Hilbert spaces, states and linear operators

This module defines types of Hilbert spaces, state vectors and linear operators acting in these spaces. It also implements the space autopartition algorithm described in [Computer Physics Communications 200, March 2016, 274-284 (section 4.2)] (https://doi.org/10.1016/j.cpc.2015.10.023).

Mapping from a compound operator index IndicesType to a linear integer index. It provides a dictionary-like iteration interface, supports indexed read-only access, functions keys(), values() and pairs(), as well as the operator in.

Fields

• map_index_n::DataStructures.SortedDict{Vector{Union{Int64, String}}, Int64}: The IndicesType -> linear index map

insert!(
    soi::KeldyshED.Hilbert.SetOfIndices,
    indices::Vector{Union{Int64, String}}
) -> DataStructures.SortedDict{Vector{Union{Int64, String}}, Int64, Base.Order.ForwardOrdering}

Insert a new compound operator index indices into soi.

insert!(
    soi::KeldyshED.Hilbert.SetOfIndices,
    indices...
) -> DataStructures.SortedDict{Vector{Union{Int64, String}}, Int64, Base.Order.ForwardOrdering}

Insert a new compound operator index built out of arguments indices... into soi.

reversemap(
    soi::KeldyshED.Hilbert.SetOfIndices
) -> Vector{Vector{Union{Int64, String}}}

Build and return the reverse map Int -> IndicesType out of a SetOfIndices object.

matching_indices(
    soi_from::KeldyshED.Hilbert.SetOfIndices,
    soi_to::KeldyshED.Hilbert.SetOfIndices
) -> Vector{Int64}

For each compound index in the set soi_from, find the linear index of that compound index within soi_to and collect the found linear indices.

primitive type UInt64 <: Unsigned 64

Fermionic Fock state encoded as a sequence of zeros (unoccupied single-particle states) and ones (occupied states) in the binary representation of an integer.

translate(
    fs::UInt64,
    bit_map::Vector{Int64};
    reverse
) -> UInt64

Reshuffle single-particle states (bits) of a Fock state fs according to a given map bit_map. bit_map is understood as a reverse map if reverse = true.

Abstract supertype for Hilbert space types.

A Hilbert space spanned by all fermionic Fock states generated by a given set of creation/annihilation operators. This type supports vector-like iteration over Fock states, read-only indexed access and the operator in.

Fields

• soi::KeldyshED.Hilbert.SetOfIndices: Set of compound operator indices used to generate this space

• dim::UInt64: Dimension of the space

FullHilbertSpace(
    soi::KeldyshED.Hilbert.SetOfIndices
) -> KeldyshED.Hilbert.FullHilbertSpace

Make a Hilbert space generated by creation/annihilation operators carrying compound indices from a given set soi.

getindex(
    fhs::KeldyshED.Hilbert.FullHilbertSpace,
    soi_indices::Set{Vector{Union{Int64, String}}}
) -> UInt64

Return the Fock state $|\psi\rangle \propto c^\dagger_i c^\dagger_j c^\dagger_k \ldots |0\rangle$, where the compound indices $i,j,k,\ldots$ form a given set soi_indices and the creation operators act in a Hilbert space fhs.

⊗(
    H_A::KeldyshED.Hilbert.FullHilbertSpace,
    H_B::KeldyshED.Hilbert.FullHilbertSpace
) -> KeldyshED.Hilbert.FullHilbertSpace

Construct a direct product of full Hilbert spaces $H_A \otimes H_B$ under the assumption that the sets of indices generating $H_A$ and $H_B$ are disjoint.

/(
    H_AB::KeldyshED.Hilbert.FullHilbertSpace,
    H_A::KeldyshED.Hilbert.FullHilbertSpace
) -> KeldyshED.Hilbert.FullHilbertSpace

Construct a quotient space $H_{AB} / H_A$ under the assumption that the sets of indices generating $H_{AB}$ and $H_A$ satisfy $soi(H_A) \subseteq soi(H_{AB})$.

product_basis_map(
    H_A::KeldyshED.Hilbert.FullHilbertSpace,
    H_B::KeldyshED.Hilbert.FullHilbertSpace,
    H_AB::KeldyshED.Hilbert.FullHilbertSpace
) -> Matrix{Int64}

Given three Hilbert spaces $H_A$, $H_B$ and $H_{AB}$, construct all product Fock states $|\psi\rangle_{AB} = |\psi\rangle_A ⊗ |\psi\rangle_B$, where $|\psi\rangle_A ∈ H_A, |\psi\rangle_B ∈ H_B, |\psi\rangle_{AB} ∈ H_{AB}$. Return a mapping (an integer-valued matrix) $i, j \mapsto k$, where $i$, $j$ and $k$ are linear indices of $|\psi\rangle_A$, $|\psi\rangle_B$ and $|\psi\rangle_{AB}$ within their respective spaces.

