Bose-Hubbard Model — Mott Insulator Phase Diagram

Mean-field phase diagram of the Bose-Hubbard model showing Mott insulator lobes (integer filling n=1,2,3) in the (J/U, μ/U) plane and the superfluid-insulator phase boundary computed from the mean-field condition.

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H = -J Σ (a†ᵢaⱼ + h.c.) + U/2 Σ nᵢ(nᵢ-1) - μ Σ nᵢ MF boundary (n-lobe): zJ·[ n/(E₀ⁿ - E₀ⁿ⁻¹) + (n+1)/(E₀ⁿ - E₀ⁿ⁺¹) ] = 1 E₀ⁿ = Un(n-1)/2 - μn
Mott insulator (MI) lobes: gapped, integer filling, ψ=⟨a⟩=0. Superfluid (SF): ψ≠0, gapless. The lobe tip marks the quantum critical point. 3D: (J/U)_c ≈ 0.034 for n=1.