TJ operators

TensorKitTensors.TJOperators.S_exchange — Function

S_exchange(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the spin exchange operator S⋅S.

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TensorKitTensors.TJOperators.S_min — Function

S_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
S⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the spin-minus operator (only defined for Trivial spin symmetry).

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TensorKitTensors.TJOperators.S_min_S_plus — Function

S_min_S_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
S⁻S⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator S⁻S⁺. The only nonzero matrix element corresponds to |↓,↑⟩ <-- |↑,↓⟩.

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TensorKitTensors.TJOperators.S_plus — Function

S_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
S⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the spin-plus operator (only defined for Trivial spin symmetry).

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TensorKitTensors.TJOperators.S_plus_S_min — Function

S_plus_S_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
S⁺S⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator S⁺S⁻. The only nonzero matrix element corresponds to |↑,↓⟩ <-- |↓,↑⟩.

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TensorKitTensors.TJOperators.S_x — Function

S_x(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
Sˣ(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the one-body spin-1/2 x-operator on the electrons (only defined for Trivial spin symmetry).

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TensorKitTensors.TJOperators.S_y — Function

S_y(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
Sʸ(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the one-body spin-1/2 y-operator on the electrons (only defined for Trivial spin symmetry).

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TensorKitTensors.TJOperators.S_z — Function

S_z(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
Sᶻ(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the one-body spin-1/2 z-operator on the electrons.

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TensorKitTensors.TJOperators.d_min_d_min — Function

d_min_d_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
d⁻d⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e_{1,↓} e_{2,↓}$ that annihilates a spin-down particle at both sites. The only nonzero matrix element corresponds to |0,0⟩ <-- |↓,↓⟩.

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TensorKitTensors.TJOperators.d_min_d_plus — Function

d_min_d_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
d⁻d⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e_{1,↓}, e†_{2,↓}$ that annihilates a spin-down electron at the first site and creates a spin-down electron at the second. The only nonzero matrix element corresponds to |0↓⟩ <-- |↓0⟩.

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TensorKitTensors.TJOperators.d_min_u_min — Function

d_min_u_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
d⁻u⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e_{1,↓} e_{2,↑}$ that annihilates a spin-down particle at the first site and a spin-up particle at the second site. The only nonzero matrix element corresponds to |0,0⟩ <-- |↓,↑⟩.

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TensorKitTensors.TJOperators.d_num — Function

d_num(particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
nꜜ(particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the one-body operator that counts the number of spin-down electrons.

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TensorKitTensors.TJOperators.d_plus_d_min — Function

d_plus_d_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
d⁺d⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e†_{1,↓}, e_{2,↓}$ that creates a spin-down electron at the first site and annihilates a spin-down electron at the second. The only nonzero matrix element corresponds to |↓0⟩ <-- |0↓⟩.

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TensorKitTensors.TJOperators.d_plus_d_plus — Function

d_plus_d_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
d⁺d⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e†_{1,↓} e†_{2,↓}$ that annihilates a spin-down particle at both sites.

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TensorKitTensors.TJOperators.d_plus_u_plus — Function

d_plus_u_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
d⁺u⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e†_{1,↓} e†_{2,↑}$ that annihilates a spin-down particle at the first site and a spin-up particle at the second site.

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TensorKitTensors.TJOperators.e_hopping — Function

e_hopping([elt::Type{<:Number}], [particle_symmetry::Type{<:Sector}], [spin_symmetry::Type{<:Sector}]; slave_fermion::Bool = false)
e_hop([elt::Type{<:Number}], [particle_symmetry::Type{<:Sector}], [spin_symmetry::Type{<:Sector}]; slave_fermion::Bool = false)

Return the two-body operator that describes a particle that hops between the first and the second site.

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TensorKitTensors.TJOperators.e_min_e_plus — Function

e_min_e_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
e⁻e⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator that annihilates a particle at the first site and creates a particle at the second. This is the sum of u_min_u_plus and d_min_d_plus.