Sets of indices $soi(H_A)$, $soi(H_B)$ must be disjoint and satisfy $soi(H_A) \subseteq soi(H_{AB})$, $soi(H_B) \subseteq soi(H_{AB})$.

factorized_basis_map(
    H_AB::KeldyshED.Hilbert.FullHilbertSpace,
    H_A::KeldyshED.Hilbert.FullHilbertSpace
) -> Vector{Tuple{Int64, Int64}}

Given a Hilbert space $H_{AB}$ and its divisor $H_A$, factorize all Fock states $|\psi\rangle_{AB} \in H_{AB}$ into a direct product $|\psi\rangle_A \otimes |\psi\rangle_B$, where $|\psi\rangle_A \in H_A$, $|\psi\rangle_B \in H_{AB} / H_A$. Return a mapping (a vector of integer pairs) $k \mapsto (i, j)$, where $i$, $j$ and $k$ are linear indices of $|\psi\rangle_A$, $|\psi\rangle_B$ and $|\psi\rangle_{AB}$ within their respective spaces.

Sets of indices $soi(H_{AB})$, $soi(H_A)$ must satisfy $soi(H_A) \subseteq soi(H_{AB})$.

Subspace of a Hilbert space, as a list of basis Fock states.

This type supports vector-like iteration over Fock states, read-only indexed access and the operator in.

Fields

• fock_states::Vector{UInt64}: List of all Fock states spanning the space

• fock_to_index::Dict{UInt64, Int64}: Reverse map to quickly find the index of a basis Fock state

insert!(
    hss::KeldyshED.Hilbert.HilbertSubspace,
    fs::UInt64
) -> Int64

Insert a basis Fock state fs into a subspace hss.

getstateindex(
    fhs::KeldyshED.Hilbert.FullHilbertSpace,
    fs::UInt64
) -> Int64

Find the index of a given Fock state fs within a full Hilbert space fhs.

getstateindex(
    hss::KeldyshED.Hilbert.HilbertSubspace,
    fs::UInt64
) -> Int64

Find the index of a given Fock state fs within a subspace hss.

Abstract supertype of quantum state types

Quantum state in a Hilbert space/subspace implemented as a vector of amplitudes.

The amplitudes can be accessed using the indexing interface and iterated over. States support addition/subtraction and multiplication/division by a constant scalar.

Fields

• hs::KeldyshED.Hilbert.HilbertSpace: Hilbert space this state belongs to

• amplitudes::Vector: Amplitudes of basis states contributing to this state

StateVector{HSType, S}(hs::HSType) where {HSType, S}

Create a vector-based state in a Hilbert space/subspace hs with all amplitudes set to zero.

Quantum state in a Hilbert space/subspace implemented as a (sparse) dictionary of amplitudes.

The amplitudes can be accessed using the indexing interface and iterated over. States support addition/subtraction and multiplication/division by a constant scalar.

Fields

• hs::KeldyshED.Hilbert.HilbertSpace: Hilbert space this state belongs to

• amplitudes::Dict{Int64}: Non-vanishing amplitudes of basis states contributing to this state. Each element of this dictionary is a pair (index of the basis state within hs, amplitude).

