API reference

RAS_DMFT

RAS_DMFT.AbstractPoles — Type

AbstractPoles{A, B}

Supertype which represents a function on the real axis as a collection of poles.

The locations are described by A, the weights/amplitudes by B.

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RAS_DMFT.AbstractPolesContinuedFraction — Type

AbstractPolesContinuedFraction{A, B} <: AbstractPoles{A, B}

Supertype which represents a (block) function on the real axis as a continued fraction.

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RAS_DMFT.AbstractPolesSum — Type

AbstractPolesSum{A, B} <: AbstractPoles{A, B}

Supertype which represents a (block) function on the real axis as a sum of poles.

The canonical representation requires the locations to be strictly increasing (sorted with no duplicates).

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RAS_DMFT.NaturalImpurityOrbital — Type

NaturalImpurityOrbital{T<:Real}

Natural impurity orbital representation of a single-particle Hamiltonian.

The ordering is [i, b, v_1, ..., v_(n_v), c_1, ..., c_(n_c)], where i is the impurity, b its mirror, v_1...v_(n_v) the fully occupied valence chain, and c_1...c_(n_c) the empty conduction chain.

Fields

H_ib: 2×2 impurity-mirror Hamiltonian block
i_v: Impurity ↔ valence coupling (i↔v₁)
i_c: Impurity ↔ conduction coupling (i↔c₁)
b_v: Mirror ↔ valence coupling (b↔v₁)
b_c: Mirror ↔ conduction coupling (b↔c₁)
e_v: Valence on-site energies
t_v: Valence hoppings (v₁↔v₂↔...)
e_c: Conduction on-site energies
t_c: Conduction hoppings (c₁↔c₂↔...)

See also

natural_impurity_orbital to construct.

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RAS_DMFT.PolesContinuedFraction — Type

PolesContinuedFraction{A <: Real, B <: Real} <: AbstractPolesContinuedFraction{A, B}

Representation of poles on the real axis as a continued fraction with locations $a_i$ of type A and amplitudes $b_i$ of type B.

The scale factor $s$ of type B rescales the whole object.

\[P(z) = \frac{s^2}{z - a_1 - \frac{b_1^2}{z - a_2 - …}}\]

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RAS_DMFT.PolesContinuedFraction — Method

PolesContinuedFraction(
    locs::AbstractVector{A}, amps::AbstractVector{B}, scl=one(B)
) where {A,B}

Create a new instance of PolesContinuedFraction by supplying locations locs, amplitudes amps, and scale scl.

By default the scale is set to $1$.

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RAS_DMFT.PolesContinuedFractionBlock — Type

PolesContinuedFractionBlock{A <: Number, B <: Number} <:
    AbstractPolesContinuedFraction{A, B}

Representation of poles on the real axis with as a continued fraction with locations $A_i$ of type A and amplitudes $B_i$ of type B.

All matrices must be Hermitian. The scale factor $S$ rescales the whole object.

\[P(z) = S \frac{1}{z - A_1 - B_1 \frac{1}{z - A_2 - …} B_1} S\]

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RAS_DMFT.PolesContinuedFractionBlock — Method

PolesContinuedFractionBlock(
    locs::AbstractVector{<:AbstractMatrix{<:A}},
    amps::AbstractVector{<:AbstractMatrix{<:B}},
    [scl::AbstractMatrix{<:B},]
) where {A,B}

Create a new instance of PolesContinuedFractionBlock by supplying locations locs, amplitudes amps, and scale scl.

By default the scale is set to the identity matrix $I$.

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RAS_DMFT.PolesSum — Type

PolesSum{A <: Real, B <: Number} <: AbstractPolesSum{A, B}

Representation of poles on the real axis with locations $a_i$ of type A and weights $w_i$ of type B

\[P(z) = ∑_i \frac{w_i}{z-a_i}.\]

For a block variant see PolesSumBlock.

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RAS_DMFT.PolesSum — Method

PolesSum(P::PolesSumBlock, i::Integer, j::Integer)

Take the $P_{i,j}$ element.

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RAS_DMFT.PolesSum — Method

PolesSum(locs::AbstractVector{A}, wgts::AbstractVector{B}) where {A,B}

Create a new instance of PolesSum by supplying the locations locs and weights wgts.

julia> locs = 0:5;

julia> wgts = 5:10;

julia> P = PolesSum(locs, wgts)
PolesSum{Int64, Int64} with 6 poles

julia> locations(P) == locs
true

julia> weights(P) == wgts
true

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RAS_DMFT.PolesSumBlock — Type

PolesSumBlock{A <: Real, B <: Number} <: AbstractPolesSum{A, B}

Representation of block of poles on the real axis with locations $a_i$ of type A and weights $W_i$ of type Matrix{B}.

\[P(z) = ∑_i \frac{W_i}{z-a_i}.\]

For $W_i$ to be physical, it needs to be Hermitian ($W_i^† = W_i$) and positive semidefinite $W_i ⪰ 0$. The latter is not enforced at construction to reduce computational cost.

For a scalar variant see PolesSum.

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RAS_DMFT.PolesSumBlock — Method

PolesSumBlock(locs::AbstractVector{A}, amps::AbstractMatrix{B}) where {A, B}

Create a new instance of PolesSumBlock by supplying locations locs and amplitudes amps.

