Bose-Hubbard: Mott Insulator Phase Diagram

Mean-field phase boundary between Mott lobes and superfluid in 2D optical lattice

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Bose-Hubbard Hamiltonian: H = −J Σ(a†ᵢaⱼ) + (U/2)Σnᵢ(nᵢ−1) − μΣnᵢ. Mean-field boundary: zJ/U = 1/[ψ⁻¹] where ψ = Σₙ [n+1/(En+1−E0−μ) + n/(En−1−E0+μ)]. Mott lobes (red) have integer filling; superfluid (blue) has phase coherence.