Monte Carlo study of the two-dimensional Bose-Hubbard model

One of the most promising applications of ultracold gases in optical lattices is the possibility to use them as quantum emulators of more complex condensed matter systems. We provide benchmark calculations, based on exact quantum Monte Carlo simulations, for the emulator to be tested against. We report results for the ground state phase diagram of the two-dimensional Bose-Hubbard model at unity filling factor. We precisely trace out the critical behavior of the system and resolve the region of small insulating gaps, $\Delta ⪡J$. The critical point is found to be ${\left(J/U\right)}_c=0.05974\left(3\right)$, in perfect agreement with the high-order strong-coupling expansion method of Elstner and Monien [Phys. Rev. B 59, 12184 (1999)]. In addition, we present data for the effective mass of particle and hole excitations inside the insulating phase and obtain the critical temperature for the superfluid-normal transition at unity filling factor.

Figure 1

(Color online) Phase diagram of the first MI-SF lobe. Solid circles are numerical data, with error bars shown but barely visible. The inset is a blowup of the region close to the tip. Dashed lines represent the critical region as calculated from finite size scaling.

Figure 2

(Color online) Finite size scaling of the energy gap at the tip of the lobe. Lines represent linear fits used to extract the critical point. The critical point can be directly read from the intersection of the curves: ${\left(J/U\right)}_c=0.05974\left(3\right)$.

Figure 3

(Color online) Effective mass for particle (circles) and hole (squares) excitations as a function of $J/U$. The exact results at $J/U=0$ are $m_{+}=0.25/J$ and $m_{-}=0.5/J$. By dashed lines we show the lowest order in $J/U$ correction to the effective masses. Close to the critical point the two curves overlap, directly demonstrating the emergence of the particle-hole symmetry. At $J/U=0.059$, the sound velocity is $c/J=4.8\pm 0.2$.

Figure 4

(Color online) Finite-temperature phase diagram at filling factor $n=1$. Solid circles are simulation results (the dotted line is to guide the eye); error bars are plotted. Dashed lines are analytical results for the weakly interacting gas [see Eq. (5)] and for the strongly interacting gas close ${\left(J/U\right)}_c$.

Figure 5

Sketch of the proposed cooling protocol. On top, a shallow trap and a deep trap are superimposed. At the bottom, the corresponding densities of states are sketched. The dashed line represents the Fermi level. Adiabatically displacing the shallow trap results in displacing entropic particles with consequent dramatic decrease of entropy per particle inside the deep trap.