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

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

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)

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).

Related

• 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.

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Verified hubs

In the Verified Atlas, these 2 models register 5 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.

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