Fermion

 module U₁SpinlessFermion

Prepare some commonly used objects for U₁ spinless fermions.

Fields

 pspace::VectorSpace

Local d = 2 Hilbert space.

 Z::TensorMap

Rank-2 fermion parity operator Z = (-1)^n.

 n::TensorMap

Rank-2 particle number operator.

 FdagF::NTuple{2, TensorMap}

Two rank-3 operators of hopping c^dag c.

 FFdag::NTuple{2, TensorMap}

Two rank-3 operators of hopping c c^dag.

 ΔdagΔ::NTuple{4, TensorMap}

Four operators of pairing Δ^dag Δ, where Δᵢⱼ = cᵢcⱼ is the pairing operator. Rank = (3, 4, 4, 3). Sign convention: (i,j,k,l) gives cᵢ^dag cⱼ^dag cₖ cₗ = - cⱼ^dag cᵢ^dag cₖ cₗ = - Δᵢⱼ^dag Δₖₗ.

 const U1SpinlessFermion = U₁SpinlessFermion

 module U₁SU₂Fermion

Prepare some commonly used objects for U₁×SU₂ fermions.

Nothing is exported, please call U₁SU₂Fermion.xxx to use xxx.

Fields

 pspace::VectorSpace

Local d = 4 Hilbert space.

 Z::TensorMap

Rank-2 fermion parity operator Z = (-1)^n.

 n::TensorMap

Rank-2 particle number operator n = n↑ + n↓.

 nd::TensorMap

Rank-2 double occupancy operator nd = n↑n↓.

 SS::NTuple{2, TensorMap}

Two rank-3 operators of Heisenberg S⋅S interaction.

 SSS::NTuple{3, TensorMap}

Three operators of chiral operator imag(S⋅(S×S)). Rank = (3, 4, 3). SSS = imag(S⋅(S×S)) = -im * S⋅(S×S) –> S⋅(S×S) = im * SSS NOTICE: The chiral operator S⋅(S×S) is a pure imaginary operator under the current basis. Thus define SSS as the imaginary part of S⋅(S×S) to reduce the computational overhead.

 FdagF::NTuple{2, TensorMap}

Two rank-3 operators of hopping c↑^dag c↑ + c↓^dag c↓.

 FFdag::NTuple{2, TensorMap}

Two rank-3 operators of hopping c↑ c↑^dag + c↓ c↓^dag.

 ΔₛdagΔₛ::NTuple{4, TensorMap}

Four operators of singlet pairing correlation Δₛ^dagΔₛ, where Δₛ = (c↓c↑ - c↑c↓)/√2. Rank = (3, 4, 4, 3).

 ΔₜdagΔₜ::NTuple{4, TensorMap}

Four operators of triplet pairing correlation Δₜ^dag⋅Δₜ, where Δₜ is the triplet pairing operator that carries 2 charge and 1 spin quantum numbers. Rank = (3, 4, 4, 3).

 Δₛ::NTuple{2, TensorMap}
 Δₛdag::NTuple{2, TensorMap}

Singlet pairing operators Δₛ and Δₛ^dag. Rank = (4, 3). Note the first operator has nontrivial left bond index.

 CpCm::NTuple{2, TensorMap}

Two rank-3 operators of C+C- correlation where C+ = c↑^dag c↓^dag and C- = C+^dag = c↓c↑. Note both operators are bosonic.

 const U1SU2Fermion = U₁SU₂Fermion

 module ℤ₂SU₂Fermion

Prepare some commonly used objects for ℤ₂×SU₂ fermions. Basis convention in (0, 0) sector is {|0⟩, |↑↓⟩}.

Each operator has the same name and behavior as U₁SU₂Fermion, details please see U₁SU₂Fermion.

 const Z2SU2Fermion = ℤ₂SU₂Fermion

 module U₁U₁Fermion

Prepare the local space of d = 4 spin-1/2 fermions with U₁ charge and U₁ spin symmetry.

Fields

 pspace::VectorSpace

Local d = 4 Hilbert space.

 Z::TensorMap

Rank-2 fermion parity operator Z = (-1)^n.

 n₊::TensorMap
 n₋::TensorMap
 n::TensorMap

Rank-2 particle number operators. ₊ and ₋ denote spin up and down as ↑ and ↓ cannot be used in variable names.

 nd::TensorMap

Rank-2 double occupancy operator nd = n↑n↓.

 Sz::TensorMap

Rank-2 spin-z operator Sz = (n↑ - n↓)/2.

 S₊₋::NTuple{2, TensorMap}
 S₋₊::NTuple{2, TensorMap}

Two rank-3 operators of S₊₋ and S₋₊ interaction. Note Heisenberg S⋅S = SzSz + (S₊₋ + S₋₊)/2.

 FdagF₊::NTuple{2, TensorMap}
 FdagF₋::NTuple{2, TensorMap}

Two rank-3 operators of hopping c↑^dag c↑ and c↓^dag c↓.

 FFdag₊::NTuple{2, TensorMap}
 FFdag₋::NTuple{2, TensorMap}

Two rank-3 operators of hopping c↑ c↑^dag and c↓ c↓^dag.

 ΔdagΔ₊₊::NTuple{4, TensorMap}
 ΔdagΔ₋₋::NTuple{4, TensorMap}
 ΔdagΔ₊₋::NTuple{4, TensorMap}

Rank-4 operators of pairing correlation. Note ΔdagΔ₊₋ means c↑^dag c↓^dag c↓ c↑.

 EdagE₊::NTuple{4, TensorMap}
 EdagE₋::NTuple{4, TensorMap}

Rank-4 operators of triplet exciton correlation. Note EdagE₊ means c↑^dag c↓ c↑ c↓^dag so (i, j, i, j) gives the correlation of the same pair.

 const U1U1Fermion = U₁U₁Fermion

 module U₁SU₂tJFermion

Prepare some commonly used objects for U₁×SU₂ tJ fermions, i.e. local d = 3 Hilbert space without double occupancy.

Behaviors of all operators are the same as U₁SU₂Fermion up to the projection, details please see U₁SU₂Fermion.

 const U1SU2tJFermion = U₁SU₂tJFermion

 module U₁U₁tJFermion

Prepare some commonly used objects for U₁×U₁ tJ fermions, i.e. local d = 3 Hilbert space without double occupancy.

Behaviors of all operators are the same as U₁U₁Fermion up to the projection, details please see U₁U₁Fermion.

 const U1U1tJFermion = U₁U₁tJFermion

 module ℤ₂SU₂tJFermion

Prepare some commonly used objects for ℤ₂×SU₂ tJ fermions, i.e. local d = 3 Hilbert space without double occupancy.

Behaviors of all operators are the same as ℤ₂SU₂Fermion up to the projection, details please see ℤ₂SU₂Fermion.

 const Z2SU2tJFermion = ℤ₂SU₂tJFermion