Hubbard operators

Creation, annihilation, number and spin operators for Hubbard-type models, defined on the four-dimensional local space of a spinful fermion, with independent symmetries for the particle number and for the spin.

Conventions

Basis ordering

A Hubbard site carries four states,

\[|0⟩, \quad |↑⟩ = e^†_↑ |0⟩, \quad |↓⟩ = e^†_↓ |0⟩, \quad |↑↓⟩ = e^†_↑ e^†_↓ |0⟩,\]

of which the two singly occupied ones are fermionic. The local space is therefore always graded by FermionParity, and since a graded space groups its basis vectors per sector, in the order of sectors(V), the index order is not the order in which the states are listed above but the parity-sorted one:

\[\underbrace{|0⟩,\; |↑↓⟩}_{\text{parity even}}, \quad \underbrace{|↑⟩,\; |↓⟩}_{\text{parity odd}}\]

This is the order in which block(op, FermionParity(0)) and block(op, FermionParity(1)) list their rows and columns — with Trivial particle and spin symmetry both blocks are $2 × 2$, so there is no single $4 × 4$ matrix — and it is the reference order of basis_transform, which is consequently the identity for Trivial symmetry.

For the symmetric versions the dense order follows the sector order of the target space and is in general not the reference order. Rather than relying on it, read it off from the basis transformation: numbering the reference states $(1, 2, 3, 4) = (|0⟩, |↑↓⟩, |↑⟩, |↓⟩)$, column $j$ of basis_transform(P, S) has its single nonzero entry in the row that is the dense index of reference state $j$:

U = convert(Array, basis_transform(U1Irrep, SU2Irrep))
i_up = findfirst(==(1), U[:, 3])   # dense index of |↑⟩

Symmetry sectors

The particle-number and the spin symmetry are chosen independently. The local sector type is always FermionParity ⊠ (particle) ⊠ (spin) with the Trivial factors omitted and the spin factor last, so e.g. hubbard_space(U1Irrep, SU2Irrep) is Vect[FermionParity ⊠ U1Irrep ⊠ SU2Irrep]((0, 0, 0) => 1, (1, 1, 1//2) => 1, (0, 2, 0) => 1). The supported sectors are Trivial, U1Irrep and SU2Irrep on both axes; anything else, Z2Irrep included, throws an ArgumentError.

The particle symmetry labels the parity-even pair (empty, doubly occupied):

Symmetry	Physical meaning	Sector label	Charges of the local states
`Trivial`	none	—	empty and doubly occupied share the parity-even sector, with multiplicity 2
`U1Irrep`	particle-number conservation	$n ∈ \{0, 1, 2\}$	empty $↦ 0$, singly occupied $↦ 1$, doubly occupied $↦ 2$
`SU2Irrep`	$η$-pairing SU(2)	$η ∈ \{0, 1/2\}$	(doubly occupied, empty) is the $η = 1/2$ doublet, ordered by descending $η^z = (n-1)/2$; singly occupied has $η = 0$

The spin symmetry labels the parity-odd pair (the two singly occupied states):

Symmetry	Physical meaning	Sector label	Charges of the local states
`Trivial`	none	—	the two singly occupied states share the parity-odd sector, with multiplicity 2
`U1Irrep`	$S^z$ conservation	$m ∈ \{-1/2, 0, +1/2\}$	$↑ ↦ +1/2$, $↓ ↦ -1/2$, empty and doubly occupied $↦ 0$
`SU2Irrep`	full spin SU(2)	$s ∈ \{0, 1/2\}$	the singly occupied states form the $s = 1/2$ doublet (descending $m$); empty and doubly occupied are $s = 0$

Charge conventions

The U1Irrep particle charge is the particle number $n$, not the particle-hole symmetric $n - 1$, and the U1Irrep spin charge is $S^z = ±1/2$, not $2S^z = ±1$. The $η$-pairing SU2Irrep is built on $η^z = (n-1)/2$ instead, so the particle number itself is not an $η$-SU(2) scalar: this is why e_num and ud_num are unavailable with SU2Irrep particle symmetry, while the particle-hole symmetric half_ud_num $= (n_↑ - 1/2)(n_↓ - 1/2)$ — which equals $+1/4$ on the $η$ doublet and $-1/4$ on the singly occupied states — is available for every symmetry combination.

Staggered gauge for SU(2) particle symmetry

The $η$-pairing SU(2) only commutes with the Hubbard Hamiltonian after the staggered gauge transformation $e_{j,σ} → i^j e_{j,σ}$ on a bipartite lattice. Accordingly, with SU2Irrep particle symmetry the operators of this module act on site $k$ with the additional gauge factor $G^{k-1}$, where $G = i^n = \mathrm{diag}(1, -1, i, i)$ in the reference order above. Operators that commute with $G$ are returned without it, so they stay real-representable and remain elementwise equal to their Trivial counterparts. Of the operators available with SU2Irrep particle symmetry, e_hopping is the only one that does not commute with $G$: it is therefore genuinely complex, requires a complex eltype, and is the gauge-transformed version of its Trivial counterpart — it generates the same physics on a bipartite lattice, but is not elementwise equal to it.

