Mean-field theory for Bose-Hubbard model under a magnetic field

We consider the superfluid-insulator transition for cold bosons under an effective magnetic field. We investigate how the applied magnetic field affects the Mott transition within mean-field theory and find that the critical hopping strength ${(t∕U)}_c$ increases with the applied field. The increase in the critical hopping follows the bandwidth of the Hofstadter butterfly at the given value of the magnetic field. We also calculate the magnetization and superfluid density within mean-field theory.

Figure 1

(Color online) Phase diagram of Bose atoms confined in a 2D optical lattice for the magnetic flux $\phi =0$ (left panel) and 0.1 (right panel). In the figure the first three Mott lobes are depicted, for on-site particle numbers 1, 2, and 3.

Figure 2

(Color online) Phase diagram for the magnetic flux $\phi =0$ (right panel) and 0.4 (left panel). On the $z$ axis the variance $\sigma \left(\rho \right)$ of the Bose density $\rho$ is plotted. For the MI phase the variance of the 2D density is zero, and the Bose charge has no fluctuation.

Figure 3

(Color online) Phase diagram of Bose atoms confined in a 2D optical lattice in the $t\text{−}\phi$ plane for different values of the chemical potential, corresponding to the MI phase with $\rho =1$ (top) and $\rho =2$ (bottom). The curves correspond to the critical values $t_c$ of the SF-MI transition. For $t>t_c$ the system is in the superfluid phase and for $t<t_c$ in the MI phase. The phase diagram is periodic in the magnetic flux with $\Delta \phi =1$ and symmetric around the flux value $\phi =0.5$.

Figure 4

Single particle energy bands as a function of magnetic flux per plaquette, the Hofstadter butterfly (Ref. 22).

Figure 5

(Color online) SF order parameter ${\Psi }_n$ for different lattice sites at MI-SF transition. The chemical potential is $\mu =0.5$ corresponding to MI with $\rho =1$ and $\mu =1.5$ corresponding to MI with $\rho =2$. The blue line with crosses represents the surface average value $\Psi$. For $t>t_c$ the second derivative $d^2\Psi ∕d{t}^2$ changes sign for the points $n=4,5$. The magnetic flux is $\phi =0.1$ and the superfluid order parameter has the periodicity ${\Psi }_n={\Psi }_{n+10}$. For the SF phase (at $t>t_c$) the lattice sites $\mathit{\text{integer}}\times 10+4$ and $\mathit{\text{integer}}\times 10+5$ exhibit very low values of the SF order parameter and low variance of $\rho$.

Figure 6

(Color online) ${\rho }_s∕\rho$ as a function of $t_c∕t$ for different values of the magnetic flux. For $\phi =0$ our curve is similar to Fig. 1 of Ref. 23. The change of slope at $t\simeq >t_c$ for $\phi \ne 0$ is related to the surface oscillations of the charged bosons. When the magnetic field is present, the superfluid phase exhibits surface region where the quantum fluctuations of SF density and the order parameter are small (compared to nearby regions; see Fig. 5) meaning that the local phase is closer to an insulator. It exhibits a lower ratio ${\rho }_s∕\rho$ compared with the case $\phi =0$.

Figure 7

(Color online) Magnetization as a function of $t$ for $\phi =0.1,0.2,0.3,0.4$ (from top to bottom). In the MI phase, for $t<t_c$, the magnetization is zero. Our numerical approach is more reliable at lower values of $t$, and not too small $\phi$, where our computational supercell size is small and the number fluctuations at each lattice site are limited. MI and superfluid phases and $t_c$ are identified for $\phi =0.1$ on the figure. The value of $t_c$ for other $\phi$ can be identified as the point where magnetization becomes nonzero on the corresponding curve.