Utility Functions

cdot(a, b)

Return the complex dot product of a and b

cinv(M)

Calculate the inverse of the complex matrix M.

ckron(a, b)
ckron(A, B)

Return the complex Kronecker(outer) product of vectors a and b, i.e. a ⊗ b†, or of two matrices A and B, i.e. A ⊗ B.

cmvmul(A, x)

Return the matrix-vector product of A and x

cmvmul_block(P::PauliMatrix, x::SVector)

Return the matrix-vector product of the block diagonal matrix containing A₊ and A₋ and the vector x

cmvmul_color(A, x)

Return the matrix-vector product of A and x, where A only acts on the color structure of x.

cmvmul_d(A, x)

Return the matrix-vector product of the adjoint of A and x

cmvmul_d_color(A, x)

Return the matrix-vector product of A† and x, where A† only acts on the colo structure of x.

cmvmul_spin_proj(A, x, ::Val{ρ}, ::Val{is_adjoint}=Val(false))

Return A * (1 ± γᵨ) * x where γᵨ is the ρ-th Euclidean gamma matrix in the Chiral basis. x is assumed to be a 4xN component complex vector. The third argument is ρ wrapped in a Val and must be within the range [-4,4]. Its sign determines the sign in front of the γᵨ matrix. If is_adjoint is true, A† is used instead of A.

cvmmul(x, A)

Return the vector-matrix product of x† and A.

cvmmul_color(x, A)

Return the matrix-vector product of x† and A, where A only acts on the color structure of x†.

cvmmul_d(x, A)

Return the vector-matrix product of x and the adjoint of A. x is implicitly assumed to be a column vector and therefore the adjoint of x is used

cvmmul_d_color(x, A)

Return the matrix-vector product of x† and A†, where A† only acts on the color structure of x†.

exp_iQ(Q::SU{3,9,T}) where {T}
exp_iQ(e::ExpiQCoeffs{T}) where {T}

Compute the exponential of a traceless Hermitian 3x3 matrix Q or return the exp_iQ field of the ExpiQCoeffs{T}-object e.
From Morningstar & Peardon (2008) arXiv:hep-lat/0311018v1

exp_iQ_coeffs(Q::SU{3,9,T}) where {T}

Return a ExpiQCoeffs object that contains the exponential of Q and all parameters obtained in the Cayley-Hamilton algorithm that are needed for Stout force recursion.

gaussian_TA_mat(::Type{T}) where {T}

Generate a normally distributed traceless anti-Hermitian 3x3 matrix with precision T.

gen_SU2_matrix(ϵ, ::Type{T}) where {T}

Generate a Matrix X ∈ SU(2) with precision T near the identity with spread ϵ.
From Gattringer C. & Lang C.B. (Springer, Berlin Heidelberg 2010)

gen_SU3_matrix(ϵ, ::Type{T}) where {T}

Generate a Matrix X ∈ SU(3) with precision T near the identity with spread ϵ.
From Gattringer C. & Lang C.B. (Springer, Berlin Heidelberg 2010)

kenney_laub(M::SMatrix{3,3,Complex{T},9}) where {T}

Compute the SU(3) matrix closest to M using the Kenney-Laub algorithm.

move(s::SiteCoords, μ, steps, lim)

Move a site s in the direction μ by steps steps with periodic boundary conditions. The maximum extent of the lattice in the direction μ is lim.

mpi_init()

Check whether MPI has been initialized and if not, initialize it.

multr(A::SMatrix{N,N,Complex{T},N²}, B::SMatrix{N,N,Complex{T},N²}) where {N,N²,T}

Calculate the trace of the product of two complex NxN matrices A and B of precision T.

rand_SU3(::Type{T}) where {T}

Generate a random Matrix X ∈ SU(3) with precision T.

spin_proj(x, ::Val{ρ})

Return (1 ± γᵨ) * x where γᵨ is the ρ-th Euclidean gamma matrix in the Chiral basis. and x is a 4xN component complex vector. The second argument is ρ wrapped in a Val and must be within the range [-4,4]. Its sign determines the sign in front of the γᵨ matrix.

spintrace(a, b)

Return the complex Kronecker(outer) product of vectors a and b, summing over dirac indices, i.e. ∑ ᵨ aᵨ ⊗ bᵨ†

spintrace_pauli(P::PauliMatrix, ::Val{i})

switch_sides(site::CartesianIndex, NX, NY, NZ, NT, NV)

Return the cartesian index equivalent to site but with opposite parity. E.g., switch_sides((1, 1, 1, 1), 4, 4, 4, 4, 256) = (1, 1, 1, 3) and reverse

σμν_spin_mul(x, ::Val{i})

Return σμν * x where σμν = i/2 * [γμ, γν] with the gamma matrices in the Chiral basis and x is a 4xN component complex vector. The last argument specifies which of the 6 independent component of σμν to multiply with.