Vectorize fusiontree manipulations #261
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| Original file line number | Diff line number | Diff line change |
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| # BASIC MANIPULATIONS: | ||
| #---------------------------------------------- | ||
| # -> rewrite generic fusion tree in basis of fusion trees in standard form | ||
| # -> only depend on Fsymbol | ||
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| """ | ||
| insertat(f::FusionTree{I, N₁}, i::Int, f₂::FusionTree{I, N₂}) | ||
| -> <:AbstractDict{<:FusionTree{I, N₁+N₂-1}, <:Number} | ||
|
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| 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`. | ||
| """ | ||
| function insertat(f₁::FusionTree{I}, i::Int, f₂::FusionTree{I, 0}) where {I} | ||
| # this actually removes uncoupled line i, which should be trivial | ||
| (f₁.uncoupled[i] == f₂.coupled && !f₁.isdual[i]) || | ||
| throw(SectorMismatch("cannot connect $(f₂.uncoupled) to $(f₁.uncoupled[i])")) | ||
| coeff = one(sectorscalartype(I)) | ||
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| uncoupled = TupleTools.deleteat(f₁.uncoupled, i) | ||
| coupled = f₁.coupled | ||
| isdual = TupleTools.deleteat(f₁.isdual, i) | ||
| if length(uncoupled) <= 2 | ||
| inner = () | ||
| else | ||
| inner = TupleTools.deleteat(f₁.innerlines, max(1, i - 2)) | ||
| end | ||
| if length(uncoupled) <= 1 | ||
| vertices = () | ||
| else | ||
| vertices = TupleTools.deleteat(f₁.vertices, max(1, i - 1)) | ||
| end | ||
| f = FusionTree(uncoupled, coupled, isdual, inner, vertices) | ||
| return fusiontreedict(I)(f => coeff) | ||
| end | ||
| function insertat(f₁::FusionTree{I}, i, f₂::FusionTree{I, 1}) where {I} | ||
| # identity operation | ||
| (f₁.uncoupled[i] == f₂.coupled && !f₁.isdual[i]) || | ||
| throw(SectorMismatch("cannot connect $(f₂.uncoupled) to $(f₁.uncoupled[i])")) | ||
| coeff = one(sectorscalartype(I)) | ||
| isdual′ = TupleTools.setindex(f₁.isdual, f₂.isdual[1], i) | ||
| f = FusionTree{I}(f₁.uncoupled, f₁.coupled, isdual′, f₁.innerlines, f₁.vertices) | ||
| return fusiontreedict(I)(f => coeff) | ||
| end | ||
| function insertat(f₁::FusionTree{I}, i, f₂::FusionTree{I, 2}) where {I} | ||
| # elementary building block, | ||
| (f₁.uncoupled[i] == f₂.coupled && !f₁.isdual[i]) || | ||
| throw(SectorMismatch("cannot connect $(f₂.uncoupled) to $(f₁.uncoupled[i])")) | ||
| uncoupled = f₁.uncoupled | ||
| coupled = f₁.coupled | ||
| inner = f₁.innerlines | ||
| b, c = f₂.uncoupled | ||
| isdual = f₁.isdual | ||
| isdualb, isdualc = f₂.isdual | ||
| if i == 1 | ||
| uncoupled′ = (b, c, tail(uncoupled)...) | ||
| isdual′ = (isdualb, isdualc, tail(isdual)...) | ||
| inner′ = (uncoupled[1], inner...) | ||
| vertices′ = (f₂.vertices..., f₁.vertices...) | ||
| coeff = one(sectorscalartype(I)) | ||
| f′ = FusionTree(uncoupled′, coupled, isdual′, inner′, vertices′) | ||
