Magnetic phase transition in coherently coupled Bose gases in optical lattices

We describe the ground state of a gas of bosonic atoms with two coherently coupled internal levels in a deep optical lattice in a one-dimensional geometry. In the single-band approximation this system is described by a Bose-Hubbard Hamiltonian. The system has a superfluid and a Mott insulating phase that can be either paramagnetic or ferromagnetic. We characterize the quantum phase transitions at unit filling by means of a density-matrix renormalization-group technique and compare the results with a mean-field approach and an effective spin Hamiltonian. The presence of the ferromagnetic Ising-like transition modifies the Mott lobes. In the Mott insulating region the system maps to the ferromagnetic spin-1/2 $XXZ$ model in a transverse field and the numerical results compare very well with the analytical results obtained from the spin model. In the superfluid regime quantum fluctuations strongly modify the phase transition with respect to the well-established mean-field three-dimensional classical bifurcation.

Figure 1

Shown on top is the MI-SF phase transition predicted by the mean-field approach (dashed lines) and the DMRG approach (symbols). In the inset we show a typical finite-size scaling of ${\mu }^{+}$ and ${\mu }^{-}$ in the superfluid regime for $U_{ab}/U=1.8$. We characterize a charge gapless phase, i.e., superfluidity, by ${\mu }^{+}-{\mu }^{-}=0$ in the thermodynamic limit. The bottom shows the associated NP-FM transition calculated with the DMRG approach. All curves correspond to $J_{\mathrm{\Omega }}/U=0.1$ and solid lines are drawn as a guide for the eye.

Figure 2

Shown on top is the NP-FM transition in the MI phase for different values of the linear coupling $J_{\mathrm{\Omega }}/U$ for $U_{ab}/U=6$. The inset shows a comparison between numerical results and the critical exponent $1/8$ of the ITF for $J_{\mathrm{\Omega }}=0.1$. The bottom shows the NP-FM transition point calculated with the DMRG approach (symbols) and using the expression $J^2(1/U-1/U_{ab})=J_{\mathrm{\Omega }}/2$ (solid lines) [see the text and Eq. (4) for more details] for two values of $U_{ab}/U$ in the MI phase.

Figure 3

Shown on top is the behavior of $C_x\left(50\right)$ (open symbols) and $C_z\left(50\right)$ (closed symbols) across the MI-SF transition for $J_{\mathrm{\Omega }}/U=0.1$ (see Fig. 1). The bottom shows the long-range behavior of $C_z\left(i\right)$ in the SF (left) and MI (right) phases close to the phase transition.