Hubbard1D — 1D Hubbard Chain (Lieb–Wu Bethe Ansatz)

Status: Phase 1 (v0.18.x)

Phase 1 exposes the Lieb–Wu (1968) closed-form integrals at half filling only ($\mu = U/2$). General filling and finite-temperature QTM observables are deferred to Phase 2; off-half-filling fetches raise DomainError.

Hamiltonian

\[H = -t \sum_{i, \sigma} \bigl(c_{i,\sigma}^\dagger c_{i+1,\sigma} + \text{h.c.}\bigr) + U \sum_i n_{i,\uparrow} n_{i,\downarrow} - \mu \sum_i n_i, \qquad t > 0,\ U > 0.\]

Half filling corresponds to $\mu = U/2$ by particle-hole symmetry; in this regime the chain is a Mott insulator for any $U > 0$ (no Mott transition — the celebrated Lieb–Wu result).

Phase 1 coverage

Quantity	Infinite (half filling)	Off half filling
`GroundStateEnergyDensity`	Lieb–Wu integral	`DomainError`
`ChargeGap`	Lieb–Wu integral	`DomainError`
`SpinGap`	$0$ exactly	`DomainError`

Closed-form integrals (Lieb–Wu 1968)

Ground-state energy per site

\[\frac{E_0}{N} = -4 t^2 \int_0^\infty d\omega \, \frac{J_0(\omega)\, J_1(\omega)}{\omega \, \bigl[1 + \exp(\omega U / 2t)\bigr]}.\]

Mott (charge) gap

\[\Delta_c = \frac{16 t^2}{U} \int_1^\infty d\omega \, \frac{\sqrt{\omega^2 - 1}}{\sinh(2\pi t \omega / U)}.\]

Spin gap

\[\Delta_s = 0 \quad \text{exactly} \quad (\text{rigorous gapless spinons}).\]

Both integrals are evaluated numerically with QuadGK.quadgk at relative tolerance $10^{-12}$, well below the test atol of $10^{-8}$.

Asymptotic limits (test anchors)

Limit	$E_0/N$	$\Delta_c$
$U/t \to 0$ (free fermion)	$-4t/\pi$	$0$ (exponentially small)
$U/t \to \infty$ (Heisenberg AFM)	$-4 t^2 \log 2 / U$	$U - 4t + 8 t^2 \log 2 / U$

The $U \to \infty$ limit reproduces the Hulthén ground-state energy of the Heisenberg AFM chain with effective coupling $J = 4 t^2 / U$.

Quick-look code

using QAtlas

m = Hubbard1D(; t=1.0, U=4.0, μ=2.0)   # half filling

QAtlas.fetch(m, GroundStateEnergyDensity(), Infinite())   # ≈ -0.5737293678984494
QAtlas.fetch(m, ChargeGap(),               Infinite())    # ≈ 1.2867270220129354
QAtlas.fetch(m, SpinGap(),                 Infinite())    # 0.0 exactly

# Off half filling raises DomainError (Phase 2 territory)
QAtlas.fetch(Hubbard1D(; t=1, U=6, μ=2),
             GroundStateEnergyDensity(), Infinite())   # DomainError

References

E. H. Lieb, F. Y. Wu, "Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension", Phys. Rev. Lett. 20, 1445 (1968).
F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper, V. E. Korepin, The One-Dimensional Hubbard Model, Cambridge University Press (2005).
E. H. Lieb, F. Y. Wu, "The one-dimensional Hubbard model: a reminiscence", Physica A 321, 1 (2003).

Heisenberg1D — the $U/t \to \infty$ effective-spin reduction of the half-filled Hubbard chain (Hulthén 1938).
XXZ1D — the same Bethe-ansatz machinery applied to the anisotropic spin-½ chain (issue #108 — TBA/NLIE).
Lieb tight-binding lattice — Lieb's other flat-band lattice result (1989); unrelated to the Lieb–Wu Hubbard solution despite the shared surname.

