Spiral spin textures of a bosonic Mott insulator with SU(3) spin-orbit coupling

We study the Mott phase of three-component bosons, with one particle per site, in an optical lattice by mapping it onto an SU(3) spin model. In the simplest case of full SU(3) symmetry, one obtains a ferromagnetic Heisenberg model. Introducing an SU(3) analog of spin-orbit coupling, additional spin-spin interactions are generated. We first consider the scenario of spin-dependent hopping phases, leading to Dzyaloshinskii-Moriya-type interactions. They result in the formation of spiral spin textures, which in one dimension can be understood by a local unitary transformation. Applying classical Monte Carlo simulations, we extend our study to two-dimensional systems, and systems with “true” spin-orbit coupling, involving spin-changing hoppings.

Figure 1

SU(2) vs SU(3). Transitions in a three-level system might span a SU(2) algebra with a single raising (lowering) operator $S^{\pm }$, as depicted in (a), but most generally span a SU(3) algebra with three pairs of raising lowering operators, $T^{\pm },U^{\pm }$, and $V^{\pm }$.

Figure 2

Spin textures in the presence of spin-dependent tunneling phases. Simulating annealing results for a $30\times 30$ lattice with periodic boundary for different tunneling phases $\alpha$. We plot projections of the local SU(3) spin vectors $\lambda$ into different spin planes. The ${\lambda }^{\left(3\right)}$- and ${\lambda }^{\left(8\right)}$-component is not plotted, as it is zero. Since the system behaves as a ferromagnet along $y$ direction, we only show the behavior along $x$. In (a), despite nonzero $\alpha$, the system remains ferromagnetic in all spin components. In (b)–(d), only the ${\lambda }^{\left(4\right)}$- and ${\lambda }^{\left(5\right)}$-components behave as a ferromagnet, while the other spin components show spiral order. In (b), we plot the case with only one spiral, while in (c), we plot the case with maximum winding number, that is, $N/2$ spirals, or antiferromagnetic order. In (d), we give an example for an incommensurate winding number, $m=7$. For all cases, we plot the corresponding unit cell.

Figure 3

System periodicity in the presence of a spin-dependent tunneling phase $\alpha$. We consider a system with 30 sites along the direction of nontrivial hopping and plot the winding number of the spirals and the corresponding size of the unit cell as a function of $\alpha$. The figure represents our Monte Carlo results for a two-dimensional system, cf. Fig. 2, but also illustrates the essence of the staircase formula, Eq. (20).

Figure 4

Spin textures in the presence of off-diagonal spin-orbit coupling. The plots show the projection of the local SU(3) spin vectors $\lambda$ into different spin planes, for a Mott system with the spin-orbit coupling of Eq. (25). We have studied a lattice of 30 sites in each direction, but here we only plot the unit cell. In (a), we restrict the dynamics to the $x$ direction, while (b) shows the results for the two-dimensional system.