### ### "THE BEER-WARE LICENSE": ### Alberto Ramos wrote this file. As long as you retain this ### notice you can do whatever you want with this stuff. If we meet some ### day, and you think this stuff is worth it, you can buy me a beer in ### return. ### ### file: GroupSU3.jl ### created: Sun Jul 11 17:23:02 2021 ### # # Use memory efficient representation: Only store # first two rows. Third row constructed on the fly. # # a.u31 = conj(a.u12*a.u23 - a.u13*a.u22) # a.u32 = conj(a.u13*a.u21 - a.u11*a.u23) # a.u33 = conj(a.u11*a.u22 - a.u12*a.u21) # using Random import Base.:*, Base.:+, Base.:-,Base.:/,Base.:\,Base.one,Base.zero import Random.rand struct SU3{T} <: Group u11::Complex{T} u12::Complex{T} u13::Complex{T} u21::Complex{T} u22::Complex{T} u23::Complex{T} end inverse(a::SU3{T}) where T <: AbstractFloat = SU3{T}(conj(a.u11),conj(a.u21),(a.u12*a.u23 - a.u13*a.u22), conj(a.u12),conj(a.u22),(a.u13*a.u21 - a.u11*a.u23)) dag(a::SU3{T}) where T <: AbstractFloat = inverse(a) tr(a::SU3{T}) where T <: AbstractFloat = a.u11+a.u22+conj(a.u11*a.u22 - a.u12*a.u21) Base.one(::Type{SU3{T}}) where T <: AbstractFloat = SU3{T}(one(T),zero(T),zero(T),zero(T),one(T),zero(T)) Random.rand(rng::AbstractRNG, ::Random.SamplerType{SU3{T}}) where T <: AbstractFloat = exp(SU3alg{T}(randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T))) function Base.:*(a::SU3{T},b::SU3{T}) where T <: AbstractFloat bu31 = conj(b.u12*b.u23 - b.u13*b.u22) bu32 = conj(b.u13*b.u21 - b.u11*b.u23) bu33 = conj(b.u11*b.u22 - b.u12*b.u21) return SU3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*bu31, a.u11*b.u12 + a.u12*b.u22 + a.u13*bu32, a.u11*b.u13 + a.u12*b.u23 + a.u13*bu33, a.u21*b.u11 + a.u22*b.u21 + a.u23*bu31, a.u21*b.u12 + a.u22*b.u22 + a.u23*bu32, a.u21*b.u13 + a.u22*b.u23 + a.u23*bu33) end function Base.:/(a::SU3{T},b::SU3{T}) where T <: AbstractFloat bu31 = (b.u12*b.u23 - b.u13*b.u22) bu32 = (b.u13*b.u21 - b.u11*b.u23) bu33 = (b.u11*b.u22 - b.u12*b.u21) return SU3{T}(a.u11*conj(b.u11) + a.u12*conj(b.u12) + a.u13*conj(b.u13), a.u11*conj(b.u21) + a.u12*conj(b.u22) + a.u13*conj(b.u23), a.u11*(bu31) + a.u12*(bu32) + a.u13*(bu33), a.u21*conj(b.u11) + a.u22*conj(b.u12) + a.u23*conj(b.u13), a.u21*conj(b.u21) + a.u22*conj(b.u22) + a.u23*conj(b.u23), a.u21*(bu31) + a.u22*(bu32) + a.u23*(bu33)) end function Base.:\(a::SU3{T},b::SU3{T}) where T <: AbstractFloat au31 = (a.u12*a.u23 - a.u13*a.u22) au32 = (a.u13*a.u21 - a.u11*a.u23) bu31 = conj(b.u12*b.u23 - b.u13*b.u22) bu32 = conj(b.u13*b.u21 - b.u11*b.u23) bu33 = conj(b.u11*b.u22 - b.u12*b.u21) return SU3{T}(conj(a.u11)*b.u11 + conj(a.u21)*b.u21 + (au31)*bu31, conj(a.u11)*b.u12 + conj(a.u21)*b.u22 + (au31)*bu32, conj(a.u11)*b.u13 + conj(a.u21)*b.u23 + (au31)*bu33, conj(a.u12)*b.u11 + conj(a.u22)*b.u21 + (au32)*bu31, conj(a.u12)*b.u12 + conj(a.u22)*b.u22 + (au32)*bu32, conj(a.u12)*b.u13 + conj(a.u22)*b.u23 + (au32)*bu33) end function isgroup(a::SU3{T}) where T <: AbstractFloat tol = 1.0E-10 g = a/a if ( (abs(g.u11 - 