I would like to know if the energy gap between ground state and the first excited state in a Bose-Hubbard model have been determined? In particular I would like to know how does it scale with the system size? So far I couldn't find any numerical data for 1D, 2D or 3D.
Thank you @Behnam Farid. It turns out the gap is inversely proportional to the total number of particles in the power of 1/3 for a 3D cubic lattice ( Capogrosso-Sansone, NV Prokof'ev, and BV Svistunov, Phys. Rev. B 75, 134302 (2007).)
Yes you are right @Behnam Farid. My reasoning was the following. If they consider commensurate filling with n=1, it means that the number of sites is equal to the total number of particles: N = L^3. That is why I put N^{-1/3} for the gap.
Indeed @Behnam Farid. I didn't edit my first comment. You can check above. Do you agree that the gap is proportional to N^{-1/3} in the case of paper [2]?
Thank you very much! I wonder if you know how does the situation looks like in the Mott insulator phase (n = 1)? If there is no hoping (t = 0) the gap is \Delta E = U, so it does not depend on the lattice size. Do you know how does the gap scale with lattice size when we use perturbation theory in hoping term of the Hamiltonian (strong-coupling expansion)? Do you know any paper where I can find this information?