hc

Adding this is equivalent to adding the hermitian conjugate.

FockHilbertSpace

A type representing a Fock Hilbert space with a given set of modes and Fock states.

FockNumber

A type representing a Fock state as the bitstring of an integer.

struct FockSymmetry{IF,FI,QN,QNfunc} <: AbstractSymmetry

FockSymmetry represents a symmetry that is diagonal in fock space, i.e. particle number conservation, parity, spin conservation.

Fields

• basisstates::IF: A vector of Fock numbers, which are integers representing the occupation of each mode.
• state_indexdict::FI: A dictionary mapping Fock states to indices.
• qntofockstates::Dictionary{QN,Vector{Int}}: A dictionary mapping quantum numbers to Fock states.
• conserved_quantity::QNfunc: A function that computes the conserved quantity from a fock number.

FockSymmetryFunction{F} <: AbstractSymmetry

FockSymmetryFunction represents a symmetry defined by a function that maps Fock states to quantum numbers. The function should return 'missing' for states which should be discarded from the hilbert space.

HC

Represents the Hermitian conjugate.

JordanWignerOrdering

A type representing the ordering of fermionic modes.

NoSymmetry

A symmetry type indicating no symmetry constraints.

NumberConservation

A symmetry type representing conservation of total fermion number.

ParityConservation

A symmetry type representing conservation of fermion parity.

SimpleFockHilbertSpace

A type representing a simple Fock Hilbert space with all fock states included.

SymmetricFockHilbertSpace

A type representing a Fock Hilbert space with fockstates organized by their quantum number.

basisstates(H)

Return an iterator over all basis states for the given Hilbert space H.

bdg_hilbert_space(labels)

This hilbert space uses Nambu states to describe non-interacting systems with superconductive pairing. Matrix representations of quadratic operators take the form
[H Δ; -Δ* -H*]
where H is hermitian and Δ is antisymmetric.

embed(m, Hsub => H; complement=complementary_subsystem(H, Hsub), kwargs...)

Compute the embedding of a matrix m in the basis Hsub into the basis H. Fermionic phase factors are included if the two spaces are fermionic Hilbert spaces.

embed(Hsub => H; kwargs...)

Compute the embedding map from Hsub into H. Fermionic phase factors are included if the two spaces are fermionic Hilbert spaces.

embedding_unitary(partition, basisstates, jw)

Compute the unitary matrix that maps between the tensor embedding and the fermionic embedding in the physical subspace. 
# Arguments
- `partition`: A partition of the labels in `jw` into disjoint sets.
- `basisstates`: The basis states in the basis
- `jw`: The Jordan-Wigner ordering.

eval_in_basis(a, f)

Evaluate an expression with fermions in a basis f.

Examples

@fermions a
f = fermions(hilbert_space(1:2))
FermionicHilbertSpaces.eval_in_basis(a[1]'*a[2] + hc, f)

extend(m, H => Hbar, Hout = tensor_product((H, Hbar)); kwargs...)

Extend an operator or state m from Hilbert space H into a disjoint space Hbar.

extend(H => Hbar, Hout = tensor_product((H, Hbar)); kwargs...)

Compute the extend map from H into a disjoint space Hbar.

fermion_sparse_matrix(fermion_number, H::AbstractFockHilbertSpace)

Constructs a sparse matrix of size representing a fermionic annihilation operator at bit position fermion_number on the Hilbert space H.

fermions(H)

Return a dictionary of fermionic annihilation operators for the Hilbert space H.

fixed_particle_number_fockstates(M, n)

Generate a list of Fock states with n occupied fermions in a system with M different fermions.

focksymmetry(basisstates, qn)

Constructs a FockSymmetry object that represents the symmetry of a many-body system.

