Symmetry sectors and fusion trees

abstract type Sector end

Abstract type for representing the (isomorphism classes of) simple objects in (unitary and pivotal) (pre-)fusion categories, e.g. the irreducible representations of a finite or compact group. Subtypes I<:Sector as the set of labels of a GradedSpace.

Every new I<:Sector should implement the following methods:

• one(::Type{I}): unit element of I
• conj(a::I): $a̅$, conjugate or dual label of $a$
• ⊗(a::I, b::I): iterable with unique fusion outputs of $a ⊗ b$ (i.e. don't repeat in case of multiplicities)
• Nsymbol(a::I, b::I, c::I): number of times c appears in a ⊗ b, i.e. the multiplicity
• FusionStyle(::Type{I}): UniqueFusion(), SimpleFusion() or GenericFusion()
• BraidingStyle(::Type{I}): Bosonic(), Fermionic(), Anyonic(), ...
• Fsymbol(a::I, b::I, c::I, d::I, e::I, f::I): F-symbol: scalar (in case of UniqueFusion/SimpleFusion) or matrix (in case of GenericFusion)
• Rsymbol(a::I, b::I, c::I): R-symbol: scalar (in case of UniqueFusion/SimpleFusion) or matrix (in case of GenericFusion)

and optionally

• dim(a::I): quantum dimension of sector a
• frobeniusschur(a::I): Frobenius-Schur indicator of a
• Bsymbol(a::I, b::I, c::I): B-symbol: scalar (in case of UniqueFusion/SimpleFusion) or matrix (in case of GenericFusion)
• twist(a::I) -> twist of sector a

and optionally, if FusionStyle(I) isa GenericFusion

• vertex_ind2label(i::Int, a::I, b::I, c::I) -> a custom label for the ith copy of c appearing in a ⊗ b

Furthermore, iterate and Base.IteratorSize should be made to work for the singleton type SectorValues{I}.

struct SectorValues{I<:Sector}

Singleton type to represent an iterator over the possible values of type I, whose instance is obtained as values(I). For a new I::Sector, the following should be defined

• Base.iterate(::SectorValues{I}[, state]): iterate over the values
• Base.IteratorSize(::Type{SectorValues{I}}): HasLenght(), SizeUnkown() or IsInfinite() depending on whether the number of values of type I is finite (and sufficiently small) or infinite; for a large number of values, SizeUnknown() is recommend because this will trigger the use of GenericGradedSpace.

If IteratorSize(I) == HasLength(), also the following must be implemented:

• Base.length(::SectorValues{I}): the number of different values
• Base.getindex(::SectorValues{I}, i::Int): a mapping between an index i and an instance of I
• findindex(::SectorValues{I}, c::I): reverse mapping between a value c::I and an index i::Integer ∈ 1:length(values(I))

FusionStyle(a::Sector) -> ::FusionStyle
FusionStyle(I::Type{<:Sector}) -> ::FusionStyle

Return the type of fusion behavior of sectors of type I, which can be either

• UniqueFusion(): single fusion output when fusing two sectors;
• SimpleFusion(): multiple outputs, but every output occurs at most one, also known as multiplicity free (e.g. irreps of $SU(2)$);
• GenericFusion(): multiple outputs that can occur more than once (e.g. irreps of $SU(3)$).

There is an abstract supertype MultipleFusion of which both SimpleFusion and GenericFusion are subtypes. Furthermore, there is a type alias MultiplicityFreeFusion for those fusion types which do not require muliplicity labels, i.e. MultiplicityFreeFusion = Union{UniqueFusion,SimpleFusion}.

BraidingStyle(::Sector) -> ::BraidingStyle
BraidingStyle(I::Type{<:Sector}) -> ::BraidingStyle

Return the type of braiding and twist behavior of sectors of type I, which can be either

• Bosonic(): symmetric braiding with trivial twist (i.e. identity)
• Fermionic(): symmetric braiding with non-trivial twist (squares to identity)
• Anyonic(): general $R_(a,b)^c$ phase or matrix (depending on SimpleFusion or GenericFusion fusion) and arbitrary twists

Note that Bosonic and Fermionic are subtypes of SymmetricBraiding, which means that braids are in fact equivalent to crossings (i.e. braiding twice is an identity: isone(Rsymbol(b,a,c)*Rsymbol(a,b,c)) == true) and permutations are uniquely defined.

abstract type AbstractIrrep{G<:Group} <: Sector end

Abstract supertype for sectors which corresponds to irreps (irreducible representations) of a group G. As we assume unitary representations, these would be finite groups or compact Lie groups. Note that this could also include projective rather than linear representations.