| return fusiontreedict(I)(f′ => coeff) | ||
| end | ||
| uncoupled′ = TupleTools.insertafter(TupleTools.setindex(uncoupled, b, i), i, (c,)) | ||
| isdual′ = TupleTools.insertafter(TupleTools.setindex(isdual, isdualb, i), i, (isdualc,)) | ||
| inner_extended = (uncoupled[1], inner..., coupled) | ||
| a = inner_extended[i - 1] | ||
| d = inner_extended[i] | ||
| e′ = uncoupled[i] | ||
| if FusionStyle(I) isa MultiplicityFreeFusion | ||
| local newtrees | ||
| for e in a ⊗ b | ||
| coeff = conj(Fsymbol(a, b, c, d, e, e′)) | ||
| iszero(coeff) && continue | ||
| inner′ = TupleTools.insertafter(inner, i - 2, (e,)) | ||
| f′ = FusionTree(uncoupled′, coupled, isdual′, inner′) | ||
| if @isdefined newtrees | ||
| push!(newtrees, f′ => coeff) | ||
| else | ||
| newtrees = fusiontreedict(I)(f′ => coeff) | ||
| end | ||
| end | ||
| return newtrees | ||
| else | ||
| local newtrees | ||
| κ = f₂.vertices[1] | ||
| λ = f₁.vertices[i - 1] | ||
| for e in a ⊗ b | ||
| inner′ = TupleTools.insertafter(inner, i - 2, (e,)) | ||
| Fmat = Fsymbol(a, b, c, d, e, e′) | ||
| for μ in axes(Fmat, 1), ν in axes(Fmat, 2) | ||
| coeff = conj(Fmat[μ, ν, κ, λ]) | ||
| iszero(coeff) && continue | ||
| vertices′ = TupleTools.setindex(f₁.vertices, ν, i - 1) | ||
| vertices′ = TupleTools.insertafter(vertices′, i - 2, (μ,)) | ||
| f′ = FusionTree(uncoupled′, coupled, isdual′, inner′, vertices′) | ||
| if @isdefined newtrees | ||
| push!(newtrees, f′ => coeff) | ||
| else | ||
| newtrees = fusiontreedict(I)(f′ => coeff) | ||
| end | ||
| end | ||
| end | ||
| return newtrees | ||
| end | ||
| end | ||
| function insertat(f₁::FusionTree{I, N₁}, i, f₂::FusionTree{I, N₂}) where {I, N₁, N₂} | ||
| F = fusiontreetype(I, N₁ + N₂ - 1) | ||
| (f₁.uncoupled[i] == f₂.coupled && !f₁.isdual[i]) || | ||
| throw(SectorMismatch("cannot connect $(f₂.uncoupled) to $(f₁.uncoupled[i])")) | ||
| T = sectorscalartype(I) | ||
| coeff = one(T) | ||
| if length(f₁) == 1 | ||
| return fusiontreedict(I){F, T}(f₂ => coeff) | ||
| end | ||
| if i == 1 | ||
| uncoupled = (f₂.uncoupled..., tail(f₁.uncoupled)...) | ||
| isdual = (f₂.isdual..., tail(f₁.isdual)...) | ||
| inner = (f₂.innerlines..., f₂.coupled, f₁.innerlines...) | ||
| vertices = (f₂.vertices..., f₁.vertices...) | ||
| coupled = f₁.coupled | ||
| f′ = FusionTree(uncoupled, coupled, isdual, inner, vertices) | ||
| return fusiontreedict(I){F, T}(f′ => coeff) | ||
| else # recursive definition | ||
| N2 = length(f₂) | ||
| f₂′, f₂′′ = split(f₂, N2 - 1) | ||
| local newtrees::fusiontreedict(I){F, T} | ||
| for (f, coeff) in insertat(f₁, i, f₂′′) | ||
| for (f′, coeff′) in insertat(f, i, f₂′) | ||
| if @isdefined newtrees | ||
| coeff′′ = coeff * coeff′ | ||
| newtrees[f′] = get(newtrees, f′, zero(coeff′′)) + coeff′′ | ||
| else | ||
| newtrees = fusiontreedict(I){F, T}(f′ => coeff * coeff′) | ||
| end | ||
| end | ||
| end | ||
| return newtrees | ||
| end | ||
| end | ||
|
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||
