Kernels · ACFlow: The User Guide

Build kernel functions.

The ACFlow toolkit supports twelve types of kernel functions. They are:

Fermionic imaginary time kernel (ktype = "fermi", grid = "ftime")
Fermionic fragment imaginary time kernel (ktype = "fermi", grid = "fpart")
Fermionic Matsubara kernel (ktype = "fermi", grid = "ffreq")
Fermionic fragment Matsubara kernel (ktype = "fermi", grid = "ffrag")
Bosonic imaginary time kernel (ktype = "boson", grid = "btime")
Bosonic fragment imaginary time kernel (ktype = "boson", grid = "bpart")
Bosonc Matsubara kernel (ktype = "boson", grid = "bfreq")
Bosonc fragment Matsubara kernel (ktype = "boson", grid = "bfrag")
Symmetric bosonic imaginary time kernel (ktype = "bsymm", grid = "btime")
Symmetric bosonic fragment imaginary time kernel (ktype = "bsymm", grid = "bpart")
Symmetric bosonic Matsubara kernel (ktype = "bsymm", grid = "bfreq")
Symmetric bosonic fragment Matsubara kernel (ktype = "bsymm", grid = "bfrag")

Note that the MaxEnt, StochAC, and StochSK solvers rely on the make_kernel() function to provide the kernel function. However, the kernel function or matrix used in the StochOM and StochPX solvers are implemented in their own calc_lambda() functions.

Making Kernels

ACFlow.build_kernel — Function

build_kernel(am::AbstractMesh, fg::FermionicImaginaryTimeGrid)

Try to build fermionic kernel function in imaginary time axis.

Arguments

am -> Real frequency mesh.
fg -> Imaginary time grid.

Returns

kernel -> Kernel function, K(τ,ω).

build_kernel(am::AbstractMesh, fg::FermionicFragmentTimeGrid)

Try to build fermionic kernel function in imaginary time axis. Note that fg contains incomplete imaginary time data.

Arguments

am -> Real frequency mesh.
fg -> Imaginary time grid.

Returns

kernel -> Kernel function, K(τ,ω).

build_kernel(am::AbstractMesh, fg::FermionicMatsubaraGrid)

Try to build fermionic kernel function in Matsubara frequency axis. This function support the so-called preblur algorithm.

Arguments

am -> Real frequency mesh.
fg -> Matsubara frequency grid.

Returns

kernel -> Kernel function, K(iωₙ,ω).

build_kernel(am::AbstractMesh, fg::FermionicFragmentMatsubaraGrid)

Try to build fermionic kernel function in Matsubara frequency axis. This function support the so-called preblur algorithm.

Arguments

am -> Real frequency mesh.
fg -> Matsubara frequency grid.

Returns

kernel -> Kernel function, K(iωₙ,ω).

build_kernel(am::AbstractMesh, bg::BosonicImaginaryTimeGrid)

Try to build bosonic kernel function in imaginary time axis.

Arguments

am -> Real frequency mesh.
bg -> Imaginary time grid.

Returns

kernel -> Kernel function, K(τ,ω).

build_kernel(am::AbstractMesh, bg::BosonicFragmentTimeGrid)

Try to build bosonic kernel function in imaginary time axis. Note that bg contains incomplete imaginary time data.

Arguments

am -> Real frequency mesh.
bg -> Imaginary time grid.

Returns

kernel -> Kernel function, K(τ,ω).

build_kernel(am::AbstractMesh, bg::BosonicMatsubaraGrid)

Try to build bosonic kernel function in Matsubara frequency axis.

Arguments

am -> Real frequency mesh.
bg -> Matsubara frequency grid.

Returns

kernel -> Kernel function, K(iωₙ,ω).

build_kernel(am::AbstractMesh, bg::BosonicFragmentMatsubaraGrid)

Try to build bosonic kernel function in Matsubara frequency axis.

Arguments

am -> Real frequency mesh.
bg -> Matsubara frequency grid.

Returns

kernel -> Kernel function, K(iωₙ,ω).

ACFlow.build_kernel_symm — Function

build_kernel_symm(am::AbstractMesh, bg::BosonicImaginaryTimeGrid)

Try to build bosonic kernel function in imaginary time axis (just for correlator of Hermitian operator only).

Arguments

am -> Real frequency mesh.
bg -> Imaginary time grid.

Returns

kernel -> Kernel function, K(τ,ω).

build_kernel_symm(am::AbstractMesh, bg::BosonicFragmentTimeGrid)

Try to build bosonic kernel function in imaginary time axis (just for correlator of Hermitian operator only). Note that bg contains incomplete imaginary time data.

Arguments

am -> Real frequency mesh.
bg -> Imaginary time grid.

Returns

kernel -> Kernel function, K(τ,ω).

build_kernel_symm(am::AbstractMesh, bg::BosonicMatsubaraGrid)

Try to build bosonic kernel function in Matsubara frequency axis (just for correlator of Hermitian operator only). This function support the so-called preblur algorithm.

Arguments

am -> Real frequency mesh.
bg -> Matsubara frequency grid.

Returns

kernel -> Kernel function, K(iωₙ,ω).

build_kernel_symm(am::AbstractMesh, bg::BosonicFragmentMatsubaraGrid)

Try to build bosonic kernel function in Matsubara frequency axis (just for correlator of Hermitian operator only). This function support the so-called preblur algorithm.

Arguments

am -> Real frequency mesh.
bg -> Matsubara frequency grid.

Returns

kernel -> Kernel function, K(iωₙ,ω).

Utilities

ACFlow.make_blur — Function

make_blur(am::AbstractMesh, A::Vector{F64}, blur::F64)

Try to blur the given spectrum A, which is defined in am. And blur is the blur parameter.

Arguments

am -> Real frequency mesh.
A -> Spectral function.
blur -> Blur parameter. It must be larger than 0.0.

Returns

A -> It is updated in this function.

ACFlow.make_singular_space — Function

make_singular_space(kernel::Matrix{F64})

Perform singular value decomposition for the input matrix kernel.

kernel = U Σ Vᵀ

Supposed that kernel is a m × n matrix, then U is m × m, Σ is m × n, and V is n × n. For Σ, only the diagonal elements are non-zero.

Arguments

kernel -> Fermionic or bosonic kernel matrix.

Returns

U -> A m × n matrix.
S -> Diagonal elements of Σ.
V -> A n × n matrix.

ACFlow.make_gauss_peaks — Function

make_gauss_peaks(blur::F64)

Try to generate a gaussian peak along a linear mesh, whose energy range is [-5 * blur, +5 * blur]. The number of mesh points is fixed to 201.

Arguments

blur -> This parameter is used to control the width of gaussian peak.

Returns

bmesh -> A linear mesh in [-5 * blur, 5 * blur].
gaussian -> A gaussian peak at bmesh.