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Advanced Usage: Automated Symmetry Reduction

In many cases, the numerical effort of computing functions in the Matsubara domain can be drastically reduced by the use of symmetries. For one-particle Green's functions $G_{i_1 i_2}(i\omega)$, for example, hermicity of the Hamiltonian dictates that $G_{i_1 i_2}(i\omega) = G^{\star}_{i_2 i_1}(-i\omega)$, relating positive and negative Matsubara frequencies. We offer an automated way to compute the set of irreducible (i.e. unrelatable by symmetries) MatsubaraFunction components, as is illustrated in the following example

ξ = 0.5
T = 1.0
N_= 128
g = MatsubaraGrid(T, N_, Fermion)
f = MatsubaraFunction(g, 1)

for v in g
    f[v, 1] = 1.0 / (im * value(v) - ξ)
end 

# complex conjugation acting on Green's function
function conj(
    w :: Tuple{MatsubaraFrequency},
    x :: Tuple{Int64}
    ) :: Tuple{Tuple{MatsubaraFrequency}, Tuple{Int64}, MatsubaraOperation}

    return (-w[1],), (x[1],), MatsubaraOperation(false, true)
end 

# compute the symmetry group 
SG = MatsubaraSymmetryGroup([MatsubaraSymmetry{1, 1}(conj)], f)

# obtain another Green's function by symmetrization
function init(
    w :: Tuple{MatsubaraFrequency},
    x :: Tuple{Int64}
    ) :: ComplexF64

    return f[w, x...]
end 

InitFunc = MatsubaraInitFunction{1, 1, ComplexF64}(init)
h        = MatsubaraFunction(g, 1)
SG(h, InitFunc)
@assert h == f

Functions and Functors for symmetry reduction

MatsubaraFunctions.MatsubaraSymmetryGroup — Type
MatsubaraSymmetryGroup{GD, SD, DD, Q <: Number}

MatsubaraSymmetryGroup type with fields:

  • classes :: Vector{Vector{Tuple{Int64, MatsubaraOperation}}} : collections of symmetry equivalent elements
  • speedup :: Float64 : expected speedup from the symmetry reduction
source
MatsubaraFunctions.get_reduced — Function
function get_reduced(
    SG :: MatsubaraSymmetryGroup{GD, SD, DD, Q},
    f  :: MatsubaraFunction{GD, SD, DD, Q}
    )  :: Vector{Q} where {GD, SD, DD, Q <: Number}

Calculate symmetry reduced representation of MatsubaraFunction

source
MatsubaraFunctions.init_from_reduced! — Function
function init_from_reduced!(
    SG   :: MatsubaraSymmetryGroup{GD, SD, DD, Q},
    f    :: MatsubaraFunction{GD, SD, DD, Q},
    fvec :: AbstractVector{Q}
    )    :: Nothing where {GD, SD, DD, Q <: Number}

Initialize MatsubaraFunction from symmetry reduced representation

source

I/O to HDF5 files