| """ | ||
| split(f::FusionTree{I, N}, M::Int) | ||
| -> (::FusionTree{I, M}, ::FusionTree{I, N-M+1}) | ||
|
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| 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)`. | ||
| """ | ||
| @inline function split(f::FusionTree{I, N}, M::Int) where {I, N} | ||
| if M > N || M < 0 | ||
| throw(ArgumentError("M should be between 0 and N = $N")) | ||
| elseif M === N | ||
| (f, FusionTree{I}((f.coupled,), f.coupled, (false,), (), ())) | ||
| elseif M === 1 | ||
| isdual1 = (f.isdual[1],) | ||
| isdual2 = TupleTools.setindex(f.isdual, false, 1) | ||
| f₁ = FusionTree{I}((f.uncoupled[1],), f.uncoupled[1], isdual1, (), ()) | ||
| f₂ = FusionTree{I}(f.uncoupled, f.coupled, isdual2, f.innerlines, f.vertices) | ||
| return f₁, f₂ | ||
| elseif M === 0 | ||
| u = leftunit(f.uncoupled[1]) | ||
| f₁ = FusionTree{I}((), u, (), ()) | ||
| uncoupled2 = (u, f.uncoupled...) | ||
| coupled2 = f.coupled | ||
| isdual2 = (false, f.isdual...) | ||
| innerlines2 = N >= 2 ? (f.uncoupled[1], f.innerlines...) : () | ||
| if FusionStyle(I) isa GenericFusion | ||
| vertices2 = (1, f.vertices...) | ||
| return f₁, FusionTree{I}(uncoupled2, coupled2, isdual2, innerlines2, vertices2) | ||
| else | ||
| return f₁, FusionTree{I}(uncoupled2, coupled2, isdual2, innerlines2) | ||
| end | ||
| else | ||
| uncoupled1 = ntuple(n -> f.uncoupled[n], M) | ||
| isdual1 = ntuple(n -> f.isdual[n], M) | ||
| innerlines1 = ntuple(n -> f.innerlines[n], max(0, M - 2)) | ||
| coupled1 = f.innerlines[M - 1] | ||
| vertices1 = ntuple(n -> f.vertices[n], M - 1) | ||
|
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| uncoupled2 = ntuple(N - M + 1) do n | ||
| return n == 1 ? f.innerlines[M - 1] : f.uncoupled[M + n - 1] | ||
| end | ||
| isdual2 = ntuple(N - M + 1) do n | ||
| return n == 1 ? false : f.isdual[M + n - 1] | ||
| end | ||
| innerlines2 = ntuple(n -> f.innerlines[M - 1 + n], N - M - 1) | ||
| coupled2 = f.coupled | ||
| vertices2 = ntuple(n -> f.vertices[M - 1 + n], N - M) | ||
|
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| f₁ = FusionTree{I}(uncoupled1, coupled1, isdual1, innerlines1, vertices1) | ||
| f₂ = FusionTree{I}(uncoupled2, coupled2, isdual2, innerlines2, vertices2) | ||
| return f₁, f₂ | ||
| end | ||
| end | ||
|
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||
| """ | ||
| merge(f₁::FusionTree{I, N₁}, f₂::FusionTree{I, N₂}, c::I, μ = 1) | ||
| -> <:AbstractDict{<:FusionTree{I, N₁+N₂}, <:Number} | ||
|
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| 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. | ||
| """ | ||
| function merge(f₁::FusionTree{I, N₁}, f₂::FusionTree{I, N₂}, c::I) where {I, N₁, N₂} | ||
| if FusionStyle(I) isa GenericFusion | ||
| throw(ArgumentError("vertex label for merging required")) | ||
| end | ||
| return merge(f₁, f₂, c, 1) | ||
| end | ||
| function merge(f₁::FusionTree{I, N₁}, f₂::FusionTree{I, N₂}, c::I, μ) where {I, N₁, N₂} | ||