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TensorKitTensors.TJOperators.e_num — Function

e_num(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
n(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the one-body operator that counts the number of particles.

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TensorKitTensors.TJOperators.e_plus_e_min — Function

e_plus_e_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
e⁺e⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator that creates a particle at the first site and annihilates a particle at the second. This is the sum of u_plus_u_min and d_plus_d_min.

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TensorKitTensors.TJOperators.h_num — Function

h_num(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)
nʰ(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool=false)

Return the one-body operator that counts the number of holes.

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TensorKitTensors.TJOperators.singlet_min — Function

singlet_min(elt, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
singlet⁻(elt, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the adjoint of singlet_plus operator, which is $(-e_{1,↑} e_{2,↓} + e_{1,↓} e_{2,↑}) / sqrt(2)$

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TensorKitTensors.TJOperators.singlet_plus — Function

singlet_plus(elt, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
singlet⁺(elt, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body singlet operator $(e^†_{1,↑} e^†_{2,↓} - e^†_{1,↓} e^†_{2,↑}) / sqrt(2)$, which creates the singlet state when acting on vaccum.

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TensorKitTensors.TJOperators.tj_space — Function

tj_space(particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the local hilbert space for a t-J-type model with the given particle and spin symmetries. The possible symmetries are

Particle number: Trivial, U1Irrep;
Spin: Trivial, U1Irrep, SU2Irrep.

Setting slave_fermion = true switches to the slave-fermion basis.

basis states for slave_fermion = false: |0⟩ = |vac⟩ (vacuum), |↑⟩ = (c↑)†|vac⟩, |↓⟩ = (c↓)†|vac⟩
basis states for slave_fermion = true: (cσ = h† bσ; holon h is fermionic, spinon b_σ is bosonic): |0⟩ = h†|vac⟩, |↑⟩ = (b↑)†|vac⟩, |↓⟩ = (b↓)†|vac⟩

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TensorKitTensors.TJOperators.u_min_d_min — Function

u_min_d_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
u⁻d⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e_{1,↑} e_{2,↓}$ that annihilates a spin-up particle at the first site and a spin-down particle at the second site. The only nonzero matrix element corresponds to |0,0⟩ <-- |↑,↓⟩.

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TensorKitTensors.TJOperators.u_min_u_min — Function

u_min_u_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
u⁻u⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e_{1,↑} e_{2,↑}$ that annihilates a spin-up particle at both sites. The only nonzero matrix element corresponds to |0,0⟩ <-- |↑,↑⟩.

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TensorKitTensors.TJOperators.u_min_u_plus — Function

u_min_u_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
u⁻u⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e_{1,↑}, e†_{2,↑}$ that annihilates a spin-up electron at the first site and creates a spin-up electron at the second. The only nonzero matrix element corresponds to |0↑⟩ <-- |↑0⟩.

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TensorKitTensors.TJOperators.u_num — Function

u_num(particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
nꜛ(particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the one-body operator that counts the number of spin-up electrons.

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TensorKitTensors.TJOperators.u_plus_d_plus — Function

u_plus_d_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
u⁺d⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e†_{1,↑} e†_{2,↓}$ that annihilates a spin-up particle at the first site and a spin-down particle at the second site.

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TensorKitTensors.TJOperators.u_plus_u_min — Function

u_plus_u_min(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
u⁺u⁻(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e†_{1,↑}, e_{2,↑}$ that creates a spin-up electron at the first site and annihilates a spin-up electron at the second. The only nonzero matrix element corresponds to |↑0⟩ <-- |0↑⟩.

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TensorKitTensors.TJOperators.u_plus_u_plus — Function

u_plus_u_plus(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)
u⁺u⁺(elt::Type{<:Number}, particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}; slave_fermion::Bool = false)

Return the two-body operator $e†_{1,↑} e†_{2,↑}$ that annihilates a spin-up particle at both sites.

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