Actual concrete implementations of those irreps can be obtained as Irrep[G], or via their actual name, which generically takes the form (asciiG)Irrep, i.e. the ASCII spelling of the group name followed by Irrep.

All irreps have BraidingStyle equal to Bosonic() and thus trivial twists.

Trivial

Singleton type to represent the trivial sector, i.e. the trivial representation of the trivial group. This is equivalent to Rep[ℤ₁], or the unit object of the category Vect of ordinary vector spaces.

struct ZNIrrep{N} <: AbstractIrrep{ℤ{N}}
ZNIrrep{N}(n::Integer)
Irrep[ℤ{N}](n::Integer)

Represents irreps of the group $ℤ_N$ for some value of N<64. (We need 2*(N-1) <= 127 in order for a ⊗ b to work correctly.) For N equals 2, 3 or 4, ℤ{N} can be replaced by ℤ₂, ℤ₃, ℤ₄. An arbitrary Integer n can be provided to the constructor, but only the value mod(n, N) is relevant.

Fields

• n::Int8: the integer label of the irrep, modulo N.

struct U1Irrep <: AbstractIrrep{U₁}
U1Irrep(charge::Real)
Irrep[U₁](charge::Real)

Represents irreps of the group $U₁$. The irrep is labelled by a charge, which should be an integer for a linear representation. However, it is often useful to allow half integers to represent irreps of $U₁$ subgroups of $SU₂$, such as the Sz of spin-1/2 system. Hence, the charge is stored as a HalfInt from the package HalfIntegers.jl, but can be entered as arbitrary Real. The sequence of the charges is: 0, 1/2, -1/2, 1, -1, ...

Fields

• charge::HalfInt: the label of the irrep, which can be any half integer.

struct SU2Irrep <: AbstractIrrep{SU₂}
SU2Irrep(j::Real)
Irrep[SU₂](j::Real)

Represents irreps of the group $SU₂$. The irrep is labelled by a half integer j which can be entered as an abitrary Real, but is stored as a HalfInt from the HalfIntegers.jl package.

Fields

• j::HalfInt: the label of the irrep, which can be any non-negative half integer.

struct CU1Irrep <: AbstractIrrep{CU₁}
CU1Irrep(j, s = ifelse(j>zero(j), 2, 0))
Irrep[CU₁](j, s = ifelse(j>zero(j), 2, 0))

Represents irreps of the group $U₁ ⋊ C$ ($U₁$ and charge conjugation or reflection), which is also known as just O₂.

Fields

• j::HalfInt: the value of the $U₁$ charge.
• s::Int: the representation of charge conjugation.

They can take values:

• if j == 0, s = 0 (trivial charge conjugation) or s = 1 (non-trivial charge conjugation)
• if j > 0, s = 2 (two-dimensional representation)

ProductSector{T<:SectorTuple}

Represents the Deligne tensor product of sectors. The type parameter T is a tuple of the component sectors. The recommended way to construct a ProductSector is using the deligneproduct (⊠) operator on the components.

FermionParity <: Sector

Represents sectors with fermion parity. The fermion parity is a ℤ₂ quantum number that yields an additional sign when two odd fermions are exchanged.

Fields

• isodd::Bool: indicates whether the fermion parity is odd (true) or even (false).

See also: FermionNumber, FermionSpin

const FermionNumber = U1Irrep ⊠ FermionParity
FermionNumber(a::Int)

Represents the fermion number as the direct product of a $U₁$ irrep a and a fermion parity, with the restriction that the fermion parity is odd if and only if a is odd.

See also: U1Irrep, FermionParity

const FermionSpin = SU2Irrep ⊠ FermionParity
FermionSpin(j::Real)

Represents the fermion spin as the direct product of a $SU₂$ irrep j and a fermion parity, with the restriction that the fermion parity is odd if 2 * j is odd.

See also: SU2Irrep, FermionParity

struct FibonacciAnyon <: Sector
FibonacciAnyon(s::Symbol)

Represents the anyons of the Fibonacci modular fusion category. It can take two values, corresponding to the trivial sector FibonacciAnyon(:I) and the non-trivial sector FibonacciAnyon(:τ) with fusion rules $τ ⊗ τ = 1 ⊕ τ$.

Fields

• isone::Bool: indicates whether the sector corresponds the to trivial anyon :I (true), or the non-trivial anyon :τ (false).

struct IsingAnyon <: Sector
IsingAnyon(s::Symbol)

Represents the anyons of the Ising modular fusion category. It can take three values, corresponding to the trivial sector IsingAnyon(:I) and the non-trivial sectors IsingAnyon(:σ) and IsingAnyon(:ψ), with fusion rules $ψ ⊗ ψ = 1$, $σ ⊗ ψ = σ$, and $σ ⊗ σ = 1 ⊕ ψ$.

