FermionConservation

A symmetry type representing conservation of total fermion number.

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 consvervation.

Fields

• focknumbers::IF: A vector of Fock numbers, which are integers representing the occupation of each mode.
• focktoinddict::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.

HC

Represents the Hermitian conjugate.

IndexConservation

A symmetry type representing conservation of the numbers of modes which contains a specific index or set of indices.

JordanWignerOrdering

A type representing the ordering of fermionic modes.

NoSymmetry

A symmetry type indicating no symmetry constraints.

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.

embedding(m, H, Hnew)

Compute the fermionic embedding of a matrix m in the basis H into the basis Hnew.

embedding_unitary(partition, fockstates, 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.
- `fockstates`: The fock 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)

extension(m, H, Hbar[, phase_factors])

Extend an operator or state m from Hilbert space 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.

fermion_to_majorana(expr)

Convert symbolic fermions to symbolic majoranas.

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

Compute the fermionic 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.

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.

focknumbers(H)

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

focksymmetry(focknumbers, qn)

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

Arguments

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

hilbert_space(labels[, symmetry, focknumbers])

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_to_fermion(expr)

Convert symbolic majoranas to symbolic fermions.

majoranas(H)

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

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::AbstractMatrix, H::AbstractHilbertSpace, Hout::AbstractHilbertSpace, phase_factors)

Compute the fermionic partial trace of a matrix m in basis H, leaving only the subsystems specified by labels. The result is stored in mout, and Hout determines the ordering of the basis states.

partial_trace(m::AbstractMatrix,  bHfull::AbstractHilbertSpace, Hsub::AbstractHilbertSpace)

Compute the partial trace of a matrix m, leaving the subsystem defined by the basis bsub.

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.

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(ms, Hs, H::AbstractHilbertSpace, phase_factors=true)

Compute the ordered product of the fermionic embeddings of the matrices ms in the spaces Hs into the space H.

tensor_product(Hs)

Compute the tensorproduct product hilbert spaces Hs. The symmetry of the resulting basis is computed by promotesymmetry.

@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.