StateDict{HSType, S}(hs::HSType) where {HSType, S}

Create a dictionary-based state in a Hilbert space/subspace hs with zero non-vanishing amplitudes.

similar(
    sv::KeldyshED.Hilbert.StateVector{HSType, S}
) -> KeldyshED.Hilbert.StateVector

Create a vector-based state similar to sv with all amplitudes set to zero.

similar(
    sd::KeldyshED.Hilbert.StateDict{HSType, S}
) -> KeldyshED.Hilbert.StateDict

Create a dictionary-based state similar to sd with zero non-vanishing amplitudes.

dot(
    sv1::KeldyshED.Hilbert.StateVector,
    sv2::KeldyshED.Hilbert.StateVector
) -> Any

Compute the scalar product of two vector-based states sv1 and sv2.

dot(
    sd1::KeldyshED.Hilbert.StateDict{HSType, S},
    sd2::KeldyshED.Hilbert.StateDict{HSType, S}
) -> Any

Compute the scalar product of two dictionary-based states sd1 and sd2.

project(
    sv::KeldyshED.Hilbert.StateVector{HSType, S},
    target_space
) -> KeldyshED.Hilbert.StateVector

Project a vector-based state sv from one Hilbert space onto another Hilbert space/subspace target_space.

project(
    sd::KeldyshED.Hilbert.StateDict{HSType, S},
    target_space
) -> KeldyshED.Hilbert.StateVector

Project a dictionary-based state sd from one Hilbert space onto another Hilbert space/subspace target_space.

show(io::IO, sv::KeldyshED.Hilbert.StateVector{HSType, S})

Print a vector-based state.

show(io::IO, sd::KeldyshED.Hilbert.StateDict{HSType, S})

Print a dictionary-based state.

Quantum-mechanical operator acting on states in a Hilbert space.

Operator{HSType, S}(op_expr::OperatorExpr{S}, soi::SetOfIndices) where {HSType, S}

Make a linear operator acting on a Hilbert space/subspace out of a [polynomial expression] (@ref Main.KeldyshED.Operators.OperatorExpr) op_expr. The set soi is used to establish a correspondence between compound indices of creation/annihilation operators met in op_expr and the single-particle states – bits of FockState.

*(
    op::KeldyshED.Hilbert.Operator,
    st::KeldyshED.Hilbert.State
) -> Union{KeldyshED.Hilbert.StateDict, KeldyshED.Hilbert.StateVector}

Act with an operator op on a state st and return the resulting state.

Partition of a Hilbert space into a set of disjoint subspaces invariant under action of a given Hermitian operator (Hamiltonian).

A detailed description of the algorithm can be found in Section 4.2 of [Computer Physics Communications 200, March 2016, 274-284] (https://doi.org/10.1016/j.cpc.2015.10.023).

SpacePartition supports iteration over pairs (index of a basis state of the Hilbert space, index of the invariant subspace that state belongs to).

Fields

• hs::KeldyshED.Hilbert.HilbertSpace: Full Hilbert space subject to partitioning

• subspaces::DataStructures.IntDisjointSets: Disjoint set of subspaces

• root_to_index::Dict{Int64, Int64}: Map root index to subspace index

• matrix_elements::SparseArrays.SparseMatrixCSC{ScalarType, Int64} where ScalarType<:Number: Matrix elements of the Hamiltonian

SpacePartition{HSType, S}(hs::HSType,
                          H::OperatorType,
                          store_matrix_elements::Bool = true) where {
    HSType <: HilbertSpace, S <: Number, OperatorType <: Operator}

Create a SpacePartition structure by performing Phase I of the automatic partitioning algorithm, i.e. discovering the invariant subspaces of the Hermitian operator H acting on the Hilbert space hs.

If store_matrix_elements = true, the field matrix_elements of the created structure contains a sparse-matrix representation of H.

merge_subspaces!(sp, op)
merge_subspaces!(sp, op, store_matrix_elements)

Merge some of the invariant subspaces in sp to ensure that the resulting subspaces are also invariant w.r.t. a given operator op.

If store_matrix_elements = true, a sparse-matrix representation of op is returned.

merge_subspaces!(sp, Cd, C)
merge_subspaces!(sp, Cd, C, store_matrix_elements)

Perform Phase II of the automatic partition algorithm.

Merge some of the invariant subspaces in sp to ensure that a given operator Cd and its Hermitian conjugate C generate only one-to-one connections between the subspaces.

If store_matrix_elements = true, a tuple of two sparse matrices representing Cd and C is returned.

numsubspaces(sp::KeldyshED.Hilbert.SpacePartition) -> Int64

Return the number of subspaces in a space partition sp.

getindex(
    sp::KeldyshED.Hilbert.SpacePartition,
    index
) -> Int64

Find the invariant subspace within a partition sp, the state with a given index belongs to.