Two-point correlators

Provide N-body response and correlation functions

LindhardΩnFiniteTemperature(dim::Int, q::T, n::Int, μ::T, kF::T, β::T, m::T, spin) where {T <: AbstractFloat}

Compute the polarization function of free electrons at a given frequency. Relative Accuracy is about ~ 1e-6

Arguments

• dim: dimension
• q: external momentum, q<1e-4 will be treated as q=0
• n: externel Matsubara frequency, ωn=2π*n/β
• μ: chemical potential
• kF: Fermi momentum
• β: inverse temperature
• m: mass
• spin : number of spins

3D Imaginary-time effective interaction derived from random phase approximation. Return $dW_0(q, τ)/v_q$ where $v_q$ is the bare Coulomb interaction and $dW_0$ is the dynamic part of the effective interaction.

The total effective interaction can be recoverd using,

\[W_0(q, τ) = v_q δ(τ) + dW_0(q, τ).\]

The dynamic contribution is the fourier transform of,

\[dW_0(q, iω_n)=v_q^2 Π(q, iω_n)/(1-v_q Π(q, iω_n))\]

Note that this dynamic contribution $dW_0'' diverges at small q. For this reason, this function returns$dW0/vq``

Arguments

• vqinv: inverse bare interaction as a function of q
• qgrid: one-dimensional array of the external momentum q
• τgrid: one-dimensional array of the imaginary-time
• dim: dimension
• μ: chemical potential
• kF: Fermi momentum
• β: inverse temperature
• spin : number of spins
• mass: mass

freePropagatorT(type, τ, ω, β)

Imaginary-time propagator.

Arguments

• type: symbol :fermi, :bose
• τ: the imaginary time, must be (-β, β]
• ω: dispersion ϵ_k-μ
• β = 1.0: the inverse temperature

freePropagatorΩ(type, n, ω, β=1.0)

Matsubara-frequency kernel of different type

Arguments

• type: symbol :fermi, :bose, :corr
• n: index of the Matsubara frequency
• ω: dispersion ϵ_k-μ
• β: the inverse temperature