# The critical point of Bose-Hubbard model

The Hamiltonian of Bose-Hubbard model reads as $$H=-t\sum\limits_{<i,j>}b_i^{\dagger}b_j+h.c.+\frac{U}{2}\sum\limits_{i}n_i(n_i-1)-\mu\sum\limits_in_i$$. In the limit $t\ll U$, the ground states is Mott insulator (MI) states; in the limit $t\gg U$, the ground state is superfluid (SF) state. How to calculate the critical value of $\frac{t}{U}$ of the phase transition between SF and MI?

• Apparently, in this reference, some results are obtained with mean Field solutions, see formula (13) page 4 – Trimok Jun 18 '13 at 8:02
• There's no reason why the critical value should be computable, in general. Of course, it's probably possible to get some value through various approximations. Actually, you don't even specify the dimension, and the critical value will clearly depend on that information. If you mean 3 dimensions, then you have to realize that one does not even know how to compute the critical value in the 3d Ising model! – Yvan Velenik Jun 18 '13 at 12:15
• I think Quantum Monte Carlo is believed to provide accurate results for this question, but I cannot locate a good paper in brief Googling. – BebopButUnsteady Jun 18 '13 at 14:36

Within mean-field, the critical point can be calculated as an analytical function of $z$, the number of nearest neighbours for each lattice point. See for example the Physical Review B paper of Fisher, Grinstein, Weichman and Fisher (1989), or the book 'Quantum Phase Transitions' by Sachdev. Within mean-field, the way to do it is to calculate the particle and hole contours of the Mott-lobe separately (by Landau's criterion of the superfluid order parameter vanishing via a second order transition) and then evaluate the point where the two transition lines touch.
However, strong coupling expansions provide very reliable behaviour of the Mott insulating lobes as the multicritical point is reached; see the Physical Review B paper by Freericks and Monien (1996). This method can nicely capture the numerically difficult Berezinskii-Kosterlitz-Thouless transition for one-dimensional bosons as well. This work was extended to much higher order by Elstner and Monien (1999) in Physical Review B. These are the most accurate values of the critical point in my opinion. Briefly put, the way to go about it in strong coupling expansions is to calculate the gap to excitations in the Mott phase as an extrapolated series in $t/U$ and evaluate the point where the gap disappears.
For the square lattice, $\frac{t}{U}_c \approx 0.059$ and for the 1D chain, $\frac{t}{U}_c \approx 0.26$. You may refer the above papers for more accurate values.