| if !(c in f₁.coupled ⊗ f₂.coupled) | ||
| throw(SectorMismatch("cannot fuse sectors $(f₁.coupled) and $(f₂.coupled) to $c")) | ||
| end | ||
| if μ > Nsymbol(f₁.coupled, f₂.coupled, c) | ||
| throw(ArgumentError("invalid fusion vertex label $μ")) | ||
| end | ||
| f₀ = FusionTree{I}((f₁.coupled, f₂.coupled), c, (false, false), (), (μ,)) | ||
| f, coeff = first(insertat(f₀, 1, f₁)) # takes fast path, single output | ||
| @assert coeff == one(coeff) | ||
|
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| return insertat(f, N₁ + 1, f₂) | ||
| end | ||
| function merge(f₁::FusionTree{I, 0}, f₂::FusionTree{I, 0}, c::I, μ) where {I} | ||
| Nsymbol(f₁.coupled, f₂.coupled, c) == μ == 1 || | ||
| throw(SectorMismatch("cannot fuse sectors $(f₁.coupled) and $(f₂.coupled) to $c")) | ||
| return fusiontreedict(I)(f₁ => Fsymbol(c, c, c, c, c, c)[1, 1, 1, 1]) | ||
| end | ||
|
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| # flip a duality flag of a fusion tree | ||
|
Member
There was a problem hiding this comment.
Member
There was a problem hiding this comment. Also, I don't think this is a very basic manipulation, as it involves both duality and braiding (twists), so not sure it is really in its place here.
Member
There was a problem hiding this comment. Ok, I am going to leave this as a TODO and think about it later. I forgot what the original constraints and design considerations were for |
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| function flip((f₁, f₂)::FusionTreePair{I, N₁, N₂}, i::Int; inv::Bool = false) where {I, N₁, N₂} | ||
| @assert 0 < i ≤ N₁ + N₂ | ||
| if i ≤ N₁ | ||
| a = f₁.uncoupled[i] | ||
| χₐ = frobenius_schur_phase(a) | ||
| θₐ = twist(a) | ||
| if !inv | ||
| factor = f₁.isdual[i] ? χₐ * θₐ : one(θₐ) | ||
| else | ||
| factor = f₁.isdual[i] ? one(θₐ) : conj(χₐ * θₐ) | ||
| end | ||
| isdual′ = TupleTools.setindex(f₁.isdual, !f₁.isdual[i], i) | ||
| f₁′ = FusionTree{I}(f₁.uncoupled, f₁.coupled, isdual′, f₁.innerlines, f₁.vertices) | ||
| return SingletonDict((f₁′, f₂) => factor) | ||
| else | ||
| i -= N₁ | ||
| a = f₂.uncoupled[i] | ||
| χₐ = frobenius_schur_phase(a) | ||
| θₐ = twist(a) | ||
| if !inv | ||
| factor = f₂.isdual[i] ? conj(χₐ) * one(θₐ) : θₐ | ||
| else | ||
| factor = f₂.isdual[i] ? conj(θₐ) : χₐ * one(θₐ) | ||
| end | ||
| isdual′ = TupleTools.setindex(f₂.isdual, !f₂.isdual[i], i) | ||
| f₂′ = FusionTree{I}(f₂.uncoupled, f₂.coupled, isdual′, f₂.innerlines, f₂.vertices) | ||
| return SingletonDict((f₁, f₂′) => factor) | ||
| end | ||
| end | ||
| function flip((f₁, f₂)::FusionTreePair{I, N₁, N₂}, ind; inv::Bool = false) where {I, N₁, N₂} | ||
| f₁′, f₂′ = f₁, f₂ | ||
| factor = one(sectorscalartype(I)) | ||
| for i in ind | ||
| (f₁′, f₂′), s = only(flip((f₁′, f₂′), i; inv)) | ||
| factor *= s | ||
| end | ||
| return SingletonDict((f₁′, f₂′) => factor) | ||
| end | ||
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