Arguments

• basisstates: The basisstates to iterate over
• qn: A function that takes an integer representing a fock state and returns corresponding quantum number.

generalized_kron(ms, Hs, H::AbstractHilbertSpace=tensor_product(Hs))

Compute the tensor product of matrices or vectors in ms with respect to the spaces Hs, respectively. Return a matrix in the space H, which defaults to the tensor_product product of Hs.

hilbert_space(labels[, symmetry, basisstates])

Construct a Hilbert space from a set of labels, with optional symmetry and Fock number specification.

jwstring(site, focknbr)

Parity of the number of fermions to the right of site.

majorana_hilbert_space(labels, qn)

Represents a hilbert space for majoranas. labels must be an even number of unique labels.

majoranas(H)

Return a dictionary of Majorana operators for the Hilbert space H.

matrix_representation(op, H::AbstractFockHilbertSpace)

Return the matrix representation of the symbolic operator op on the hilbert space H.

number_conservation(sectors=missing, weight_function=label -> true)

Constructs a UninstantiatedNumberConservations symmetry object that represents conservation of fermion number. 'sectors' can be an integer or a collection of integers specifying the allowed fermion numbers. 'weight_function' is a function that takes a label and returns an integer weight (which can be negative) indicating how that label contributes to the fermion number.

numberoperator(H)

Return the number operator for the Hilbert space H.

parityoperator(H)

Return the fermionic parity operator for the Hilbert space H.

partial_trace!(mout, m, H::AbstractHilbertSpace, Hsub::AbstractHilbertSpace, phase_factors::Bool, complement, extend_state=StateExtender((Hsub, complement), H))

Compute the partial trace of m from H to Hsub.

partial_trace(m, H => Hsub; phase_factors=use_phase_factors(H) && use_phase_factors(Hsub), complement=complementary_subsystem(H, Hsub))

Compute the partial trace of m from H to Hsub. Fermionic phase factors are included if both H and Hsub are Fermionic, unless specified otherwise in kwargs.

partial_trace(H => Hsub; kwargs...)

Compute the partial trace map from H to Hsub, represented by a sparse matrix of dimension dim(Hsub)^2 x dim(H)^2 that can be multiplied with a vectorized density matrix.

removefermion(digitposition, f::FockNumber)

Return (newfocknbr, fermionstatistics) where newfocknbr is the state obtained by removing a fermion at digitposition from f and fermionstatistics is the phase from the Jordan-Wigner string, or 0 if the operation is not allowed.

single_particle_hilbert_space(labels)

A hilbert space suitable for non-interacting systems with fermion number conservation. Matrix representations of symbolic operators give the single particle hamiltonian, without any contribution from the identity matrix.

subregion(modes, H::AbstractHilbertSpace)

Return a subregion of the Hilbert space H that is spanned by the modes in modes. Only substates in H are included.

tensor_product(Hs)

Return the tensor product space of hilbert spaces Hs.

tensor_product(ms, Hs, H::AbstractHilbertSpace; kwargs...)

Compute the ordered product of the fermionic embeddings of the matrices ms in the spaces Hs into the space H. kwargs can be passed a bool phase_factors and a hilbert space complement.

togglefermions(digitpositions, daggers, f::FockNumber)

Return (newfocknbr, fermionstatistics) where newfocknbr is the state obtained by toggling fermions at digitpositions with daggers in the Fock state f, and fermionstatistics is the phase from the Jordan-Wigner string. If the operation puts two fermions one the same site, the resulting state is undefined.

@fermions a b ...

Create one or more fermion species with the given names. Indexing into fermions species gives a concrete fermion. Fermions in one @fermions block anticommute with each other, and commute with fermions in other @fermions blocks.

Examples:

• @fermions a b creates two species of fermions that anticommute:
  • a[1]' * a[1] + a[1] * a[1]' == 1
  • a[1]' * b[1] + b[1] * a[1]' == 0
• @fermions a; @fermions b creates two species of fermions that commute with each other:
  • a[1]' * a[1] + a[1] * a[1]' == 1
  • a[1] * b[1] - b[1] * a[1] == 0

See also @majoranas, FermionicHilbertSpaces.eval_in_basis.

@majoranas a b ...

Create one or more Majorana species with the given names. Indexing into Majorana species gives a concrete Majorana. Majoranas in one @majoranas block anticommute with each other, and commute with Majoranas in other @majoranas blocks.

Examples:

• @majoranas a b creates two species of Majoranas that anticommute:
  • a[1] * a[1] + a[1] * a[1] == 1
  • a[1] * b[1] + b[1] * a[1] == 0
• @majoranas a; @majoranas b creates two species of Majoranas that commute with each other:
  • a[1] * a[1] + a[1] * a[1] == 1
  • a[1] * b[1] - b[1] * a[1] == 0

See also @fermions, FermionicHilbertSpaces.eval_in_basis.