API

spectral_to_imaginary_time_correlation_function(;
    # KEYWORD ARGUMENTS
    τ::AbstractVector{T},
    β::T,
    spectral_function::Function,
    kernel_function::Function,
    tol::T = 1e-10,
    bounds = (-Inf, +Inf),
    normalize::Bool = false
) where {T<:AbstractFloat}

Calculate and return the imaginary-time correlation function

\[C(\tau) = \int_{-\infty}^{\infty} d\omega \ K_\beta(\omega, \tau) \ A(\omega).\]

on a grid of $\tau$ (τ) values, given a spectral function $A(\omega)$ (spectral_function) and kernel function $K_\beta(\omega,\tau)$ (kernel_function). This integral is evaluated within a specified tolerance tol.

Arguments

• τ::AbstractVector{T}: Vector of imaginary time such that τ[end] = β equal the inverse temperature.
• spectral_function::Function: The spectral function $A(\omega)$ that takes a single argument.
• kernel_function::Function: The kernel function $K_\beta(\omega,\tau)$ that takes three arguments as shown.
• tol::T = 1e-10: Specified precision with which $C(\tau)$ is evaluated
• bounds = (-Inf, +Inf): Bounds on integration domain.
• normalize::Bool = false: Whether the spectral function needs to be normalized to unity.

spectral_to_matsubara_correlation_function(;
    # KEYWORD ARGUMENTS
    n::AbstractVector{Int},
    β::T,
    spectral_function::Function,
    kernel_function::Function,
    tol::T = 1e-10,
    bounds = (-Inf, 0.0, +Inf),
    normalize::Bool = false
) where {T<:AbstractFloat}

Calculate and return the Matsubara correlation function

\[C({\rm i} \omega_n) = \int_{-\infty}^{\infty} d\omega \ K_\beta(\omega, \omega_n) \ A(\omega).\]

for a vector of $n \in \mathbb{Z}$ values, given a spectral function $A(\omega)$ (spectral_function) and kernel function $K_\beta(\omega,\omega_n)$ (kernel_function). This integral is evaluated within a specified tolerance tol.

Note that the kernel function should be called as kernel_function(ω, n, β) where n specifies the Matsubara frequency, which is evaluated internally as either $\omega_n = (2n+1)\pi/\beta$ or $\omega_n = 2n\pi/\beta$ depending on whether the kernel function is fermionic of bosonic respectively.

Arguments

• n::AbstractVector{Int}: Vector of integers specifying Matsubara frequencies for which $C({\rm i}\omega_n)$ will be evaluated.
• spectral_function::Function: The spectral function $A(\omega)$ that takes a single argument.
• kernel_function::Function: The kernel function $K_\beta(\omega,\omega_n)$ that takes three arguments as shown.
• tol::T = 1e-10: Specified precision with which $C({\rm i}\omega_n)$ is evaluated.
• bounds = (-Inf, +Inf): Bounds on integration domain.
• normalize::Bool = false: Whether the spectral function needs to be normalized to unity.

add_noise(;
    # KEYWORD ARGUMENTS
    Cτ_exact::AbstractVector{T},
    τ::AbstractVector{T},
    σ::T,
    ξ::T,
    sum_rule::Function = C0 -> 1 - C0,
    noise_type::Symbol = :TruncatedNormal
) where {T<:AbstractFloat}

add_noise(
    # ARGUMENTS
    N_samples::Int;
    # KEYWORD ARGUMENTS
    Cτ_exact::AbstractVector{T},
    τ::AbstractVector{T},
    σ::T,
    ξ::T,
    sum_rule::Function = C0 -> 1 - C0,
    noise_type::Symbol = :TruncatedNormal
) where {T<:AbstractFloat}

Add noise to an imaginary time correlation function $C(\tau)$ that is exponentially correlated in imaginary time.

Arguments

• N_samples (optional): Number of random samples to generate. If passed this function returns a matrix where the columsn correspond to the different samples.

Keyword Arguments

• Cτ_exact::AbstractVector{T}: Vector containing the exact values for $C(\tau)$.
• τ::AbstractVector{T}: Vector specifying the imaginary time $\tau$ grid that $C(\tau)$ is evaluated on. Assumes that the last element equals the inverse temperature, i.e. τ[end] = β.
• σ::T: Standard deviation of the noise; controls the typical amplitude of the error.
• ξ::T: Correlation length associated with the noise in imaginary time.
• sum_rule::Function = C0 -> 1 - C0: Enforces sum rule, or bounday condition in imaginay time. Default behavior assumes a fermionic correlation function, enforcing that $C(\beta) = 1 - C(0)$.
• noise_type::Symbol = :TruncatedNormal: Distribution that the noise is sampled from prior to correlations being introduced in imaginary time. Available options are noise_type ∈ (:TruncatedNormal, :Gamma, :Normal).

