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.
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.
We investigate whether a previously unknown two-dimensional non-Abelian quasiparticle type can exist as a multiplicity-free unitary modular tensor category (UMTC) of rank at most four. Existence is understood at the mathematical and model level, as a skeletal UMTC over the complex numbers with exact algebraic F, R, S, and T data, rather than as a claim of a new material or Standard-Model particle.An exhaustive enumeration yields 1, 2, 4, and 10 commutative multiplicity-free based fusion rings of ranks 1, 2, 3, and 4, respectively. Exact residual-zero checks of the pentagon and hexagon equations, unitarity, Verlinde relations, and projective SL(2,Z) identities are carried out for the Rowell–Stong–Wang prime theories with explicitly available F-symbols, including Fibonacci and Ising theories as positive controls. Verification is performed independently using a fusion-tree implementation that does not rely on discovery caches.Fusion rings outside the known catalog produce no modular ribbon data in a conductor-≤16 scan, with Rep(S3) providing an explicit non-modular counterexample. Combined with the published classification of all UMTCs of rank at most four, the results show that there is no previously unknown multiplicity-free UMTC within this cutoff. A Levin–Wen parent Hamiltonian is also constructed from Fibonacci F-symbols; its bulk excitations are described by the Drinfeld center Z(Fib), rather than by a chiral Fibonacci theory alone.The results are mathematical and model-level statements. They do not establish the experimental realization of a new quasiparticle, material, or fundamental particle.
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 D2=1964.590) and 12 (with D2=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)⟩
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)⟩, 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)21 or a certain modular subcategory of C(so5,9,ejπi/9) with gcd(18,j)=1.
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.
FINDING: Modular tensor categories (MTCs) provide the algebraic backbone for topological quantum computation, with S and T matrices encoding anyon braiding and fusion rules. | MATH: MTC data: fusion rules Nijk, S-matrix (modular transformation τ→−1/τ), T-matrix (diagonal, θa=e2πiha topological spins), satisfying S2=C (charge conjugation), (ST)3=e2πic/8S2 (where c is the chiral central charge). The entries of S and T lie in a cyclotomic field Q(ζN), ζN=e2πi/N. For the Fibonacci anyon MTC: S=ϕ+21(1ϕϕ−1), T=diag(1,e4πi/5), where ϕ=(1+5)/2. | CONNECTION: **Direct hit**: The Fibonacci MTC's S-matrix contains ϕ=1.618 and ϕ+2=ϕ2+1=3.618, whose inverse square root yields 1/3.618≈0.5257. The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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) 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.