WignerSymbols

Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan coefficients and Racah's symbols.

What's new in v2

WignerSymbols.jl was updated to version 2.0 on June 16th, 2021. This is the first major update in several years. The most important change is that WignerSymbols.jl is now completely thread safe, i.e. you can request Wigner symbols from different threads simultaneously. The computation of the Wigner symbols is not in itself multithreaded (this may be added in the future).

WignerSymbols.jl does no longer store the Wigner 3j and 6j symbols in a Dict cache, but rather in an LRU cache from [LRUCache.jl](https://github.com/JuliaCollections/ LRUCache.jl). Hence, it no longer stores all Wigner symbols ever computed, but only the most recent ones, and it that sense this is a (softly) breaking release. By default, it stores the $10^6$ most recent ones, which is probably equivalent to storing all of them in most use cases. This number can be changed via the interface

WignerSymbols.set_buffer3j_size(; maxsize = ...)
WignerSymbols.set_buffer6j_size(; maxsize = ...)

Thus note that there are separate cache buffers for 3j symbols (or Clebsch-Gordan coefficients, or Racah V coefficients) and 6j symbols (or Racah W coefficients).

For the underlying prime factorizations on which WignerSymbols.jl is based (which are also cached), a custom type GrowingList was implemented that can be expanded indefinitely in a thread-safe way. While there is some overhead in making the caches thread safe, these should mostly be compensated (except for maybe in compilation time) by overall improvements throughout the library, being more careful about unnecessary computations and about memory consumption for temporary variables. These changes also rely on Base.unsafe_rational which is only available since Julia 1.5, which is now required and thus provides another good reason for increasing the major version of WignerSymbols.jl. In tests for generating all Wigner symbols up to a maximal angular momentum value, WignerSymbols version 2 outperforms version 1.x with about ten to tweny percent.

Installation

Install with the new package manager via ]add WignerSymbols or

using Pkg
Pkg.add("WignerSymbols")

Available functions

While the following function signatures are probably self-explanatory, you can query help for them in the Julia REPL to get further details.

• wigner3j(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₁) -> ::T
• wigner6j(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, j₄, j₅, j₆) -> ::T
• clebschgordan(T::Type{<:Real} = RationalRoot{BigInt}, j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) -> ::T
• racahV(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) -> ::T
• racahW(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, J, j₃, J₁₂, J₂₃) -> ::T
• δ(j₁, j₂, j₃) -> ::Bool
• Δ(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃) -> ::T

The package relies on HalfIntegers.jl to support and use arithmetic with half integer numbers, and, since v1.1, on RationalRoots.jl to return the result exactly as the square root of a Rational{BigInt}, which will then be automatically converted to a suitable floating point value upon further arithmetic, using the AbstractIrrational interface from Julia Base.

Implementation

Largely based on reading the paper (but not the code):

[1] H. T. Johansson and C. Forssén, SIAM Journal on Scientific Compututing 38 (2016) 376-384 (arXiv:1504.08329)

with some additional modifications to further improve efficiency for large j (angular momenta quantum numbers).

In particular, 3j and 6j symbols are computed exactly, in the format √(r) * s where r and s are exactly computed as Rational{BigInt}, using an intermediate representation based on prime number factorization. This exact representation is captured by the RationalRoot type. For further calculations, these values probably need to be converted to a floating point type. Because of this exact representation, all of the above functions can be called requesting BigFloat precision for the result.

Most intermediate calculations (prime factorizations of numbers and their factorials, conversion between prime powers and BigInts) are cached to improve the efficiency, but this can result in large use of memory when querying Wigner symbols for large values of j.

Also uses ideas from

[2] J. Rasch and A. C. H. Yu, SIAM Journal on Scientific Compututing 25 (2003), 1416–1428

for caching the computed 3j and 6j symbols.

Todo

• Wigner 9-j symbols, as explained in [1] and based on

  [3] L. Wei, New formula for 9-j symbols and their direct calculation, Computers in Physics, 12 (1998), 632–634.