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First commit with layout for Latttice GPU code
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Alberto Ramos
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146
src/Groups/GroupSU2.jl
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src/Groups/GroupSU2.jl
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###
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### "THE BEER-WARE LICENSE":
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### Alberto Ramos wrote this file. As long as you retain this
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### notice you can do whatever you want with this stuff. If we meet some
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### day, and you think this stuff is worth it, you can buy me a beer in
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### return. <alberto.ramos@cern.ch>
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###
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### file: GroupSU2.jl
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### created: Sun Jul 11 17:23:12 2021
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###
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#
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# SU(2) group elements represented trough Cayley-Dickson
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# construction
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# https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
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import Base.:*, Base.:+, Base.:-,Base.:/,Base.:\
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struct SU2 <: Group
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t1::ComplexF64
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t2::ComplexF64
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end
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SU2() = SU2(1.0, 0.0)
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inverse(b::SU2) = SU2(conj(b.t1), -b.t2)
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dag(a::SU2) = inverse(a)
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norm(a::SU2) = sqrt(abs2(a.t1) + abs2(a.t2))
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tr(g::SU2) = 2.0*real(a.t1)
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"""
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function normalize(a::SU2)
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Return a normalized element of `SU(2)`
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"""
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function normalize(a::SU2)
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dr = sqrt(abs2(a.t1) + abs2(a.t2))
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if (dr == 0.0)
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return SU2(0.0)
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end
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return SU2(a.t1/dr,a.t2/dr)
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end
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Base.:+(a::SU2,b::SU2) = SU2(a.t1+b.t1,a.t2+b.t2)
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Base.:-(a::SU2,b::SU2) = SU2(a.t1-b.t1,a.t2-b.t2)
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Base.:*(a::SU2,b::SU2) = SU2(a.t1*b.t1-a.t2*conj(b.t2),a.t1*b.t2+a.t2*conj(b.t1))
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Base.:/(a::SU2,b::SU2) = SU2(a.t1*conj(b.t1)+a.t2*conj(b.t2),-a.t1*b.t2+a.t2*b.t1)
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Base.:\(a::SU2,b::SU2) = SU2(conj(a.t1)*b.t1+a.t2*conj(b.t2),conj(a.t1)*b.t2-a.t2*conj(b.t1))
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Base.:+(a::SU2) = SU2(a.t1,a.t2)
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Base.:-(a::SU2) = SU2(-a.t1,-a.t2)
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struct SU2alg <: Algebra
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t1::Float64
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t2::Float64
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t3::Float64
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end
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SU2alg(x::Real) = SU2alg(x,0.0,0.0)
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SU2alg(v::Vector) = SU2alg(v[1],v[2],v[3])
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projalg(g::SU2) = SU2alg(imag(g.t1), real(g.t2), imag(g.t2))
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dot(a::SU2alg, b::SU2alg) = a.t1*b.t1 + a.t2*b.t2 + a.t3*b.t3
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Base.:+(a::SU2alg) = SU2alg(a.t1,a.t2,a.t3)
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Base.:-(a::SU2alg) = SU2alg(-a.t1,-a.t2,-a.t3)
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Base.:+(a::SU2alg,b::SU2alg) = SU2alg(a.t1+b.t1,a.t2+b.t2,a.t3+b.t3)
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Base.:-(a::SU2alg,b::SU2alg) = SU2alg(a.t1-b.t1,a.t2-b.t2,a.t3-b.t3)
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Base.:*(a::SU2alg,b::Real) = SU2alg(a.t1*b,a.t2*b,a.t3*b)
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Base.:*(b::Real,a::SU2alg) = SU2alg(a.t1*b,a.t2*b,a.t3*b)
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Base.:/(a::SU2alg,b::Real) = SU2alg(a.t1/b,a.t2/b,a.t3/b)
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"""
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function Base.exp(a::SU2alg, t::Number=1)
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Computes `exp(a)`
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"""
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function Base.exp(a::SU2alg)
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rm = sqrt( a.t1^2+a.t2^2+a.t3^2 )/2.0
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if (abs(rm) < 0.05)
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rms = rm^2/2.0
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ca = 1.0 - rms *(1.0 - (rms/6.0 )*(1.0 - rms/15.0))
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sa = 0.5 - rms/6.0*(1.0 - (rms/10.0)*(1.0 - rms/21.0))
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else
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ca = cos(rm)
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sa = sin(rm)/(2.0*rm)
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end
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return SU2(complex(ca,sa*a.t1),complex(sa*a.t2,sa*a.t3))
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end
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function Base.exp(a::SU2alg, t::Number)
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rm = t*sqrt( a.t1^2+a.t2^2+a.t3^2 )/2.0
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if (abs(rm) < 0.05)
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rms = rm^2/2.0
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ca = 1.0 - rms *(1.0 - (rms/6.0 )*(1.0 - rms/15.0))
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sa = t*(0.5 - rms/6.0*(1.0 - (rms/10.0)*(1.0 - rms/21.0)))
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else
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ca = cos(rm)
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sa = t*sin(rm)/(2.0*rm)
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end
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return SU2(complex(ca,sa*a.t1),complex(sa*a.t2,sa*a.t3))
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end
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"""
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function expm(g::SU2, a::SU2alg)
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Computes `exp(a)*g`
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"""
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function expm(g::SU2, a::SU2alg)
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rm = sqrt( a.t1^2+a.t2^2+a.t3^2 )/2.0
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if (abs(rm) < 0.05)
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rms = rm^2/2.0
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ca = 1.0 - rms *(1.0 - (rms/6.0 )*(1.0 - rms/15.0))
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sa = 0.5 - rms/6.0*(1.0 - (rms/10.0)*(1.0 - rms/21.0))
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else
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ca = cos(rm)
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sa = sin(rm)/(2.0*rm)
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end
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return SU2(complex(ca,sa*a.t1)*g.t1-complex(sa*a.t2,sa*a.t3)*conj(g.t2),
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complex(ca,sa*a.t1)*g.t2+complex(sa*a.t2,sa*a.t3)*conj(g.t1))
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end
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"""
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function expm(g::SU2, a::SU2alg, t::Float64)
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Computes `exp(t*a)*g`
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"""
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function expm(g::SU2, a::SU2alg, t::Float64)
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rm = t*sqrt( a.t1^2+a.t2^2+a.t3^2 )/2.0
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if (abs(rm) < 0.05)
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rms = rm^2/2.0
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ca = 1.0 - rms *(1.0 - (rms/6.0 )*(1.0 - rms/15.0))
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sa = t*(0.5 - rms/6.0*(1.0 - (rms/10.0)*(1.0 - rms/21.0)))
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else
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ca = cos(rm)
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sa = t*sin(rm)/(2.0*rm)
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end
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return SU2(complex(ca,sa*a.t1)*g.t1-complex(sa*a.t2,sa*a.t3)*conj(g.t2),
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complex(ca,sa*a.t1)*g.t2+complex(sa*a.t2,sa*a.t3)*conj(g.t1))
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end
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