Fields

• s::Symbol: the label of the represented anyon, which can be :I, :σ, or :ψ.

struct FusionTree{I, N, M, L, T}

Represents a fusion tree of sectors of type I<:Sector, fusing (or splitting) N uncoupled sectors to a coupled sector. It actually represents a splitting tree, but fusion tree is a more common term.

Fields

• uncoupled::NTuple{N,I}: the uncoupled sectors coming out of the splitting tree, before the possible 𝑍 isomorphism (see isdual).
• coupled::I: the coupled sector.
• isdual::NTuple{N,Bool}: indicates whether a 𝑍 isomorphism is present (true) or not (false) for each uncoupled sector.
• innerlines::NTuple{M,I}: the labels of the M=max(0, N-2) inner lines of the splitting tree.
• vertices::NTuple{L,T}: the L=max(0, N-1) labels of type T of the vertices of the splitting tree. If FusionStyle(I) isa MultiplicityFreeFusion, then T = Nothing.

const Irrep

A constant of a singleton type used as Irrep[G] with G<:Group a type of group, to construct or obtain a concrete subtype of AbstractIrrep{G} that implements the data structure used to represent irreducible representations of the group G.

one(::Sector) -> Sector
one(::Type{<:Sector}) -> Sector

Return the unit element within this type of sector.

dual(a::Sector) -> Sector

Return the conjugate label conj(a).

Nsymbol(a::I, b::I, c::I) where {I<:Sector} -> Integer

Return an Integer representing the number of times c appears in the fusion product a ⊗ b. Could be a Bool if FusionStyle(I) == UniqueFusion() or SimpleFusion().

Fsymbol(a::I, b::I, c::I, d::I, e::I, f::I) where {I<:Sector}

Return the F-symbol $F^{abc}_d$ that associates the two different fusion orders of sectors a, b and c into an ouput sector d, using either an intermediate sector $a ⊗ b → e$ or $b ⊗ c → f$:

a-<-μ-<-e-<-ν-<-d                                     a-<-λ-<-d
    ∨       ∨       -> Fsymbol(a,b,c,d,e,f)[μ,ν,κ,λ]      ∨
    b       c                                             f
                                                          v
                                                      b-<-κ
                                                          ∨
                                                          c

If FusionStyle(I) is UniqueFusion or SimpleFusion, the F-symbol is a number. Otherwise it is a rank 4 array of size (Nsymbol(a, b, e), Nsymbol(e, c, d), Nsymbol(b, c, f), Nsymbol(a, f, d)).

Rsymbol(a::I, b::I, c::I) where {I<:Sector}

Returns the R-symbol $R^{ab}_c$ that maps between $c → a ⊗ b$ and $c → b ⊗ a$ as in

a -<-μ-<- c                                 b -<-ν-<- c
     ∨          -> Rsymbol(a,b,c)[μ,ν]           v
     b                                           a

If FusionStyle(I) is UniqueFusion() or SimpleFusion(), the R-symbol is a number. Otherwise it is a square matrix with row and column size Nsymbol(a,b,c) == Nsymbol(b,a,c).

Bsymbol(a::I, b::I, c::I) where {I<:Sector}

Return the value of $B^{ab}_c$ which appears in transforming a splitting vertex into a fusion vertex using the transformation

a -<-μ-<- c                                                    a -<-ν-<- c
     ∨          -> √(dim(c)/dim(a)) * Bsymbol(a,b,c)[μ,ν]           ∧
     b                                                            dual(b)

If FusionStyle(I) is UniqueFusion() or SimpleFusion(), the B-symbol is a number. Otherwise it is a square matrix with row and column size Nsymbol(a, b, c) == Nsymbol(c, dual(b), a).

dim(a::Sector)

Return the (quantum) dimension of the sector a.

frobeniusschur(a::Sector)

Return the Frobenius-Schur indicator of a sector a.

twist(a::Sector)

Return the twist of a sector a

isreal(::Type{<:Sector}) -> Bool

Return whether the topological data (Fsymbol, Rsymbol) of the sector is real or not (in which case it is complex).

vertex_labeltype(I::Type{<:Sector}) -> Type

Return the type of labels for the fusion vertices of sectors of type I.

vertex_ind2label(k::Int, a::I, b::I, c::I) where {I<:Sector}

Convert the index k of the fusion vertex (a,b)->c into a label. For FusionStyle(I) == UniqueFusion() or FusionStyle(I) == MultipleFusion(), where every fusion output occurs only once and k == 1, the default is to suppress vertex labels by setting them equal to nothing. For FusionStyle(I) == GenericFusion(), the default is to just use k, unless a specialized method is provided.

