jED

AIM <: Model
AIM(ϵₖ::Vector{Float64}, Vₖ::Vector{T}, μ::Float64, U::Float64)

Type for the Anderson impurity model. A model can be used to construct a Eigenspace given a set of Basis.

Can be created from

• bath onsite energies $\epsilon_k$,
• bath-impurity hopping $V_k$
• impurity chemical potential $\mu$
• Coulomb interaction strength on the impurity $U$

Example

julia> ϵₖ = [0.5, -5.0]
julia> Vₖ = [1.0, 1.0]
julia> U  = 1.0
julia> μ  = 0.5
julia> model = AIM(ϵₖ, Vₖ, μ, U)
AIM{3, Float64}([
-0.5 1.0 1.0; 1.0 0.5 0.0; 1.0 0.0 -5.0], [1.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0], [0.0 0.0 0.0; 0.0 0.0 0.0; 0.0 0.0 0.0])

AIMParams

Parameters for energy levels ($\epsilon_k$) and hybridization amplitudes ($V_k$) of the Anderson impurity model.

Basis

A state is represented as a BitVetor of length 64. The first 32 bit (ordering follows Julia Vector indexing) correspond to spin up for each site (index 1 corresponds to the first site, 2 to the second, etc.) and bits 33 to 64 to spin down, for site 1 to 32. Each bit is set to 1 if it is occupied by an electron of the given spin. In principle, the number of electron flavors and orbitals is free, however, for now it is fixed to 2 and 1.

A list of states should be generated using States(NSites) with NSites being the number of sites in the model.

Fields

• NFlavors : Integer, number of flavors (TODO: hardcoded to 2 for now. later: orbitals*2?!)
• NSites : Integer, number of sites
• states : Vector{State}, list of Fock states. The are sorted according to good quantum numbers. blocklist contains start and size of blocks.
• blocklist : Vector{NTuple{4,Int}}, Vector of 4-tuples. Each entry encodes a block in the following form: 1. element is the start index of the block, 2. element is the column size of the block, 3. element is the electron number (see N_el), 4. element is the spin (see S)
• cdag_ov : Vector{Tuple{UnitRange{Int},UnitRange{Int}}}, Slices of states, that contribute to an overlap with one creation operator. Needed for performance reasons in computation of Lehmann representation (see also (@ref findcdagoverlapblocks)[_find_cdag_overlap_blocks])).

Basis(NSites::Int; NFlavors::Int = 2)

Contructs a Fock basis for NSites with NFlavors.

Eigenspace

Containes Eigenvalues and Eigenvectors grouped into blocks with are each ordered by magnitude of Eigenvalues. The blocks are part of the Basis.

Fields

• evals : Eigenvalues
• evecs : Eigenvectors
• E0 : smallest Eigenvalue

Eigenspace(model::Model, basis::Basis; verbose::Bool = true, FPT::Type{FPTi} = eltype(model.tMatrix))

Constructs the Eigenspace for Model over given Basis by diagonalizing the Hamiltonian (see also calc_Hamiltonian) for each block.

Fockstate

Internal representation of a Fockstate.

Model

Abstract type for model type. Currently implemented:

• AIM

Operator

Operator for Fockstates. Holds function for transformation of Fock state and change in quantum numbers.

Fields

• f : Function, operation on Fock state
• N_inc : Integer, change in electron number
• S_inc : Integer, change in spin number

Example

julia> o1 = Operator(x->create())

Overlap

Contains information about overlap of Fock states under given Operator.

Fields

• op : Function, operation on Fock state
• ov_blocks : Vector{Int}, index is starting block, entry target
• ov_list : Vector{Int}, index is starting block, entry index in eigenvector (see Eigenspace)

CDag_sign(state,i)

Sign when annihilating an electron at position i in state. Returns: -1/1/0 for Uneven/Even permutations and 0 when the state is not occupied.

julia> CDag_sign(Fockstate{6}(Bool[1,0,1,1,0,0]),5)
-1
julia> CDag_sign(Fockstate{6}(Bool[1,0,1,1,0,0]),3)
0
julia> CDag_sign(Fockstate{7}(Bool[1,0,1,0,1,0,0]),4)
1

C_sign(state,i)

Sign when creating an electron at position i in state. Returns: -1/1/0 for Uneven/Even permutations and 0 when the state is not occupied.

julia> C_sign(Fockstate{6}(Bool[1,0,1,1,0,0]),3)
-1
julia> C_sign(Fockstate{6}(Bool[1,0,1,1,0,0]),2)
0
julia> C_sign(Fockstate{6}(Bool[1,0,1,1,0,0]),4)

GLoc(ΣImp::MatsubaraF, μ::Float64, νnGrid::FermionicMatsubaraGrid, kG::KGrid)

Compute local Green's function $\int dk [i\nu_n + \mu + \epsilon_k - \Sigma_\text{Imp}(i\nu_n)]^{-1}$. TODO: simplify -conj!!!