1.0) < tol) && (abs(g.u12) < tol) && (abs(g.u13) < tol) && (abs(g.u21) < tol) && (abs(g.u22 - 1.0) < tol) && (abs(g.u23) < tol) ) return true else return false end end struct SU3alg{T} <: Algebra t1::T t2::T t3::T t4::T t5::T t6::T t7::T t8::T end function projalg(a::SU3{T}) where T <: AbstractFloat sr3ov2::T = 0.866025403784438646763723170752 ditr = ( imag(a.u11) + imag(a.u22) + 2.0*imag(a.u11*a.u22 - a.u12*a.u21) )/3.0 m12 = (a.u12 - conj(a.u21))/2.0 m13 = (a.u13 - (a.u12*a.u23 - a.u13*a.u22) )/2.0 m23 = (a.u23 - (a.u13*a.u21 - a.u11*a.u23) )/2.0 return SU3alg{T}(imag( m12 ), imag( m13 ), imag( m23 ), real( m12 ), real( m13 ), real( m23 ), (imag(a.u11)-imag(a.u22))/2.0, sr3ov2*(ditr)) end dot(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = a.t1*b.t1 + a.t2*b.t2 + a.t3*b.t3 + a.t4*b.t4 + a.t5*b.t5 + a.t6*b.t6 + a.t7*b.t7 + a.t8*b.t8 norm2(a::SU3alg{T}) where T <: AbstractFloat = a.t1^2 + a.t2^2 + a.t3^2 + a.t4^2 + a.t5^2 + a.t6^2 + a.t7^2 + a.t8^2 norm(a::SU3alg{T}) where T <: AbstractFloat = sqrt(a.t1^2 + a.t2^2 + a.t3^2 + a.t4^2 + a.t5^2 + a.t6^2 + a.t7^2 + a.t8^2) Base.zero(::Type{SU3alg{T}}) where T <: AbstractFloat = SU3alg{T}(zero(T),zero(T),zero(T),zero(T),zero(T),zero(T),zero(T),zero(T)) Random.rand(rng::AbstractRNG, ::Random.SamplerType{SU3alg{T}}) where T <: AbstractFloat = SU3alg{T}(randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T),randn(rng,T)) Base.:+(a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1,a.t2,a.t3,a.t4,a.t5,a.t6,a.t7,a.t8) Base.:-(a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(-a.t1,-a.t2,-a.t3,-a.t4,-a.t5,-a.t6,-a.t7,-a.t8) Base.:+(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1+b.t1,a.t2+b.t2,a.t3+b.t3,a.t4+b.t4,a.t5+b.t5,a.t6+b.t6,a.t7+b.t7,a.t8+b.t8) Base.:-(a::SU3alg{T},b::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(a.t1-b.t1,a.t2-b.t2,a.t3-b.t3,a.t4-b.t4,a.t5-b.t5,a.t6-b.t6,a.t7-b.t7,a.t8-b.t8) Base.:*(a::SU3alg{T},b::Number) where T <: AbstractFloat = SU3alg{T}(b*a.t1,b*a.t2,b*a.t3,b*a.t4,b*a.t5,b*a.t6,b*a.t7,b*a.t8) Base.:*(b::Number,a::SU3alg{T}) where T <: AbstractFloat = SU3alg{T}(b*a.t1,b*a.t2,b*a.t3,b*a.t4,b*a.t5,b*a.t6,b*a.t7,b*a.t8) Base.:/(a::SU3alg{T},b::Number) where T <: AbstractFloat = SU3alg{T}(a.t1/b,a.t2/b,a.t3/b,a.t4/b,a.t5/b,a.t6/b,a.t7/b,a.t8/b) export SU3, SU3alg, inverse, dag, tr, projalg, expm, exp, norm, norm2, isgroup struct M3x3{T} u11::Complex{T} u12::Complex{T} u13::Complex{T} u21::Complex{T} u22::Complex{T} u23::Complex{T} u31::Complex{T} u32::Complex{T} u33::Complex{T} end Base.:*(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*b.u31, a.u11*b.u12 + a.u12*b.u22 + a.u13*b.u32, a.u11*b.u13 + a.u12*b.u23 + a.u13*b.u33, a.u21*b.u11 + a.u22*b.u21 + a.u23*b.u31, a.u21*b.u12 + a.u22*b.u22 + a.u23*b.u32, a.u21*b.u13 + a.u22*b.u23 + a.u23*b.u33, a.u31*b.u11 + a.u32*b.u21 + a.u33*b.u31, a.u31*b.u12 + a.u32*b.u22 + a.u33*b.u32, a.u31*b.u13 + a.u32*b.u23 + a.u33*b.u33) function Base.:*(a::SU3{T},b::M3x3{T}) where T <: AbstractFloat a.u31 = conj(a.u12*a.u23 - a.u13*a.u22) a.u32 = conj(a.u13*a.u21 - a.u11*a.u23) a.u33 = conj(a.u11*a.u22 - a.u12*a.u21) return M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*b.u31, a.u11*b.u12 + a.u12*b.u22 + a.u13*b.u32, a.u11*b.u13 + a.u12*b.u23 + a.u13*b.u33, a.u21*b.u11 + a.u22*b.u21 + a.u23*b.u31, a.u21*b.u12 + a.u22*b.u22 + a.u23*b.u32, a.u21*b.u13 + a.u22*b.u23 + a.u23*b.u33, au31*b.u11 + au32*b.u21 + au33*b.u31, au31*b.u12 + au32*b.u22 + au33*b.u32, au31*b.u13 + au32*b.u23 + au33*b.u33) end function Base.:*(a::M3x3{T},b::SU3{T}) where T <: AbstractFloat bu31 = conj(b.u12*b.u23 - b.u13*b.u22) bu32 = conj(b.u13*b.u21 - b.u11*b.u23) bu33 = conj(b.u11*b.u22 - b.u12*b.u21) return M3x3{T}(a.u11*b.u11 + a.u12*b.u21 + a.u13*bu31, a.u11*b.u12 + a.u12*b.u22 + a.u13*bu32, a.u11*b.u13 + a.u12*b.u23 + a.u13*bu33, a.u21*b.u11 + a.u22*b.u21 + a.u23*bu31, a.u21*b.u12 + a.u22*b.u22 + a.u23*bu32, a.u21*b.u13 + a.u22*b.u23 + a.u23*bu33, a.u31*b.u11 + a.u32*b.u21 + a.u33*bu31, a.u31*b.u12 + a.u32*b.u22 + a.u33*bu32, a.u31*b.u13 + a.u32*b.u23 + a.u33*bu33) end function Base.:/(a::M3x3{T},b::SU3{T}) where T <: AbstractFloat bu31 = (b.u12*b.u23 - b.u13*b.u22) bu32 = (b.u13*b.u21 - b.u11*b.u23) bu33 = (b.u11*b.u22 - b.u12*b.u21) return M3x3{T}(a.u11*conj(b.u11) + a.u12*conj(b.u12) + a.u13*conj(b.u13), a.u11*conj(b.u21) + a.u12*conj(b.u22) + a.u13*conj(b.u23), a.u11*(bu31) + a.u12*(bu32) + a.u13*(bu33), a.u21*conj(b.u11) + a.u22*conj(b.u12) + a.u23*conj(b.u13), a.u21*conj(b.u21) + a.u22*conj(b.u22) + a.u23*conj(b.u23), a.u21*(bu31) + a.u22*(bu32) + a.u23*(bu33), a.u31*conj(b.u11) + a.u32*conj(b.u12) + a.u33*conj(b.u13), a.u31*conj(b.u21) + a.u32*conj(b.u22) + a.u33*conj(b.u23), a.u31*(bu31) + a.u32*(bu32) + a.u33*(bu33)) end Base.:*(a::Number,b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a*b.u11, a*b.u12, a*bu13, a*b.u21, a*b.u22, a*bu23, a*b.u31, a*b.u32, a*bu33) Base.:*(b::M3x3{T},a::Number) where T <: AbstractFloat = M3x3{T}(a*b.u11, a*b.u12, a*bu13, a*b.u21, a*b.u22, a*bu23, a*b.u31, a*b.u32, a*bu33) Base.:+(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11+b.u11, a.u12+b.u12, a.u13+bu13, a.u21+b.u21, a.u22+b.u22, a.u23+bu23, a.u31+b.u31, a.u32+b.u32, a.u33+bu33) Base.:-(a::M3x3{T},b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(a.u11-b.u11, a.u12-b.u12, a.u13-bu13, a.u21-b.u21, a.u22-b.u22, a.u23-bu23, a.u31-b.u31, a.u32-b.u32, a.u33-bu33) Base.:-(b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(-b.u11, -b.u12, -bu13, -b.u21, -b.u22, -bu23, -b.u31, -b.u32, -bu33) Base.:+(b::M3x3{T}) where T <: AbstractFloat = M3x3{T}(b.u11, b.u12, bu13, b.u21, b.u22, bu23, b.u31, b.u32, bu33) function alg2mat(a::SU3alg{T}) where T <: AbstractFloat two::T = 2.0 rct::T = 3.46410161513775458 x8p::T = a.t8/rct x7p::T = a.t7/two u11::Complex{T} = complex(0.0, x7p + x8p) u22::Complex{T} = complex(0.0,-x7p + x8p) u33::Complex{T} = complex(0.0,-2.0*x8p) u12::Complex{T} = complex(a.t4,a.t1)/two u13::Complex{T} = complex(a.t5,a.t2)/two u23::Complex{T} = complex(a.t6,a.t3)/two