First draft of FusionTensor

[ogauthe]

This PR provides a first draft for FusionTensors. It defines a generic interface to implement non-abelian symmetries. It currently supports O(2), SU(2), any abelian group defined in SymmetrySectors. It allows to construct, permute and contract tensors.

Work to be done in future PR:

• remove FusedAxes, currently only used in FusionTensor initialization
• handle dual in a more explicit way when fusing codomain and domain fusion trees
• store unitary as a sparse matrix
• improve caching of structural data
• maybe improve detection of a charge block size
• maybe moving Clebsch-Gordan tensor definition to SymmetrySectors

Work depending on other packages

• implement linear algebra decomposition such as SVD (depends on [Enhancement] Add singular value decomposition BlockSparseArrays.jl#16)
• rewrite TensorAlgebra interface once contract preserves domain and codomain information
• move nsymbol to SymmetrySectors (define nsymbol SymmetrySectors.jl#4)

[ogauthe]

I have issues with building test both on local and on GitHub actions. Locally, I get an error

  LoadError: ArgumentError: Package LinearAlgebra not found in current path.

although the package is included in the Project.toml. I guess there was an issue with SafeTestsets configuration when I merged the new skeleton. @lkdvos do you have a clue?

[ogauthe]

From reading the comments above, I see two problems with the current design:

• length(::AbstractGradedUnitRange) does not match the size of the fusion tensor
• data_matrix does not make explicitly the pairing between a sector and a matrix block

Independently, there is a question whether axes(::FusionTensor) should return a flat tuple or a tuple of 2 tuples.

I think length(::AbstractGradedUnitRange) should return the quantum dimension and not the number of multiplets. This means rewriting some parts of GradedUnitRanges, hopefully the change can be easily implemented now that AbstractGradedUnitRange is no more an alias for BlockedUnitRange. Then we would automatically have length.(axes(ft)) == size(ft). This could also play well with ITensor/SymmetrySectors.jl#4, where we already discussed changing blocklengths(::AbstractGradedUnitRange) behavior.

Concerning data_matrix, to me the cleanest would be to have a custom type to take into account the label. Something like

struct SymmetricDataMatrix <: AbstractBlockSparseMatrix
    matrix_blocks::Vector{Pair{AbstractSector,AbstractMatrix}}
    row_axis::AbstractGradedUnitRange
    column_axis::AbstractGradedUnitRange
end

which would act very similarly to the current data_matrix, but the quantum dimension of each block would be easier to access.

[ogauthe]

I opened dedicated issues for most of the points. There are still some discussions about precise naming conventions, but I feel like this is more a question of taste than a real issue.

I think this is ready to merge as a starting point.

[mtfishman]

I opened dedicated issues for most of the points. There are still some discussions about precise naming conventions, but I feel like this is more a question of taste than a real issue.

Could you open issues for them anyway? I think it is good to keep track of these things so we remember to circle back to them.