Sector Types
This page provides an overview of the concrete sector types implemented in TensorKitSectors.jl.
Summary Table
| Sector Type | Group/Category | Fusion Style | Braiding | Infinite? | Common Use Cases |
|---|---|---|---|---|---|
Trivial | Trivial group | Unique | Bosonic | No | No symmetry |
ZNIrrep | Rep[ℤₙ] | Unique | Bosonic | No | Clock models, discrete symmetries |
ZNElement | Vec[ℤₙ] | Unique | Varies | No | Group-graded categories, discrete torsion |
U1Irrep | Rep[U₁] | Unique | Bosonic | Yes | Particle number, charge conservation |
SU2Irrep | Rep[SU₂] | Simple | Bosonic | Yes | Spin systems, angular momentum |
CU1Irrep | Rep[U₁ ⋊ ℤ₂] | Simple | Bosonic | Yes | Particle-hole symmetry, O(2) |
DNIrrep | Rep[Dₙ] | Simple | Bosonic | No | Molecular/crystal symmetries |
A4Irrep | Rep[A₄] | Generic | Bosonic | No | Tetrahedral symmetry |
HeisenbergIrrep | Rep[H_N] | Generic | Bosonic | No | Weyl-Heisenberg symmetry, projective representations |
FibonacciAnyon | Fibonacci category | Simple | Anyonic | No | Topological quantum computing |
IsingAnyon | Ising category | Simple | Anyonic | No | Majorana fermions, ν=5/2 QHE |
TambaraYamagami | ℤₙ-Tambara-Yamagami category | Simple | No | No | Self-dual spin chains |
FermionParity | fℤ₂ | Unique | Fermionic | No | Fermion parity conservation |
FermionNumber | fU₁ | Unique | Fermionic | Yes | Fermion number conservation |
FermionSpin | fSU₂ | Simple | Fermionic | Yes | Fermions with spin symmetry |
ProductSector | Product categories | Varies | Varies | Varies | Multiple simultaneous symmetries |
NamedSector | Product categories | Varies | Varies | Varies | Product sectors with name-accessible components |
TimeReversed | Inverted braiding | Varies | Varies | Varies | Time-reversal or other orientation-reversing symmetries |
Multifusion categories
Beyond fusion categories (with a unique simple unit), TensorKitSectors.jl also supports multifusion categories, whose unit is semisimple. These are signalled by UnitStyle(I) === GenericUnit() and require the additional allunits, leftunit, and rightunit methods. The exported IsingBimodule is a minimal worked example: the Ising category reinterpreted as a bimodule category over two copies of Irrep[ℤ₂]. It is intended as a reference implementation illustrating how to build such categories, rather than as a general-purpose sector, so it is not listed in the summary table above.
Other packages
TensorKitSectors.jl provides the architecture for implementing new sector types, but does not implement all possible sectors. Other packages which implement additional sector types and their topological data include:
SUNRepresentations.jl: Implements irreducible representations of SU(N) for arbitrary NCategoryData.jl: Provides a variety of fusion categories up to rank 7, based on theAnyonWikiQWignerSymbols.jl: Provides the irreps of q-deformed SU(2)