

Published Nov 1, 2023• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2311.00746

Related papers

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

Ken Kikuchi

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.

arXiv:2404.16125•Apr 24, 2024•2 citations

Classification of connected étale algebras in modular fusion categories up to rank five

Ken Kikuchi

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.

arXiv:2312.13353•Dec 20, 2023•2 citations

Classification of connected étale algebras in multiplicity-free modular fusion categories at rank six

Ken Kikuchi, Kah-Sen Kam, Fu-Hsiang Huang

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.

arXiv:2402.00403•Advances in Theoretical and Mathematical Physics•Jan 1, 2026•2 citations

Classification of modular data up to rank 12

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

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.

arXiv:2308.09670•Aug 18, 2023•3 citations

Condensable Algebras in Product Modular Tensor Categories: A Comprehensive Survey of SU(N₁)_{k₁} ⊗ SU(N₂)_{k₂}

Adam Bazih

We present a systematic computational survey of condensable algebras (connected étale algebras) in product modular tensor categories of the form SU(N₁)_{k₁} ⊗ SU(N₂)_{k₂} at small rank. Using the Verlinde formula and modular invariant classification, we identify all non-trivial condensable algebras across 10 product MTCs with rank ≤ 36. We verify the Shimizu-Yadav conditions ensuring that the local module categories are modular, and compute the condensation completion structure including domain walls and defects using the Décoppet-Yu framework. Our survey reveals a rich landscape: simple current algebras of types Z₂, Z₃, and Z₂×Z₂ dominate, with condensed phases including the Ising MTC. We also correct an error in a previous classification of SU(3)₃ ⊗ SU(2)₂, where the maximal condensable algebra has d_A = 6 (not 4), yielding the Ising MTC as the condensed phase. All computational code is included.

•May 12, 2026•0 citations

Non-invertible Symmetry Boundaries from Anyon Condensation: A Computational Classification

Adam Bazih

[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras (e.g. 1+(j=3) in SU(2)_10, within the level range covered here), and whose condensed-phase identification did not resolve the sixteen-fold way. The boundary counts below are therefore lower bounds where E-type algebras were missed; re-check against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We develop a systematic computational framework connecting anyon condensation in modular tensor categories to the classification of non-invertible symmetry boundaries in the Symmetry Topological Field Theory (SymTFT) paradigm. For each condensable algebra in a doubled MTC, we identify the corresponding symmetry boundary condition and classify the resulting non-invertible symmetry structure. We present explicit results for SU(2) Chern-Simons theories at levels k=1-8, SU(3) at levels k=1-3, and G₂ at level 1, covering 40+ modular tensor categories and identifying 100+ distinct symmetry boundary types.

•May 20, 2026•0 citations
We develop a systematic method to classify connected étale algebras AAA's in (possibly degenerate) pre-modular category B\mathcal BB. In particular, we find the category of AAA-modules, BA\mathcal B_ABA​, have ranks bounded from above by ⌊FPdim(B)⌋\lfloor\text{FPdim}(\mathcal B)\rfloor⌊FPdim(B)⌋. For demonstration, we classify connected étale algebras in some B\mathcal BB's, which appear in physics. Physically, the results constrain (or fix) ground state degeneracies of (certain) B\mathcal BB-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 B\mathcal BB's are spontaneously broken.
We classify connected étale algebras AAA's in multiplicity-free modular fusion categories B\mathcal BB's with rank(B)≤9\text{rank}(\mathcal B)\le9rank(B)≤9. We also identify categories BA\mathcal B_ABA​'s of right AAA-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 B\mathcal BB-symmetry breaking.
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 connected étale algebras AAA 's in multiplicity-free modular fusion categories (MFCs) B\mathcal{B}B's at rank six, namely rank⁡(B)=6\operatorname{rank}(\mathcal{B})=6rank(B)=6. There are eight MFCs in total and the result indicates that only so(5)2s o(5)_2so(5)2​ has nontrivial connected étale algebra. We briefly mention anyon condensation as it is used to determine the category of right AAA-modules in so(5)2s o(5)_2so(5)2​. Finally, we discuss physical applications, specifically proving spontaneous B\mathcal{B}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 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.
We present a systematic computational survey of condensable algebras (connected étale algebras) in product modular tensor categories of the form SU(N₁)_{k₁} ⊗ SU(N₂)_{k₂} at small rank. Using the Verlinde formula and modular invariant classification, we identify all non-trivial condensable algebras across 10 product MTCs with rank ≤ 36. We verify the Shimizu-Yadav conditions ensuring that the local module categories are modular, and compute the condensation completion structure including domain walls and defects using the Décoppet-Yu framework. Our survey reveals a rich landscape: simple current algebras of types Z₂, Z₃, and Z₂×Z₂ dominate, with condensed phases including the Ising MTC. We also correct an error in a previous classification of SU(3)₃ ⊗ SU(2)₂, where the maximal condensable algebra has d_A = 6 (not 4), yielding the Ising MTC as the condensed phase. All computational code is included.
[NOTE, June 2026: Computations in this paper used mtctools <= 0.2.x, whose condensable-algebra finder has since been found to have false negatives on conformal-embedding (E-type) algebras (e.g. 1+(j=3) in SU(2)_10, within the level range covered here), and whose condensed-phase identification did not resolve the sixteen-fold way. The boundary counts below are therefore lower bounds where E-type algebras were missed; re-check against mtctools 0.3.0 and the corrected release at DOI 10.5281/zenodo.20618695. The text below is the original and is retained for the record.] We develop a systematic computational framework connecting anyon condensation in modular tensor categories to the classification of non-invertible symmetry boundaries in the Symmetry Topological Field Theory (SymTFT) paradigm. For each condensable algebra in a doubled MTC, we identify the corresponding symmetry boundary condition and classify the resulting non-invertible symmetry structure. We present explicit results for SU(2) Chern-Simons theories at levels k=1-8, SU(3) at levels k=1-3, and G₂ at level 1, covering 40+ modular tensor categories and identifying 100+ distinct symmetry boundary types.