u21::Complex{T} = -conj(u12) u31::Complex{T} = -conj(u13) u32::Complex{T} = -conj(u23) return M3x3{T}(u11,u12,u13, u21,u22,u23, u31,u32,u33) end @inline function exp_iter(dch::Complex{T}, tch::T) where T <: AbstractFloat c::NTuple{22, T} = ( 1.957294106339126128e-20, 4.110317623312164853e-19, 8.220635246624329711e-18, 1.561920696858622643e-16, 2.811457254345520766e-15, 4.779477332387385293e-14, 7.647163731819816473e-13, 1.147074559772972473e-11, 1.605904383682161451e-10, 2.087675698786809894e-09, 2.505210838544171879e-08, 2.755731922398589067e-07, 2.755731922398589065e-06, 2.480158730158730158e-05, 1.984126984126984127e-04, 1.388888888888888888e-03, 8.333333333333333333e-03, 4.166666666666666666e-02, 1.666666666666666666e-01, 0.5, 1.0, 1.0 ) q0 = complex(c[1]) q1 = complex(0.0) q2 = complex(0.0) @inbounds for i in 2:length(c) qt0 = q0 qt1 = q1 q0 = complex(c[i]) + dch*q2 q1 = qt0 - tch*q2 q2 = qt1 end return q0, q1, q2 end function expm(g::SU3{T}, a::SU3alg{T}, t::Number) where T <: AbstractFloat tpw = t^2 M = alg2mat(a) Msq = M*M dch::Complex{T} = tpw*t*(M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 + M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 - M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31) tch::T = -tpw*(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0 q0, q1, q2 = exp_iter(dch, tch) q1 = t*q1 q2 = tpw*q2 g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13, q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23) return g2*g end function expm(g::SU3{T}, a::SU3alg{T}) where T <: AbstractFloat M = alg2mat(a) Msq = M*M dch::Complex{T} = M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 + M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 - M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31 tch::T = -(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0 q0, q1, q2 = exp_iter(dch, tch) g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13, q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23) return g2*g end function Base.exp(a::SU3alg{T}) where T <: AbstractFloat M = alg2mat(a) Msq = M*M dch::Complex{T} = M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 + M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 - M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31 tch::T = -(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0 q0, q1, q2 = exp_iter(dch, tch) g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13, q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23) return g2 end function Base.exp(a::SU3alg{T}, t::Number) where T <: AbstractFloat tpw = t^2 M = alg2mat(a) Msq = M*M dch::Complex{T} = tpw*t*(M.u11*M.u22*M.u33 + M.u13*M.u21*M.u32 + M.u31*M.u12*M.u23 - M.u11*M.u23*M.u32 - M.u12*M.u21*M.u33 - M.u13*M.u22*M.u31) tch::T = -tpw*(real(Msq.u11)+real(Msq.u22)+real(Msq.u33))/2.0 q0, q1, q2 = exp_iter(dch, tch) q1 = t*q1 q2 = tpw*q2 g2 = SU3{T}(q1*M.u11 + q2*Msq.u11+q0, q1*M.u12 + q2*Msq.u12, q1*M.u13 + q2*Msq.u13, q1*M.u21 + q2*Msq.u21, q1*M.u22 + q2*Msq.u22+q0, q1*M.u23 + q2*Msq.u23) return g2 end