<!– ATLAS:HUBS:START – auto-generated by docs/atlas/generate.jl. Do not edit by hand; edits between these markers are overwritten on next regen. –>

Verified hubs

In the Verified Atlas, these 2 models register 8 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

Model	Quantity	BC	Assurance	Cards
`ExtendedHubbard1D`	`ChargeGap`	`Infinite`	🔵 coherent	3
`Hubbard1D`	`ChargeGap`	`Infinite`	🔵 coherent	6
`Hubbard1D`	`Energy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`Hubbard1D`	`FreeEnergy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`Hubbard1D`	`GroundStateEnergyDensity`	`Infinite`	🔵 coherent	5
`Hubbard1D`	`LuttingerParameter`	`Infinite`	🟢 corroborated-at-p	3
`Hubbard1D`	`MassGap`	`Infinite`	🟠 uncorroborated-but-feasible	0
`Hubbard1D`	`SpinGap`	`Infinite`	🟢 corroborated-at-p	3

<!– ATLAS:HUBS:END –>

<!– ATLAS:DOCS:START – auto-generated by docs/atlas/generate.jl. Do not edit by hand; edits between these markers are overwritten on next regen. –>

API

Every fetch(::Model, …) method registered for these models — together with the model struct(s) and exported helpers — generated directly from the source (in lock-step with @register):

QAtlas.ExtendedHubbard1D — Type

ExtendedHubbard1D(; t::Real = 1.0, U::Real = 4.0, V::Real = 0.0)
    <: AbstractQAtlasModel

1D t-U-V Hubbard chain (nearest-neighbour-density extension of the standard Hubbard model):

H = -t Σ_{i, σ} (c†_{i,σ} c_{i+1,σ} + h.c.)
    + U Σ_i n_{i,↑} n_{i,↓}
    + V Σ_i n_i n_{i+1}.

Convention: t > 0 hopping, U on-site, V nearest-neighbour density-density.

Phase 1 scope (this release)

Phase 1 exposes only the V = 0 limit, at which the model reduces to the standard 1D Hubbard chain. The ChargeGap at Infinite() is delegated to Hubbard1D at half filling (μ = U/2), i.e. the Lieb–Wu (1968) integral

Δ_c = (16 t² / U) ∫_1^∞ dω  √(ω² - 1) / sinh(2π t ω / U).

Any V ≠ 0 raises DomainError. The V ≠ 0 phase diagram (CDW / SDW / BOW / phase separation, Voit 1995; Nakamura 2000) is deferred to Phase 2.

References

Lieb, Wu, Phys. Rev. Lett. 20, 1445 (1968).
Voit, Rep. Prog. Phys. 58, 977 (1995).
Nakamura, Phys. Rev. B 61, 16377 (2000).

source

QAtlas.fetch — Method

fetch(model::ExtendedHubbard1D, ::ChargeGap, ::Infinite) -> Float64

Charge (Mott) gap of the t-U-V Hubbard chain at half filling. Phase 1 only supports the V = 0 limit, where the gap reduces to the Lieb–Wu (1968) integral and is delegated to Hubbard1D at μ = U/2. Any V ≠ 0 raises DomainError.

References

Lieb, Wu, Phys. Rev. Lett. 20, 1445 (1968).
Voit, Rep. Prog. Phys. 58, 977 (1995).
Nakamura, Phys. Rev. B 61, 16377 (2000).

source

QAtlas.Hubbard1D — Type

Hubbard1D(; t::Real = 1.0, U::Real = 4.0, μ::Real = 2.0) <: AbstractQAtlasModel

1D Hubbard model

H = -t Σ_{i, σ} (c†_{i,σ} c_{i+1,σ} + h.c.)
    + U Σ_i n_{i,↑} n_{i,↓}
    - μ Σ_i n_i

Convention: t > 0 hopping, U > 0 on-site repulsion. Half filling corresponds to μ = U/2 (particle-hole-symmetric point).

Phase 1 scope (this release)

Phase 1 implements the Lieb–Wu (1968) closed-form integrals at half filling only (μ = U/2). The supported Infinite() quantities are

GroundStateEnergyDensity — E₀/N via the Lieb–Wu integral,
ChargeGap — Mott gap Δ_c via the Lieb–Wu second integral,
SpinGap — 0 exactly (rigorous gapless-spinon result).

Off half filling raises a DomainError — the general-filling (coupled-integral-equation) and finite-temperature (QTM) surfaces are deferred to Phase 2.

source

QAtlas.fetch — Method

fetch(model::Hubbard1D, ::ChargeGap, ::Infinite) -> Float64

Lieb–Wu Mott (charge) gap at half filling:

Δ_c = (16 t² / U) ∫_1^∞ dω  √(ω² - 1) / sinh(2π t ω / U).

Strictly positive for any U > 0 (no Mott transition in 1D — the chain is insulating at half filling for arbitrarily small U, the celebrated Lieb–Wu result).

