Public names index

Intermediate representation (IR) for many-body propagators.

AugmentedBasis <: AbstractBasis

Augmented basis on the imaginary-time/frequency axis.

Groups a set of additional functions, augmentations, with a given basis. The augmented functions then form the first basis functions, while the rest is provided by the regular basis, i.e.:

u[l](x) == l < naug ? augmentations[l](x) : basis.u[l-naug](x),

where naug = length(augmentations) is the number of added basis functions through augmentation. Similar expressions hold for Matsubara frequencies.

Augmentation is useful in constructing bases for vertex-like quantities such as self-energies ^{[wallerberger2021]} and when constructing a two-point kernel that serves as a base for multi-point functions ^{[shinaoka2018]}.

See also: MatsubaraConst for vertex basis ^{[wallerberger2021]}, TauConst, TauLinear for multi-point ^{[shinaoka2018]}

Bosonic statistics.

DiscreteLehmannRepresentation <: AbstractBasis

Discrete Lehmann representation (DLR) with poles selected according to extrema of IR.

This class implements a variant of the discrete Lehmann representation (DLR) 1. Instead of a truncated singular value expansion of the analytic continuation kernel $K$ like the IR, the discrete Lehmann representation is based on a "sketching" of $K$. The resulting basis is a linear combination of discrete set of poles on the real-frequency axis, continued to the imaginary-frequency axis:

 G(iv) == sum(a[i] / (iv - w[i]) for i in range(L))

Warning The poles on the real-frequency axis selected for the DLR are based on a rank-revealing decomposition, which offers accuracy guarantees. Here, we instead select the pole locations based on the zeros of the IR basis functions on the real axis, which is a heuristic. We do not expect that difference to matter, but please don't blame the DLR authors if we were wrong :-)

Fermionic statistics.

FiniteTempBasis <: AbstractBasis

Intermediate representation (IR) basis for given temperature.

For a continuation kernel K from real frequencies, ω ∈ [-ωmax, ωmax], to imaginary time, τ ∈ [0, β], this type stores the truncated singular value expansion or IR basis:

K(τ, ω) ≈ sum(u[l](τ) * s[l] * v[l](ω) for l in 1:L)

This basis is inferred from a reduced form by appropriate scaling of the variables.

Fields

• u::PiecewiseLegendrePolyVector: Set of IR basis functions on the imaginary time (tau) axis. These functions are stored as piecewise Legendre polynomials.

  To obtain the value of all basis functions at a point or a array of points x, you can call the function u(x). To obtain a single basis function, a slice or a subset l, you can use u[l].

• uhat::PiecewiseLegendreFT: Set of IR basis functions on the Matsubara frequency (wn) axis. These objects are stored as a set of Bessel functions.

  To obtain the value of all basis functions at a Matsubara frequency or a array of points wn, you can call the function uhat(wn). Note that we expect reduced frequencies, which are simply even/odd numbers for bosonic/fermionic objects. To obtain a single basis function, a slice or a subset l, you can use uhat[l].

• s: Vector of singular values of the continuation kernel

• v::PiecewiseLegendrePoly: Set of IR basis functions on the real frequency (w) axis. These functions are stored as piecewise Legendre polynomials.

  To obtain the value of all basis functions at a point or a array of points w, you can call the function v(w). To obtain a single basis function, a slice or a subset l, you can use v[l].

FiniteTempBasis{S}(β, ωmax, ε=nothing; max_size=nothing, args...)

Construct a finite temperature basis suitable for the given S (Fermionic or Bosonic) and cutoffs β and ωmax.

FiniteTempBasisSet

Type for holding IR bases and sparse-sampling objects.

An object of this type holds IR bases for fermions and bosons and associated sparse-sampling objects.

Fields

• basis_f::FiniteTempBasis: Fermion basis
• basis_b::FiniteTempBasis: Boson basis
• tau::Vector{Float64}: Sampling points in the imaginary-time domain
• wn_f::Vector{Int}: Sampling fermionic frequencies
• wn_b::Vector{Int}: Sampling bosonic frequencies
• smpltauf::TauSampling: Sparse sampling for tau & fermion
• smpltaub::TauSampling: Sparse sampling for tau & boson
• smplwnf::MatsubaraSampling: Sparse sampling for Matsubara frequency & fermion
• smplwnb::MatsubaraSampling: Sparse sampling for Matsubara frequency & boson
• sve_result::Tuple{PiecewiseLegendrePoly,Vector{Float64},PiecewiseLegendrePoly}: Results of SVE

Getters

• beta::Float64: Inverse temperature
• ωmax::Float64: Cut-off frequency

LogisticKernel <: AbstractKernel

Fermionic/bosonic analytical continuation kernel.

