In a 1D Bose-Hubbard model with hopping 1/r^α, the superfluid-Mott-insulator transition is continuous and scale-invariant for 1<α≤3, not BKT, while true long-range order in the superfluid appears only for α≤2.
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Bose-Hubbard model with power-law hopping in one dimension
In a 1D Bose-Hubbard model with hopping 1/r^α, the superfluid-Mott-insulator transition is continuous and scale-invariant for 1<α≤3, not BKT, while true long-range order in the superfluid appears only for α≤2.