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Merge pull request #7 from krivenko/dev/exact_atomic_ppgf
Exact atomic PPGF functionality.
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| # [`QInchworm.exact_atomic_ppgf`](@id QInchworm.exact_atomic_ppgf) | ||
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| ```@autodocs | ||
| Modules = [QInchworm.exact_atomic_ppgf] | ||
| ``` |
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| # QInchworm.jl | ||
| # | ||
| # Copyright (C) 2021-2024 I. Krivenko, H. U. R. Strand and J. Kleinhenz | ||
| # | ||
| # QInchworm.jl is free software: you can redistribute it and/or modify it under the | ||
| # terms of the GNU General Public License as published by the Free Software | ||
| # Foundation, either version 3 of the License, or (at your option) any later | ||
| # version. | ||
| # | ||
| # QInchworm.jl is distributed in the hope that it will be useful, but WITHOUT ANY | ||
| # WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS | ||
| # FOR A PARTICULAR PURPOSE. See the GNU General Public License for more | ||
| # details. | ||
| # | ||
| # You should have received a copy of the GNU General Public License along with | ||
| # QInchworm.jl. If not, see <http://www.gnu.org/licenses/>. | ||
| # | ||
| # Authors: Hugo U. R. Strand, Igor Krivenko | ||
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| """ | ||
| 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 | ||
| $(EXPORTS) | ||
| """ | ||
| module exact_atomic_ppgf | ||
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| using DocStringExtensions | ||
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| using LinearAlgebra: Diagonal, tr | ||
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| using Keldysh: BranchPoint, AbstractTimeGF | ||
| using KeldyshED; ked = KeldyshED; | ||
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| import Keldysh: interpolate! | ||
| import QInchworm.ppgf: partition_function, atomic_ppgf, density_matrix | ||
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| export ExactAtomicPPGF, partition_function, atomic_ppgf, density_matrix, interpolate! | ||
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| """ | ||
| $(TYPEDEF) | ||
| Exact atomic pseudo-particle Green's function type. | ||
| # Fields | ||
| $(TYPEDFIELDS) | ||
| """ | ||
| struct ExactAtomicPPGF <: AbstractTimeGF{ComplexF64, false} | ||
| "Inverse temperature" | ||
| β::Float64 | ||
| "Eigenvalues of the atomic Hamiltonian" | ||
| E::Array{Float64, 1} | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| 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`. | ||
| """ | ||
| function (P::ExactAtomicPPGF)(z::Number) | ||
| return -im * Diagonal(exp.(-im * z * P.E)) | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| 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`. | ||
| """ | ||
| function (P::ExactAtomicPPGF)(z1::BranchPoint, z2::BranchPoint) | ||
| Δz = z1.val - z2.val | ||
| return P(Δz) | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| 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`. | ||
| """ | ||
| function interpolate!(x::Matrix{ComplexF64}, | ||
| P_0::ExactAtomicPPGF, | ||
| z1::BranchPoint, | ||
| z2::BranchPoint) | ||
| Δz = z1.val - z2.val | ||
| x[:] = P_0(Δz) | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| Construct the exact atomic pseudo-particle Green's function. | ||
| # Parameters | ||
| - `β`: Inverse temperature. | ||
| - `ed`: Exact diagonalization structure describing the atomic problem. | ||
| """ | ||
| function atomic_ppgf(β::Float64, ed::ked.EDCore)::Vector{ExactAtomicPPGF} | ||
| Z = ked.partition_function(ed, β) | ||
| λ = log(Z) / β # Pseudo-particle chemical potential (enforcing Tr[i P(β)] = Tr[ρ] = 1) | ||
| P = [ ExactAtomicPPGF(β, E .+ λ) for E in energies(ed) ] | ||
| return P | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| Extract the partition function ``Z = \\mathrm{Tr}[i P_0(-i\\beta, 0)]`` from a un-normalized | ||
| pseudo-particle Green's function `P_0`. | ||
| """ | ||
| function partition_function(P_0::Vector{ExactAtomicPPGF})::ComplexF64 | ||
| return sum(P_0, init=0im) do P_s | ||
| im * tr(P_s(-im * P_s.β)) | ||
| end | ||
| end | ||
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| """ | ||
| $(TYPEDSIGNATURES) | ||
| 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. | ||
| """ | ||
| function density_matrix(P::Vector{ExactAtomicPPGF})::Vector{Matrix{ComplexF64}} | ||
| @assert !isempty(P) | ||
| return [1im * P_s(-im * P_s.β) for P_s in P] | ||
| end | ||
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| end # module exact_atomic_ppgf |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,60 @@ | ||
| # QInchworm.jl | ||
| # | ||
| # Copyright (C) 2021-2024 I. Krivenko, H. U. R. Strand and J. Kleinhenz | ||
| # | ||
| # QInchworm.jl is free software: you can redistribute it and/or modify it under the | ||
| # terms of the GNU General Public License as published by the Free Software | ||
| # Foundation, either version 3 of the License, or (at your option) any later | ||
| # version. | ||
| # | ||
| # QInchworm.jl is distributed in the hope that it will be useful, but WITHOUT ANY | ||
| # WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS | ||
| # FOR A PARTICULAR PURPOSE. See the GNU General Public License for more | ||
| # details. | ||
| # | ||
| # You should have received a copy of the GNU General Public License along with | ||
| # QInchworm.jl. If not, see <http://www.gnu.org/licenses/>. | ||
| # | ||
| # Authors: Hugo U. R. Strand, Igor Krivenko | ||
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| # Test exact atomic pseudo particle Green's function (exact_atomic_ppgf) module | ||
| # by comparing with the standard equidistant constructor in the ppgf module. | ||
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| using Test | ||
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| using Keldysh; kd = Keldysh | ||
| using KeldyshED; ked = KeldyshED; op = KeldyshED.Operators | ||
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| using QInchworm.ppgf: atomic_ppgf, partition_function | ||
| using QInchworm.exact_atomic_ppgf: exact_atomic_ppgf, partition_function | ||
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| @testset "exact_atomic_ppgf" begin | ||
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| β = 5.0 | ||
| nτ = 11 | ||
| μ = 1.337 | ||
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| H = μ * op.n(1) | ||
| soi = ked.Hilbert.SetOfIndices([[1]]) | ||
| ed = ked.EDCore(H, soi) | ||
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| P = atomic_ppgf(β, ed) | ||
| Z = partition_function(P) | ||
| @test Z ≈ 1.0 | ||
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| contour = kd.ImaginaryContour(β=β) | ||
| grid = kd.ImaginaryTimeGrid(contour, nτ) | ||
| P0 = atomic_ppgf(grid, ed) | ||
| Z0 = partition_function(P0) | ||
| @test Z == Z0 | ||
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| t0 = grid.points[1].bpoint | ||
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| for t1 in grid.points | ||
| t1 = t1.bpoint | ||
| for (p, p0) in zip(P, P0) | ||
| @test p(t1, t0) == p0(t1, t0) | ||
| end | ||
| end | ||
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| end |
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