In dimensionless variables $x = 2 τ/β - 1$, $y = β ω/Λ$, the integral kernel is a function on $[-1, 1] × [-1, 1]$:

\[ K(x, y) = \frac{e^{-Λ y (x + 1) / 2}}{1 + e^{-Λ y}}\]

LogisticKernel is a fermionic analytic continuation kernel. Nevertheless, one can model the $τ$ dependence of a bosonic correlation function as follows:

\[ ∫ \frac{e^{-Λ y (x + 1) / 2}}{1 - e^{-Λ y}} ρ(y) dy = ∫ K(x, y) ρ'(y) dy,\]

with

\[ ρ'(y) = w(y) ρ(y),\]

where the weight function is given by

\[ w(y) = \frac{1}{\tanh(Λ y/2)}.\]

MatsubaraConst <: AbstractAugmentation

Constant in Matsubara, undefined in imaginary time.

MatsubaraFreq(n)

Prefactor n of the Matsubara frequency ω = n*π/β

Struct representing the Matsubara frequency ω entering the Fourier transform of a propagator G(τ) on imaginary time τ to its Matsubara equivalent Ĝ(iω) on the imaginary-frequency axis:

        β
Ĝ(iω) = ∫  dτ exp(iωτ) G(τ)      with    ω = n π/β,
        0

where β is inverse temperature and by convention we include the imaginary unit in the frequency argument, i.e, Ĝ(iω). The frequencies depend on the statistics of the propagator, i.e., we have that:

G(τ - β) = ± G(τ)

where + is for bosons and - is for fermions. The frequencies are restricted accordingly.

• Bosonic frequency (S == Fermionic): n even (periodic in β)
• Fermionic frequency (S == Bosonic): n odd (anti-periodic in β)

MatsubaraSampling <: AbstractSampling

Sparse sampling in Matsubara frequencies.

Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary frequencies.

MatsubaraSampling(basis; positive_only=false,
                  sampling_points=default_matsubara_sampling_points(basis; positive_only),
                  factorize=true)

Construct a MatsubaraSampling object. If not given, the sampling_points are chosen as the (discrete) extrema of the highest-order basis function in Matsubara. This turns out to be close to optimal with respect to conditioning for this size (within a few percent).

By setting positive_only=true, one assumes that functions to be fitted are symmetric in Matsubara frequency, i.e.:

\[ Ĝ(iν) = conj(Ĝ(-iν))\]

or equivalently, that they are purely real in imaginary time. In this case, sparse sampling is performed over non-negative frequencies only, cutting away half of the necessary sampling space. factorize controls whether the SVD decomposition is computed.

RegularizedBoseKernel <: AbstractKernel

Regularized bosonic analytical continuation kernel.

In dimensionless variables $x = 2 τ/β - 1$, $y = β ω/Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:

\[ K(x, y) = y \frac{e^{-Λ y (x + 1) / 2}}{e^{-Λ y} - 1}\]

Care has to be taken in evaluating this expression around $y = 0$.

TauConst <: AbstractAugmentation

Constant in imaginary time/discrete delta in frequency.

TauLinear <: AbstractAugmentation

Linear function in imaginary time, antisymmetric around β/2.

TauSampling <: AbstractSampling

Sparse sampling in imaginary time.

Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary time.

TauSampling(basis; sampling_points=default_tau_sampling_points(basis), factorize=true)

Construct a TauSampling object. If not given, the sampling_points are chosen as the extrema of the highest-order basis function in imaginary time. This turns out to be close to optimal with respect to conditioning for this size (within a few percent). factorize controls whether the SVD decomposition is computed.

evaluate!(buffer::AbstractArray{T,N}, sampling, al; dim=1) where {T,N}

Like evaluate, but write the result to buffer. Please use dim = 1 or N to avoid allocating large temporary arrays internally.

evaluate(sampling, al; dim=1)

Evaluate the basis coefficients al at the sparse sampling points.

fit!(buffer::Array{S,N}, smpl::AbstractSampling, al::Array{T,N}; 
    dim=1, workarr::Vector{S}) where {S,T,N}

Like fit, but write the result to buffer. Use dim = 1 or dim = N to avoid allocating large temporary arrays internally. The length of workarr cannot be smaller than SparseIR.workarrlength(smpl, al).

fit(sampling, al::AbstractArray{T,N}; dim=1)

Fit basis coefficients from the sparse sampling points Please use dim = 1 or N to avoid allocating large temporary arrays internally.

overlap(poly::PiecewiseLegendrePoly, f; 
    rtol=eps(T), return_error=false, maxevals=10^4, points=T[])

Evaluate overlap integral of poly with arbitrary function f.

Given the function f, evaluate the integral

∫ dx f(x) poly(x)

using adaptive Gauss-Legendre quadrature.

points is a sequence of break points in the integration interval where local difficulties of the integrand may occur (e.g. singularities, discontinuities).