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SUNRepresentations.jl

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Compute Clebsch-Gordan coefficients for general SU(N) groups. Reimplementation of arXiv:1009.0437. Compatibility / interoperability with TensorKit.jl.

Installation

julia> using Pkg; Pkg.add("SUNRepresentations")

Usage

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)]

Caching Clebsch-Gordan coefficients

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 KiB

The 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/
└── ...

Conventions

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")

Extensions

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"$\overline{\textbf{10}}$"

TODO