⊠(s₁::Sector, s₂::Sector)
deligneproduct(s₁::Sector, s₂::Sector)

Given two sectors s₁ and s₂, which label an isomorphism class of simple objects in a fusion category $C₁$ and $C₂$, s1 ⊠ s2 (obtained as \boxtimes+TAB) labels the isomorphism class of simple objects in the Deligne tensor product category $C₁ ⊠ C₂$.

The Deligne tensor product also works in the type domain and for spaces and tensors. For group representations, we have Irrep[G₁] ⊠ Irrep[G₂] == Irrep[G₁ × G₂].

fusiontrees(uncoupled::NTuple{N,I}[,
    coupled::I=one(I)[, isdual::NTuple{N,Bool}=ntuple(n -> false, length(uncoupled))]])
    where {N,I<:Sector} -> FusionTreeIterator{I,N}

Return an iterator over all fusion trees with a given coupled sector label coupled and uncoupled sector labels and isomorphisms uncoupled and isdual respectively.

insertat(f::FusionTree{I, N₁}, i::Int, f₂::FusionTree{I, N₂})
-> <:AbstractDict{<:FusionTree{I, N₁+N₂-1}, <:Number}

Attach a fusion tree f₂ to the uncoupled leg i of the fusion tree f₁ and bring it into a linear combination of fusion trees in standard form. This requires that f₂.coupled == f₁.uncoupled[i] and f₁.isdual[i] == false.

split(f::FusionTree{I, N}, M::Int)
-> (::FusionTree{I, M}, ::FusionTree{I, N-M+1})

Split a fusion tree into two. The first tree has as uncoupled sectors the first M uncoupled sectors of the input tree f, whereas its coupled sector corresponds to the internal sector between uncoupled sectors M and M+1 of the original tree f. The second tree has as first uncoupled sector that same internal sector of f, followed by remaining N-M uncoupled sectors of f. It couples to the same sector as f. This operation is the inverse of insertat in the sense that if f₁, f₂ = split(t, M) ⇒ f == insertat(f₂, 1, f₁).

merge(f₁::FusionTree{I, N₁}, f₂::FusionTree{I, N₂}, c::I, μ = nothing)
-> <:AbstractDict{<:FusionTree{I, N₁+N₂}, <:Number}

Merge two fusion trees together to a linear combination of fusion trees whose uncoupled sectors are those of f₁ followed by those of f₂, and where the two coupled sectors of f₁ and f₂ are further fused to c. In case of FusionStyle(I) == GenericFusion(), also a degeneracy label μ for the fusion of the coupled sectors of f₁ and f₂ to c needs to be specified.

elementary_trace(f::FusionTree{I,N}, i) where {I<:Sector,N} -> <:AbstractDict{FusionTree{I,N-2}, <:Number}

Perform an elementary trace of neighbouring uncoupled indices i and i+1 on a fusion tree f, and returns the result as a dictionary of output trees and corresponding coefficients.

planar_trace(f::FusionTree{I,N}, q1::IndexTuple{N₃}, q2::IndexTuple{N₃}) where {I<:Sector,N,N₃}
    -> <:AbstractDict{FusionTree{I,N-2*N₃}, <:Number}

Perform a planar trace of the uncoupled indices of the fusion tree f at q1 with those at q2, where q1[i] is connected to q2[i] for all i. The result is returned as a dictionary of output trees and corresponding coefficients.

artin_braid(f::FusionTree, i; inv::Bool = false) -> <:AbstractDict{typeof(f), <:Number}

Perform an elementary braid (Artin generator) of neighbouring uncoupled indices i and i+1 on a fusion tree f, and returns the result as a dictionary of output trees and corresponding coefficients.

The keyword inv determines whether index i will braid above or below index i+1, i.e. applying artin_braid(f′, i; inv = true) to all the outputs f′ of artin_braid(f, i; inv = false) and collecting the results should yield a single fusion tree with non-zero coefficient, namely f with coefficient 1. This keyword has no effect if BraidingStyle(sectortype(f)) isa SymmetricBraiding.

braid(f::FusionTree{<:Sector, N}, levels::NTuple{N, Int}, p::NTuple{N, Int})
-> <:AbstractDict{typeof(t), <:Number}

