Compute Wigner's 3j and 6j symbols, and related quantities such as ClebschGordan coefficients and Racah's symbols.
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
WignerSymbols.set_buffer3j_size(; maxsize = ...)
WignerSymbols.set_buffer6j_size(; maxsize = ...)
Thus note that there are separate cache buffers for 3j symbols (or ClebschGordan 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
threadsafe 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.
Install with the new package manager via ]add WignerSymbols
or
using Pkg
Pkg.add("WignerSymbols")
While the following function signatures are probably selfexplanatory, 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 = m₂m₁) > ::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 = m₁m₂) > ::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.
Largely based on reading the paper (but not the code):
[1] H. T. Johansson and C. Forssén, SIAM Journal on Scientific Compututing 38 (2016) 376384 (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 BigInt
s) 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.

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