QInchworm.exact_atomic_ppgf

Exact atomic pseudo-particle Green's function module enabling exact evaluation of the atomic propagator $P_0(z)$ by evaluating the exponential $P_0(z) = -i e^{-iz \hat H_{loc}}$.

Exports

• ExactAtomicPPGF
• atomic_ppgf
• density_matrix
• interpolate!
• partition_function

struct ExactAtomicPPGF <: Keldysh.AbstractTimeGF{ComplexF64, false}

Exact atomic pseudo-particle Green's function type.

Fields

• β::Float64: Inverse temperature

• E::Vector{Float64}: Eigenvalues of the atomic Hamiltonian

Evaluate atomic propagator at the difference between imaginary time branch points.

Parameters

• z1: first branch point.
• z2: second branch point.

Returns

• Value of atomic pseudo-particle propagator $P_0(z_1 - z_2)$ as a diagonal matrix Diagonal.

Evaluate atomic propagator at complex contour time z.

Parameters

• z: scalar time.

Returns

• Value of atomic pseudo-particle propagator $P_0(z)$ as a diagonal matrix Diagonal.

interpolate!(
    x::Matrix{ComplexF64},
    P_0::QInchworm.exact_atomic_ppgf.ExactAtomicPPGF,
    z1::Keldysh.BranchPoint,
    z2::Keldysh.BranchPoint
)

In-place evaluation of the atomic propagator at the difference between imaginary time branch points.

Parameters

• x: Matrix to store the value of the atomic pseudo-particle propagator in.
• P_0: Atomic pseudo-particle propagator.
• z1: first branch point.
• z2: second branch point.

Returns

• Value of atomic pseudo-particle propagator $P_0(z_1 - z_2)$ as a diagonal matrix Diagonal.

atomic_ppgf(
    β::Float64,
    ed::KeldyshED.EDCore
) -> Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}

Construct the exact atomic pseudo-particle Green's function.

Parameters

• β: Inverse temperature.
• ed: Exact diagonalization structure describing the atomic problem.

density_matrix(
    P::Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}
) -> Vector{Matrix{ComplexF64}}

Extract the equilibrium density matrix $\rho = i P(-i\beta, 0)$ from a normalized pseudo-particle Green's function P. The density matrix is block-diagonal and is returned as a vector of blocks.

partition_function(
    P_0::Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}
) -> ComplexF64

Extract the partition function $Z = \mathrm{Tr}[i P_0(-i\beta, 0)]$ from a un-normalized pseudo-particle Green's function P_0.