Additional Information

If the correlation function $C(\tau_i)$ the corresponding noisy correlation function is given by

\[C_{\rm noisy}(\tau_i) = C(\tau_i) + \frac{\sum_j e^{-|\tau_j-\tau_i|/\xi} R_j}{\sum_j e^{-2|\tau_j-\tau_i|/\xi}},\]

where the sum is performed assuming periodic boundary conditions. If noise_type = :Normal then the random numbers of sampled according to $R_j \sim {\rm Normal}(0,\sigma)$. Otherwise,

\[R_j \sim {\rm sign}(C(\tau_j)) \cdot P(|C(\tau_j)|, \sigma) - C(\tau_j)\]

where $P(|C(\tau_j)|, \sigma)$ is either a Gamma or Truncated Normal distribution with mean and standard deviation given by $|C(\tau_j)|$ and $\sigma$ respectively.

The support for the Truncated Normal and Gamma distributions is $[0,\infty)$. Note that in the case that a Normal distribution is used it is possible for $C_{\rm noisy}(\tau_i)$ to have a different sign that $C(\tau_i)$, whereas this is not possible with the other two distributions.

add_noise!(
    # ARGUMENTS
    Cτ_noisy::AbstractVector{T};
    # KEYWORD ARGUMENTS
    Cτ_exact::AbstractVector{T},
    τ::AbstractVector{T},
    σ::T,
    ξ::T,
    sum_rule::Function = C0 -> 1 - C0,
    noise_type::Symbol = :TruncatedNormal
) where {T<:AbstractFloat}

add_noise!(
    # ARGUMENTS
    Cτ_noisy::AbstractMatrix{T};
    # KEYWORD ARGUMENTS
    Cτ_exact::AbstractVector{T},
    τ::AbstractVector{T},
    σ::T,
    ξ::T,
    sum_rule::Function = C0 -> 1 - C0,
    noise_type::Symbol = :TruncatedNormal
) where {T<:AbstractFloat}

Add noise to an imaginary time correlation function $C(\tau)$ that is exponentially correlated in imaginary time. See add_noise for more information.

Arguments

• Cτ_noisy: The array to which the noisy $C(\tau)$ data will be written. If a matrix and not a vector then the columns correspond to samples and the rows to the imaginary time slices $\tau$.

Keyword Arguments

• Cτ_exact::AbstractVector{T}: Vector containing the exact values for $C(\tau)$.
• τ::AbstractVector{T}: Vector specifying the imaginary time $\tau$ grid that $C(\tau)$ is evaluated on. Assumes that the last element equals the inverse temperature, i.e. τ[end] = β.
• σ::T: Standard deviation of the noise; controls the typical amplitude of the error.
• ξ::T: Correlation length associated with the noise in imaginary time.
• sum_rule::Function = C0 -> 1 - C0: Enforces sum rule, or bounday condition in imaginay time. Default behavior assumes a fermionic correlation function, enforcing that $C(\beta) = 1 - C(0)$.
• noise_type::Symbol = :TruncatedNormal: Distribution that the noise is sampled from prior to correlations being introduced in imaginary time. Available options are noise_type ∈ (:TruncatedNormal, :Gamma, :Normal).

kernel_mat(ω::T, ω_n::T) where {T<:AbstractFloat}

Generic Matsubara frequency kernel

\[K_\beta(\omega, \omega_n) = \frac{1}{\omega - {\rm i}\omega_n},\]

where $\omega_n = (2n+1)\pi/\beta \ (= 2n\pi/\beta)$ for fermions (bosons) with $n \in \mathbb{Z}$. This kernel assumes the sign convention

\[C(\tau) = \langle \hat{a}(\tau) \hat{a}^\dagger(0) \rangle\]

for a the Green's function, where it is assumed that $\tau \in [0,\beta)$.

kernel_tau_fermi(ω::T, τ::T, β::T) where {T<:AbstractFloat}

The imaginary time fermionic kernel

\[\begin{align*} K_\beta(\omega,\tau) & = \overbrace{\left(\frac{e^{-\tau\omega}}{1+e^{-\beta\omega}}\right)}^{\text{numerically unstable}} \\ & = \underbrace{\left( e^{\tau\omega} + e^{(\tau-\beta)\omega} \right)^{-1}}_{\text{numerically stable}}, \end{align*}\]

where it is assumed that $\tau \in [0,\beta)$.

kernel_mat_fermi(ω::T, n::Int, β::T) where {T<:AbstractFloat}

The fermionic matsubara frequency kernel

\[K_\beta(\omega, \omega_n) = \frac{1}{{\omega - {\rm i}\omega_n}},\]

where $\omega_n = (2n+1)\pi/\beta$ for fermions with $n \in \mathbb{Z}$.

kernel_tau_bose(ω::T, τ::T, β::T) where {T<:AbstractFloat}

The imaginary time bosonic kernel

\[\begin{align*} K_\beta(\omega,\tau) & = \overbrace{\left(\frac{e^{-\tau\omega}}{1-e^{-\beta\omega}}\right)}^{\text{numerically unstable}} \\ & = \underbrace{\left( e^{\tau\omega} - e^{(\tau-\beta)\omega} \right)^{-1}}_{\text{numerically stable}}, \end{align*}\]

where it is assumed that $\tau \in [0,\beta)$.

