Compute Clebsch-Gordan coefficients for general SU(N) groups. Reimplementation of arXiv:1009.0437. Compatibility / interoperability with TensorKit.jl.
julia> using Pkg; Pkg.add("SUNRepresentations")using TensorKit, SUNRepresentations
I = SUNIrrep(2, 1, 0)
println("$I ⊗ $I = $(collect(I ⊗ I))")Irrep[SU{3}]((2, 1, 0)) ⊗ Irrep[SU{3}]((2, 1, 0)) = SUNIrrep{3}[(0, 0, 0), (4, 2, 0), (3, 3, 0), (2, 1, 0), (3, 0, 0)]
As computing the Clebsch-Gordan coefficients is a relatively expensive operation, this packages automatically caches the results of the computations. To obtain information about the current status of the cache, one can call SUNRepresentations.cache_info().
Often, it may be useful to precompute a large set of coefficients (in parallel). These can
then be stored on disk and loaded when needed, or even transferred to other machines. This
can be done using the SUNRepresentations.precompute_disk_cache(N, a_max) function, which
will compute all Clebsch-Gordan coefficients for s1 ⊗ s2 -> s3, where s1 and s2 will
have Dynkin labels smaller than a_max, and s3 runs over all outputs of the fusion
product.
julia> SUNRepresentations.precompute_disk_cache(3)
CGC disk cache info:
====================
* SU(3) - Float64 - 32 entries - 134.462 KiBThe values are stored at SUNRepresentations.CGC_CACHE_PATH, which is a package-wide
scratchspace. Each file CGC/N/T/s1/s2.jld2 contains coefficients with datatype T for
the fusion of the SU(N) irreps s1 ⊗ s2 → s3, where s3 runs over all possible fusion
channels. The folder structure is as follows:
CGC/
├── 3/
│ ├── Float64/
│ │ ├── (0, 0, 0)/
│ │ │ ├── (0, 0, 0).jld2
│ │ │ ├── (1, 0, 0).jld2
│ │ │ └── ...
│ │ ├── (1, 0, 0)/
│ │ │ └── ...
│ │ └── ...
│ └── Float32/
│ └── ...
├── 4/
└── ...
By default, irreps are denoted by their N weights, which are equivalent to the number of
boxes in each row of the Young tableau, and this is also how they are stored. For example,
the fundamental representation of SU(3) is denoted by SUNIrrep(1, 0, 0), and the adjoint
representation by SUNIrrep(2, 1, 0). Nevertheless, we also support using N - 1 Dynkin
labels, which are denoted using Vector{Int}. For example, the fundamental representation
of SU(3) is denoted by SUNIrrep([1, 0]), and the adjoint representation by
SUNIrrep([1, 1]). Finally, it is also possible to use the dimensional name which is often
used in physics, e.g. SUNIrrep{3}("3") and SUNIrrep{3}("8").
The display of irreps can be changed in a persistent way by setting the display_mode
preference:
julia> using SUNRepresentations
julia> for mode in ["weight", "dynkin", "dimension"]
SUNRepresentations.display_mode(mode)
@show SUNIrrep(2,2,2,0)
end
SUNIrrep(2, 2, 2, 0) = Irrep[SU₄]((2, 2, 2, 0))
SUNIrrep(2, 2, 2, 0) = Irrep[SU₄]([0, 0, 2])
SUNIrrep(2, 2, 2, 0) = Irrep[SU₄]("10")This package supports outputting the irreps to a LaTeX format via a package extension for
Latexify.jl. To use this extension, load Latexify.jl and SUNRepresentations.jl and
then the following should work:
julia> using SUNRepresentations, Latexify
julia> latexify(SUNIrrep{4}("10⁺"))
L"$\overline{\textbf{10}}$"- Documentation