Chiral Orbital Magnetism of $p$-Orbital Bosons in Optical Lattices

Chiral magnetism is a fascinating quantum phenomena that has been found in low-dimensional magnetic materials. It is not only interesting for understanding the concept of chirality, but also important for potential applications in spintronics. Past studies show that chiral magnets require both a lack of inversion symmetry and spin-orbit coupling to induce the Dzyaloshinskii-Moriya interaction. Here we report that the combination of inversion symmetry breaking and quantum degeneracy of orbital degrees of freedom will provide a new paradigm to achieve chiral orbital magnetism. By means of density matrix renormalization group calculation, we demonstrate that chiral orbital magnetism can be found when considering bosonic atoms loaded in the $p$ band of an optical lattice in the Mott regime. The high tunability of our scheme is also illustrated through simply manipulating the inversion symmetry of the system for cold atom experimental conditions.

Figure 1

(a) Phase diagram of the effective spin model in Eq. (2) as a function of the orbital exchange interactions $D_y/J_x$ and $J_z/J_x$. For certain $J_z/J_x$ there is a threshold of the orbital DM interaction $D_y/J_x$, beyond that the chiral orbital (CHO) phase appears. Dashed lines in different colors separated CHO from other nonchiral magnetic phases divided by colored solid lines. Other parameters are $J_y/J_x=2.8$ and $h/J_x=0.2$. Panels (b),(c) and (d),(e) show the spin-spin correlation ${\mathcal{S}}_{ij}^{\gamma }$ and chiral correlation ${\mathcal{K}}_{ij}^{\gamma }$ for the chiral orbital ($J_z/J_x=1$, $D_y/J_x=2.2$) and ferromagnetic orbital (FMO) ($J_z/J_x=-3.5$, $D_y/J_x=0.85$) phases, respectively. Other parameters are the same as in (a), and the system size is $L=150$ with the open boundary condition.

Figure 2

(a) Entanglement entropy $S_E$ as a function of $D_y/J_x$ with different system size $L$, calculated with open boundary condition when $J_y/J_x=2.8$, $J_z/J_x=1$, and $h/J_x=0.2$. The peak position of $D_y/J_x$ determines the phase transition point $(D_y/J_x{)}_{\text{Cri}}$. Panel (b) shows the finite size scaling of $(D_y/J_x{)}_{\text{Cri}}$.

Figure 3

The orbital exchange interactions as a function of the orbital hybridization $t/t_y$ when $V_x/E_{Rx}=6$ and $V_y/E_{Ry}=24$. Inset: Orbital hybridization $t/t_y$ as a function of the gradient magnetic field. Here the red and blue colors label the sign (positive and negative) of orbital exchange interaction, respectively.