kernel_mat_bose(ω::T, n::Int, β::T) where {T<:AbstractFloat}

The bosonic matsubara frequency kernel

\[K_\beta(\omega, \omega_n) = \frac{1}{\omega - {\rm i}\omega_n},\]

where $\omega_n = 2n\pi/\beta$ for bosons with $n \in \mathbb{Z}$.

kernel_tau_bose_alt(ω::T, τ::T, β::T) where {T<:AbstractFloat}

The imaginary time bosonic kernel

\[\begin{align*} K_\beta(\omega,\tau) & = \overbrace{\left(\frac{\omega e^{-\tau\omega}}{1-e^{-\beta\omega}}\right)}^{\text{numerically unstable}} \\ & = \underbrace{\omega \left( e^{\tau\omega} - e^{(\tau-\beta)\omega} \right)^{-1}}_{\text{numerically stable}}, \end{align*}\]

where it is assumed that $\tau \in [0,\beta)$ and $K_\beta(0,\tau) = \beta^{-1}$.

kernel_mat_bose_alt(ω::T, n::Int, β::T) where {T<:AbstractFloat}

The bosonic matsubara frequency kernel

\[K_\beta(\omega, \omega_n) = \frac{\omega}{\omega - {\rm i}\omega_n},\]

where $\omega_n = 2n\pi/\beta$ for bosons with $n \in \mathbb{Z}$ and $K_\beta(0,0) = -1$.

kernel_tau_sym_bose(ω::T, τ::T, β::T) where {T<:AbstractFloat}

The imaginary time symmetrized bosonic kernel

\[\begin{align*} K_\beta(\omega,\tau) & = \overbrace{\left(\frac{e^{-\tau\omega} + e^{-(\beta-\tau)\omega}}{1-e^{-\beta\omega}}\right)}^\text{numerically unstable} \\ & = \underbrace{\left( e^{\tau\omega} - e^{(\tau-\beta)\omega} \right)^{-1} - \left(e^{-\tau\omega} - e^{-(\tau-\beta)\omega} \right)^{-1}}_{\text{numerically stable}}, \end{align*}\]

where it is assumed that $\tau \in [0,\beta)$.

kernel_mat_sym_bose(ω::T, n::Int, β::T) where {T<:AbstractFloat}

The symmetrized bosonic matsubara frequency kernel

\[\begin{align*} K_β(\omega, \omega_n) & = \frac{2\omega}{\omega_n^2 + \omega^2}, \end{align*}\]

where $\omega_n = 2n\pi/\beta$ for bosons with $n \in \mathbb{Z}$.

kernel_tau_sym_bose_alt(ω::T, τ::T, β::T) where {T<:AbstractFloat}

The imaginary time symmetrized bosonic kernel

\[\begin{align*} K_\beta(\omega,\tau) & = \overbrace{\omega \left(\frac{e^{-\tau\omega} + e^{-(\beta-\tau)\omega}}{1-e^{-\beta\omega}}\right)}^\text{numerically unstable} \\ & = \underbrace{\omega\left[\left( e^{\tau\omega} - e^{(\tau-\beta)\omega} \right)^{-1} - \left(e^{-\tau\omega} - e^{-(\tau-\beta)\omega} \right)^{-1}\right]}_{\text{numerically stable}}, \end{align*}\]

where it is assumed that $\tau \in [0,\beta)$ and $K_\beta(0,\tau) = 2/\beta$.

kernel_mat_sym_bose_alt(ω::T, n::Int, β::T) where {T<:AbstractFloat}

The symmetrized bosonic matsubara frequency kernel

\[\begin{align} K_β(\omega, \omega_n) & = \frac{2\omega^2}{\omega_n^2 + \omega^2}, \end{align}\]

where $\omega_n = 2n\pi/\beta$ for bosons with $n \in \mathbb{Z}$ and $K_\beta(0,0) = 2$.

fermi(ϵ::T, β::T) where {T<:AbstractFloat}

The Fermi-Dirac function

\[\begin{align*} f_\beta(\epsilon) & = \overbrace{\left( \frac{1}{e^{\beta\epsilon} + 1}\right)}^{\text{numerically unstable}} \\ & = \underbrace{\frac{1}{2}\left( 1 - \tanh\left(\frac{\beta\epsilon}{2}\right) \right)}_{\text{numerically stable}}, \end{align*}\]

where $\epsilon$ is energy and $\beta$ is inverse temperature.

bose(ϵ::T, β::T) where {T<:AbstractFloat}

The Bose-Einstein function

\[\begin{align*} n_\beta(\epsilon) & = \overbrace{\left( \frac{1}{e^{\beta\epsilon} - 1}\right)}^{\text{numerically unstable}} \\ & = \underbrace{\frac{1}{2}\left(\coth\left(\frac{\beta\epsilon}{2}\right) - 1 \right)}_{\text{numerically stable}}, \end{align*}\]

where $\epsilon$ is energy and $\beta$ is inverse temperature.