GWeiss(νnGrid::FermionicMatsubaraGrid, p::AIMParams)
GWeiss!(target::Vector, νnGrid::Vector, μ::Float64, p::AIMParams)

Computes Weiss Green's frunction from Anderson Parameters.

GWeiss(Δ::OffsetVector{ComplexF64, Vector{ComplexF64}}, μ::Float64, νnGrid::FermionicMatsubaraGrid)

Computes Weiss Green's frunction from hybridization function.

GWeiss_from_Imp(GLoc::MatsubaraF, ΣImp::MatsubaraF)

Compute Updates Weiss Green's function from impurity self-energy via $[G_\text{loc} + \Sigma_\text{Imp}]^{-1}$ (see GLoc and Σ_from_GImp).

N_do(s::Fockstate{NSites})

Number of down electrons, $N_\downarrow$ in state s.

N_el(s::AbstractVector)::Int

Total electron number of state.

N_up(s::Fockstate{NSites})

Number of up electrons, $N_\uparrow$ in state s.

S(s::AbstractVector)::Int

Total spin of state. #TODO: only implemented for flavor=2

_H_CDag_C(istate, jstate, tMatrix)

Returns the hopping contribution for states $\sum_{i,j} \langle i | T | j \rangle$, with T being the hopping matrix tmatrix and the states i and j given by istate and jstate.

_H_nn(istate, jstate, tMatrix)

Returns the hopping contribution for states $\sum_{i,j} \langle i | T | j \rangle$, with T being the hopping matrix tmatrix and the states i and j given by istate and jstate.

_block_slice(bi::Blockinfo)

Slice of continuous vector for given block bi.

_find_cdag_overlap_blocks(blocklist::Vector{Blockinfo}; S_inc=-1, N_inc = 1)

Find block index with N_el increased by 1 and S either in increased or decreased (depending on S_inc), e.g. block index of state $\langle j |$ with non-zero overlap of $c^\dagger_\uparrow |i \rangle$.

Returns: Vector{Int}, equal length to blocklist. Each entry with index i contains the index j of the block with one more electron and spin (i.e. the block for which $\langle j| c^\dagger | i \rangle$ does NOT vanish). This is stored here for performance reasons, since these overlaps are used often in the Lehman representation. TODO: do not hardcode spin up GF!!! (this forces S+1 instead of S+-1 for now)

_generate_blocks!(states::Vector{Fockstate})

Sort state list and generate list of blocks.

_overlap_cdagger_ev!(tmp::Vector{Float64}, vec1::Vector{Float64}, vec2::Vector{Float64}, perm::Vector{Int})::Float64

Computes $\langle \text{EV}_j | c^\dagger_k | \text{EV}_i \rangle$ where $\c^\dagger$ has been computed in the Fockbasis and is given as a permutation of indices for $\langle \text{EV}_j |$.

This is an internal function used to compute the overlaps.

overlaplist(basis::Basis[, i::Int, j::Int], op::Function)

Calculates the overlap between block i and j under op. Builds full overlap indices for all block if i and j are not provided.

Example

julia> basis = jED.Basis(3)
julia> op(b) = jED.create(b, 4)  # creation operator for spin down at site 1 of 3
julia> jED._overlap_list(basis, 2, 4, op)
[0, 1, 2]

ann(state::Fockstate{Length}, i::Int)::Union{Nothing,Fockstate} where Length

Returns either a new Fockstate with an electron i annihilated (see layout of Fockstate, how up and down is encoded), or nothing, if i is not occupied. See also create. See also ann_op for the Operator version.

anne_op(b::Basis, i::Int)

Operator for annihilation of state i (for basis length N, i > N annihilates spin down, otherwise spin up).

calc_E(eigenspace, β)

Calculates the total energy from eigenspace and the inverse temperature β.

calcGF1(basis::Basis, es::Eigenspace, νnGrid::AbstractVector{ComplexF64}, β::Float64; ϵ_cut::Float64=1e-16, overlap=nothing)

Computes $|\langle i | c^\dagger | j \rangle|^2 \frac{e^{-\beta E_i} + e^{-\beta E_j}}{Z (E_j - E_i + freq)}$.