Asymptotic limits

U → 0: Δ_c → 0 (exponentially small, ∝ exp(-2π t / U)).
U → ∞: Δ_c → U - 4t + 8 t² log 2 / U.

References

Lieb–Wu, PRL 20, 1445 (1968).

source

QAtlas.fetch — Method

fetch(model::Hubbard1D, ::Energy{:per_site}, ::Infinite; kwargs...) -> Float64

Lieb–Wu ground-state energy density E₀/N at half filling. Currently only implemented for μ = U/2; off-half-filling raises a DomainError.

source

QAtlas.fetch — Method

fetch(model::Hubbard1D, ::FreeEnergy, ::Infinite; beta, kwargs...) -> Float64

Per-site Helmholtz free energy of the 1D Hubbard chain at finite temperature 1/beta, computed by the JKS 1998 quantum-transfer-matrix non-linear integral equations (issue #523).

The solver uses the paper-precise eq (47) NLIE in 3 channels (b, c, c̄) on a discretised contour, then evaluates the QTM eigenvalue via paper eq (49) third form. At β → 0 the result reproduces the atomic limit atomic_free_energy(β, U, μ) to within a few percent (Stage C.10 test guards this).

Keyword arguments

beta::Real: inverse temperature (β > 0 required).
H::Real = 0: external magnetic field (couples to magnetisation).
grid_N::Int = 64, x_max::Real = 8.0: discretisation knobs.
alpha::Real = m.U / 6: contour shift in the b channel (0 < α < η = U/4).
tol::Real = 1e-6, maxiter::Int = 40: Newton convergence knobs.

Notes

Half-filling (μ = U/2) is the well-tested regime. Other fillings work as long as the NLIE converges (the Newton solver does not warn on non-convergence; the result is NaN in that case).

References

JKS 1998 = Jüttner, Klümper, Suzuki, Nucl. Phys. B 522, 471 (1998), arXiv:cond-mat/9711310.
Paper eqs (23), (47), (48), (49), (54), (55) all implemented per PDF (Stage C.22c paper-precise rewrite + Stage C.24 FE evaluator fix).

source

QAtlas.fetch — Method

fetch(model::Hubbard1D, ::GroundStateEnergyDensity, ::Infinite) -> Float64

Lieb–Wu ground-state energy density E₀/N at half filling. Currently only implemented for μ = U/2; off-half-filling raises a DomainError.

Asymptotic limits

U/t → 0: E₀/N → -4t/π (free 1D fermion).
U/t → ∞: E₀/N → -4 t² log 2 / U (Heisenberg AFM reduction).

Both are exercised in test/standalone/test_hubbard1d.jl.

References

Lieb–Wu, PRL 20, 1445 (1968).
Essler et al., The One-Dimensional Hubbard Model (Cambridge, 2005).

source

QAtlas.fetch — Method

fetch(::Hubbard1D, ::LuttingerParameter, ::Infinite; t=m.t, U=m.U, μ=m.μ) -> Float64

Luttinger-liquid parameter of the 1D Hubbard model in the free-fermion limit U = 0:

K = 1               (non-interacting spinful fermions; Voit 1995)

For U > 0, the Lieb–Wu Bethe-ansatz solution gives a non-closed-form expression for both Kρ (charge) and Kσ (spin); deferred to Phase 2 (or Phase 3). This entry exposes only the U = 0 free-fermion limit and throws DomainError for any U ≠ 0.

References

E. H. Lieb, F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).
J. Voit, Rep. Prog. Phys. 58, 977 (1995) — TLL review for Hubbard.

source

QAtlas.fetch — Method

fetch(model::Hubbard1D, ::MassGap, ::Infinite; type::Symbol=:charge, kwargs...) -> Float64

Lieb–Wu mass gap of the 1D Hubbard chain at half filling.

type = :charge (default): returns the Mott charge gap Δ_c (Lieb–Wu 1968; Ovchinnikov 1970).
type = :spin: returns the spin gap which is rigorously 0.0 (gapless spinons).

Off-half-filling raises a DomainError.

source

QAtlas.fetch — Method

fetch(model::Hubbard1D, ::SpinGap, ::Infinite) -> Float64

Spin gap of the half-filled 1D Hubbard chain. Returns 0.0 exactly: the spinon branch is gapless for any U > 0 (rigorous Lieb–Wu result; spin-charge separation in the low-energy effective theory).

References

Lieb–Wu, PRL 20, 1445 (1968).
Essler et al., The One-Dimensional Hubbard Model (Cambridge, 2005).

source

<!– ATLAS:DOCS:END –>