The $η$-pairing SU(2) and the gauge transformation that makes it a symmetry of the Hubbard model are due to C. N. Yang and S. C. Zhang, SO₄ symmetry in a Hubbard model, Mod. Phys. Lett. B 4, 759 (1990), doi:10.1142/S0217984990000933; see also C. N. Yang, η pairing and off-diagonal long-range order in a Hubbard model, Phys. Rev. Lett. 63, 2144 (1989), doi:10.1103/PhysRevLett.63.2144.

Operator overview

The two symmetry columns are independent conditions: an operator is available for exactly those combinations that satisfy both. any means all three supported sectors, Trivial, U1Irrep and SU2Irrep; every other request throws an ArgumentError.

Function	Alias	Sites	Particle symmetry	Spin symmetry
`hubbard_space`	—	—	any	any
`basis_transform`	—	—	any	any
`e_num`	`n`	1	`Trivial`, `U1Irrep`	any
`u_num`	`nꜛ`	1	`Trivial`, `U1Irrep`	`Trivial`, `U1Irrep`
`d_num`	`nꜜ`	1	`Trivial`, `U1Irrep`	`Trivial`, `U1Irrep`
`ud_num`	`nꜛꜜ`	1	`Trivial`, `U1Irrep`	any
`half_ud_num`	—	1	any	any
`h_num`	`nʰ`	1	`Trivial`, `U1Irrep`	any
`S_x`	`Sˣ`	1	any	`Trivial`
`S_y`	`Sʸ`	1	any	`Trivial`
`S_z`	`Sᶻ`	1	any	`Trivial`, `U1Irrep`
`S_plus`	`S⁺`	1	any	`Trivial`
`S_min`	`S⁻`	1	any	`Trivial`
`u_plus_u_min`	`u⁺u⁻`	2	`Trivial`, `U1Irrep`	`Trivial`, `U1Irrep`
`u_min_u_plus`	`u⁻u⁺`	2	`Trivial`, `U1Irrep`	`Trivial`, `U1Irrep`
`d_plus_d_min`	`d⁺d⁻`	2	`Trivial`, `U1Irrep`	`Trivial`, `U1Irrep`
`d_min_d_plus`	`d⁻d⁺`	2	`Trivial`, `U1Irrep`	`Trivial`, `U1Irrep`
`e_plus_e_min`	`e⁺e⁻`	2	`Trivial`, `U1Irrep`	any
`e_min_e_plus`	`e⁻e⁺`	2	`Trivial`, `U1Irrep`	any
`e_hopping`	`e_hop`	2	any	any
`u_min_d_min`	`u⁻d⁻`	2	`Trivial`	`Trivial`, `U1Irrep`
`d_min_u_min`	`d⁻u⁻`	2	`Trivial`	`Trivial`, `U1Irrep`
`u_plus_d_plus`	`u⁺d⁺`	2	`Trivial`	`Trivial`, `U1Irrep`
`d_plus_u_plus`	`d⁺u⁺`	2	`Trivial`	`Trivial`, `U1Irrep`
`u_min_u_min`	`u⁻u⁻`	2	`Trivial`	`Trivial`
`u_plus_u_plus`	`u⁺u⁺`	2	`Trivial`	`Trivial`
`d_min_d_min`	`d⁻d⁻`	2	`Trivial`	`Trivial`
`d_plus_d_plus`	`d⁺d⁺`	2	`Trivial`	`Trivial`
`singlet_plus`	`singlet⁺`	2	`Trivial`	any
`singlet_min`	`singlet⁻`	2	`Trivial`	any
`S_plus_S_min`	`S⁺S⁻`	2	any	`Trivial`, `U1Irrep`
`S_min_S_plus`	`S⁻S⁺`	2	any	`Trivial`, `U1Irrep`
`S_exchange`	`SS`	2	any	any
`singlet_plus_singlet_min_3site`	`Δ⁺ij_Δjk`	3	`Trivial`, `U1Irrep`	any
`singlet_plus_singlet_min_4site`	`Δ⁺ij_Δkl`	4	`Trivial`, `U1Irrep`	any

Note that the singlet-pair terms singlet_plus_singlet_min_3site and singlet_plus_singlet_min_4site are available with U1Irrep particle symmetry even though singlet_plus and singlet_min individually are not: only the product $Δ^† Δ$ conserves the particle number. S_y requires a complex eltype for every symmetry combination, and e_hopping requires one with SU2Irrep particle symmetry; all other operators honour any eltype, and the basis transformations have exact integer entries so that they never degrade the precision of the result.

The t-J restriction of this module, obtained by projecting out the doubly occupied state, is TJOperators.

API

TensorKitTensors.HubbardOperators.S_exchange — Method

S_exchange([elt::Type{<:Number}], [particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}])
SS([elt::Type{<:Number}], [particle_symmetry::Type{<:Sector}, spin_symmetry::Type{<:Sector}])

Return the spin exchange operator

\[𝐒_1 ⋅ 𝐒_2 = S^z_1 S^z_2 + \tfrac{1}{2}(S^+_1 S^-_2 + S^-_1 S^+_2),\]

i.e. the Heisenberg exchange interaction between the two sites.

SU2Irrep particle symmetry

Being a scalar under both the spin and the $η$-pairing SU(2), this operator is available for every symmetry combination.

Supported symmetries: particle Trivial, U1Irrep, SU2Irrep; spin Trivial, U1Irrep, SU2Irrep.