Arguments

• basis : Basis, obtained with Basis.
• es : Eigenspace, obtained with Eigenspace
• νnGrid : AbstractVector{ComplexF64}, list of Matsubara Frequencies
• β : Float64, inverse temperature.
• ϵ_cut : Float64, cutoff for $e^{-\beta E_n}$ terms in Lehrmann representation (all contributions below this threshold are disregarded)
• overlap : Overlap, precalculated overlap between blocks of basis. Obtained with Overlap

calc_Hamiltonian(model::Model, basis::Basis)

Calculates the Hamiltonian for a given

• model, see for example AIM)) in a
• basis, see Basis

calc_Z(eigenspace, β)

Calculates the partition function from eigenspace and the inverse temperature β.

create(state::Fockstate{Length}, i::Int)::Union{Nothing,Fockstate} where Length

Returns either a new Fockstate with an electron i created (see layout of Fockstate, how up and down is encoded), or nothing, if i is already occupied. See also ann. See also create_op for the Operator version.

create_op(b::Basis, i::Int)

Operator for creation of state i (for basis length N, i > N creates spin down, otherwise spin up).

operator_ni(state::Fockstate, i::Int)

Calculate n_i |ket⟩, i.e. i is the index for the number operator

Returns: (eigenval,newState)::Tuple{Int, Fockstate} with eigenval 1 (state i occupied) or 0 (state i unoccupied) and newState. In case of this operator, newState === state. The density overlap of 2 states can be calculated efficiently using overlap_ni_nj.

Example

julia> jED.operator_ni(jED.SVector{8}(Bool[1,1,1,1,0,1,0,1]), 2);
(1, Bool[1, 1, 1, 1, 0, 1, 0, 1])

overlap(bra::Fockstate, ket::Fockstate)

Calculates ⟨bra|ket⟩. This is useful when operators have been applied to one state and no efficient direct implementation for an overlap involving this operator is available.

Returns: True/False

overlap_2(bra::Vector, ket::Vector)

Computes |⟨bra|ket⟩|² for Float64 or ComplexF64 type vectors.

overlap_cdagger_c(bra::Fockstate, i::Int, ket::Fockstate, j::Int)::Int

Calculates $\langle$ bra $| c^\dagger_i c_j |$ ket $\rangle$, i.e. i is the index for the creation operator and j the index for the annihilation operator. Internally, we check, that both states have exactly two or no occupation difference. In both cases the product of C_sign and CDag_sign is returned, otherwise 0. Note, that this allows spin flips!

Returns: -1/0/1

Example

julia> t1 = jED.SVector{8}(Bool[1,1,1,1,0,1,0,1]);
julia> t2 = jED.SVector{8}(Bool[1,1,0,1,1,1,0,1]);
julia> jED.overlap_cdagger_c(t1,t2,3,3)
0
julia> jED.overlap_cdagger_c(t1,t1,2,2)
1

overlap_ni_nj(bra::Fockstate, ket::Fockstate, i::Int, j::Int)

Calculate ⟨bra| n^†i nj |ket⟩

Returns: True/False (converts to 1/0)

Δ_AIM(νnGrid::FermionicMatsubaraGrid, p::AIMParams)
Δ_AIM(νnGrid::FermionicMatsubaraGrid, p::Vector{Float64})

Computes hybridization function $\sum_p \frac{V_p^2}{i\nu_n - \epsilon_p}$ from Anderson Impurity Model Parameters with $p$ bath sites.

Δ_from_GWeiss(GWeiss::MatsubaraF, μ::Float64, νnGrid::FermionicMatsubaraGrid)

Computes hybridization function from Weiss Green's function via $\Delta^{\nu} = i\nu_n + \mu - \left(\mathcal{G}^{\nu}_0\right^{-1}$.

Σ_from_GImp(GWeiss::OffsetVector{ComplexF64, Vector{ComplexF64}}, GImp::OffsetVector{ComplexF64, Vector{ComplexF64}})

Computes self-energy from impurity Green's function (as obtained from a given impurity solver) and the Weiss Greens Function.