Perform a braiding of the uncoupled indices of the fusion tree f and return the result as a <:AbstractDict of output trees and corresponding coefficients. The braiding is determined by specifying that the new sector at position k corresponds to the sector that was originally at the position i = p[k], and assigning to every index i of the original fusion tree a distinct level or depth levels[i]. This permutation is then decomposed into elementary swaps between neighbouring indices, where the swaps are applied as braids such that if i and j cross, $τ_{i,j}$ is applied if levels[i] < levels[j] and $τ_{j,i}^{-1}$ if levels[i] > levels[j]. This does not allow to encode the most general braid, but a general braid can be obtained by combining such operations.

permute(f::FusionTree, p::NTuple{N, Int}) -> <:AbstractDict{typeof(t), <:Number}

Perform a permutation of the uncoupled indices of the fusion tree f and returns the result as a <:AbstractDict of output trees and corresponding coefficients; this requires that BraidingStyle(sectortype(f)) isa SymmetricBraiding.

repartition(f₁::FusionTree{I, N₁}, f₂::FusionTree{I, N₂}, N::Int) where {I, N₁, N₂}
-> <:AbstractDict{Tuple{FusionTree{I, N}, FusionTree{I, N₁+N₂-N}}, <:Number}

Input is a double fusion tree that describes the fusion of a set of incoming uncoupled sectors to a set of outgoing uncoupled sectors, represented using the individual trees of outgoing (f₁) and incoming sectors (f₂) respectively (with identical coupled sector f₁.coupled == f₂.coupled). Computes new trees and corresponding coefficients obtained from repartitioning the tree by bending incoming to outgoing sectors (or vice versa) in order to have N outgoing sectors.

repartition(tsrc::AbstractTensorMap{S}, N₁::Int, N₂::Int; copy::Bool=false) where {S}
    -> tdst::AbstractTensorMap{S,N₁,N₂}

Return tensor tdst obtained by repartitioning the indices of t. The codomain and domain of tdst correspond to the first N₁ and last N₂ spaces of t, respectively.

If copy=false, tdst might share data with tsrc whenever possible. Otherwise, a copy is always made.

To repartition into an existing destination, see repartition!.

transpose(f₁::FusionTree{I}, f₂::FusionTree{I},
        p1::NTuple{N₁, Int}, p2::NTuple{N₂, Int}) where {I, N₁, N₂}
-> <:AbstractDict{Tuple{FusionTree{I, N₁}, FusionTree{I, N₂}}, <:Number}

Input is a double fusion tree that describes the fusion of a set of incoming uncoupled sectors to a set of outgoing uncoupled sectors, represented using the individual trees of outgoing (t1) and incoming sectors (t2) respectively (with identical coupled sector t1.coupled == t2.coupled). Computes new trees and corresponding coefficients obtained from repartitioning and permuting the tree such that sectors p1 become outgoing and sectors p2 become incoming.

braid(f₁::FusionTree{I}, f₂::FusionTree{I},
        levels1::IndexTuple, levels2::IndexTuple,
        p1::IndexTuple{N₁}, p2::IndexTuple{N₂}) where {I<:Sector, N₁, N₂}
-> <:AbstractDict{Tuple{FusionTree{I, N₁}, FusionTree{I, N₂}}, <:Number}

Input is a fusion-splitting tree pair that describes the fusion of a set of incoming uncoupled sectors to a set of outgoing uncoupled sectors, represented using the splitting tree f₁ and fusion tree f₂, such that the incoming sectors f₂.uncoupled are fused to f₁.coupled == f₂.coupled and then to the outgoing sectors f₁.uncoupled. Compute new trees and corresponding coefficients obtained from repartitioning and braiding the tree such that sectors p1 become outgoing and sectors p2 become incoming. The uncoupled indices in splitting tree f₁ and fusion tree f₂ have levels (or depths) levels1 and levels2 respectively, which determines how indices braid. In particular, if i and j cross, $τ_{i,j}$ is applied if levels[i] < levels[j] and $τ_{j,i}^{-1}$ if levels[i] > levels[j]. This does not allow to encode the most general braid, but a general braid can be obtained by combining such operations.

permute(f₁::FusionTree{I}, f₂::FusionTree{I},
        p1::NTuple{N₁, Int}, p2::NTuple{N₂, Int}) where {I, N₁, N₂}
-> <:AbstractDict{Tuple{FusionTree{I, N₁}, FusionTree{I, N₂}}, <:Number}

Input is a double fusion tree that describes the fusion of a set of incoming uncoupled sectors to a set of outgoing uncoupled sectors, represented using the individual trees of outgoing (t1) and incoming sectors (t2) respectively (with identical coupled sector t1.coupled == t2.coupled). Computes new trees and corresponding coefficients obtained from repartitioning and permuting the tree such that sectors p1 become outgoing and sectors p2 become incoming.