Ferromagnetism in a two-component Bose-Hubbard model with synthetic spin-orbit coupling

We study the effect of the synthetic spin-orbit coupling in a two-component Bose-Hubbard model in one dimension by employing the density-matrix renormalization group method. A ferromagnetic long-range order emerges in both Mott-insulator and superfluid phases resulting from the spontaneous breaking of the $Z_2$ symmetry, when the spin-orbit coupling term becomes comparable to the hopping kinetic energy and the intercomponent interaction is smaller than the intracomponent one as well. This effect is expected to be detectable with the present realization of the synthetic spin-orbit coupling in experiments.

Figure 1

Phase diagrams with the first Mott lobe for $\lambda =0.1t,1.0t,4.0t$. For each $\lambda$, the region of the Mott-insulator phase is surrounded by the $\mu$ axis and a colorful curve connecting symbols, and the rest in the $t\text{−}\mu$ plane for the superfluid phase.

Figure 2

A $(t+\lambda ,\eta )$-phase diagram with $U^{\prime }=0.2U$. The phase diagram shows only $\eta \in [0.5,1]$ owing to its symmetry with respect to the axis of $\eta =0.5$ (see text). A star stands for a representative point in each phase.

Figure 3

Finite-size scaling of the maximum of $S_{\rho }$ to determine the critical values. Symbols represent $\eta$ with maximum $S_{\rho }$ for the chain length $L=24,32,48,64$.

Figure 4

Determining the critical value of the BKT transition between the FM-MI phase and the FM-SF phase at $\eta =0.5$. Top panel shows the charge gap ${\Delta }_c$ as a function of $t+\lambda$. Bottom panel gives a finite-size scaling analysis of ${\Delta }_c$. The critical value is estimated as $t+\lambda \simeq 0.12U$.

Figure 5

The same as Fig. 4 but for $\eta =0.95$, a transition from the PM-MI phase to the PM-SF phase. The critical value is estimated as $t+\lambda \simeq 0.115U$.

Figure 6

Excitation gaps ${\Delta }_c$, ${\Delta }_l^1$ and ${\Delta }_l^2$ are shown as a function of $1/L$ for the representative points in two Mott-insulator phases. Lines denote the results of FM-MI with symbols, and those of PM-MI without symbols otherwise.

Figure 7

Characteristic correlation functions are shown as a function of $|i-j|$ in corresponding phases. Black triangles for PM-MI phase, red dots for FM-MI phase, magenta circles for PM-SF phase, and blue squares for FM-SF phase. (a) $n_{ij}^{\tau }$ are shown for four phases. In two SF phases, $n_{ij}^{\tau }$ are shown fsackor $|i-j|\ge 10$. Blue solid and magenta dot-dashed lines fit numerical results (see text). (b) ${\mathcal{S}}_{ij}^y$ are shown in both FM-MI and FM-SF phases. They are finite in the large $|i-j|$ limit. (c) ${\mathcal{C}}_{ij}$ is shown for PM-MI phase. Algebraic dependence of ${\mathcal{C}}_{ij}$ on $|i-j|$ is found for the two PM phases, but almost shows no qualitative difference between such two phases so that we displays ${\mathcal{C}}_{ij}$ only for PM-MI phase.

Figure 8

Entanglement entropy $S_{\rho }$ are shown as a function of $lnL$ for four representative points in different phases, from which the central charges are deduced with the logarithmic law Eq. (7) and given in Table 1.