The $i$-th column of amps is interpreted as the vector $\vec{b}_i$ and the weight as the outer product $W_i = \vec{b}_i \vec{b}^†_i$.

Weights of equal locations are summed. To merge nearby locations, call merge_degenerate_poles! afterwards.

julia> locs = 0:1;

julia> amps = [1+2im 3im; 4 5+6im];

julia> P = PolesSumBlock(locs, amps)
PolesSumBlock{Int64, Complex{Int64}} with 2 poles of size 2×2

julia> locations(P) == locs
true

julia> weights(P) == [[5 4+8im; 4-8im 16], [9 18+15im; 18-15im 61]]
true

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RAS_DMFT.PolesSumBlock — Method

PolesSumBlock(locs::AbstractVector{A}, wgts::Vector{<:AbstractMatrix{B}}) where {A, B}

Create a new instance of PolesSumBlock by supplying locations locs and weights wgts.

julia> locs = 0:2;

julia> wgts = [[1 0; 0 1], [2 1; 1 2], [2 -1; -1 2]];

julia> P = PolesSumBlock(locs, wgts)
PolesSumBlock{Int64, Int64} with 3 poles of size 2×2

julia> locations(P) == locs
true

julia> weights(P) == wgts
true

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RAS_DMFT.RASWavefunction_singlet — Method

RASWavefunction_singlet(
    D::Type, L_v::Integer, L_c::Integer, V_v::Integer, V_c::Integer, excitation::Integer
)

Initialize a RASWavefunction with singlet in impurity and mirror site and all valence sites filled.

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RAS_DMFT.Wavefunction_singlet — Method

Wavefunction_singlet(
    D::Type, L_v::Integer, L_c::Integer, V_v::Integer, V_c::Integer
)

Initialize a Wavefunction with singlet in impurity and mirror site and all valence sites filled.

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RAS_DMFT._orthogonalize_states! — Method

_orthogonalize_states!(
    M1::AbstractMatrix, Q_new::AbstractMatrix, Q_old::AbstractMatrix
)

Orthogonalize Q_new against Q_old.

Overwrites M1.

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RAS_DMFT._orthonormalize_lowdin — Method

_orthonormalize_lowdin(M::AbstractMatrix)

Löwdin orthonormalization for given states M by diagonalizing their overlap matrix.

Calculate overlap matrix $S = M^† M$ and diagonalize

\[\begin{aligned} S &= U Λ U^† \\ S^{1/2} &= U Λ^{1/2} U^† \\ S^{-1/2} &= U Λ^{-1/2} U^†. \end{aligned}\]

Return $Q = M S^{-1/2}$ and $B = S^{1/2}$, such that $M = Q B$.

This serves states without a singular value decomposition, such as RASWavefunction. Forming $S$ squares the condition number $κ$ of M, so the columns of Q are orthonormal only up to about $ϵ κ^2$.

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RAS_DMFT.add_pole_at_zero! — Method

add_pole_at_zero!(P::PolesSum)

Add a pole at location zero with weight 0 inside P.

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RAS_DMFT.amplitude — Function

amplitude(P::AbstractPoles, i::Integer)

Return the amplitude B_i of P at index i, satisfying $B_i B_i^† = W_i$ for the weight W_i.

RAS_DMFT.Combinatorics

RAS_DMFT.Combinatorics.ndet — Method

ndet(nbit::Int, nup::Int, ndown::Int, excitation::Int, nfilled::Int, nempty::Int)

Number of Slater determinants for given excitation.

Parameters:

nbit::Int: number of bath sites in $|ψ_\mathrm{I}⟩$
nup::Int: number of up-electrons in $|ψ_\mathrm{I}⟩$
ndown::Int: number of down-electrons in $|ψ_\mathrm{I}⟩$
excitation::Int: number of excitations in $|ψ_\mathrm{II}⟩$
nfilled::Int: number of valence bath sites in $|ψ_\mathrm{II}⟩$
nempty::Int: number of conduction bath sites in $|ψ_\mathrm{II}⟩$

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RAS_DMFT.Combinatorics.ndet0 — Method

ndet0(nbit::Int, nup::Int, ndown::Int, ::Int, ::Int)

Number of Slater determinants for zero excitation.

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RAS_DMFT.Combinatorics.ndet0_bit — Method

ndet0_bit(nbit::Int, nup::Int, ndown::Int)

Number of Slater determinants for zero excitation in the bit component.

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RAS_DMFT.Combinatorics.ndet1 — Method

ndet1(nbit::Int, nup::Int, ndown::Int, nfilled::Int, nempty::Int)

Number of Slater determinants for single excitation.

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RAS_DMFT.Combinatorics.ndet1_bit — Method

ndet1_bit(nbit::Int, nup::Int, ndown::Int)

Number of Slater determinants for single excitation in the bit component.

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RAS_DMFT.Combinatorics.ndet2 — Method

ndet2(nbit::Int, nup::Int, ndown::Int, nfilled::Int, nempty::Int)

Number of Slater determinants for double excitation.

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RAS_DMFT.Combinatorics.ndet2_bit — Method

ndet2_bit(nbit::Int, nup::Int, ndown::Int)

Number of Slater determinants for double excitation in the bit component.

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RAS_DMFT.Combinatorics.ndet_bit — Method

ndet_bit(nbit::Int, nup::Int, ndown::Int, excitation::Int)

Number of Slater determinants for given excitation in the bit component.

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