Lieb-Liniger 1D Bose Gas

The Lieb-Liniger model describes N bosons in 1D interacting via delta-function repulsion: H = −Σ∂²_i + 2c·Σ_{i<j}δ(x_i−x_j). The dimensionless coupling γ = c/n controls the regime: γ≪1 is weakly-interacting (quasi-BEC), γ≫1 is the Tonks-Girardeau fermionized limit. The ground-state energy per particle e(γ) = n²·ε(γ) is computed via Bethe ansatz integral equations.

Parameters

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Limits:
γ→0: e ≈ n²γ (BEC-like, Bogoliubov)
γ→∞: e → π²n²/3 (free fermions, TG)
Bethe ansatz: exact at all γ
Bethe ansatz (numerical)
Weak coupling: γ − 4γ^{3/2}/(3π)
Strong coupling: π²/3·(1 − 4/γ)