Emergent Chiral Spin State in the Mott Phase of a Bosonic Kane-Mele-Hubbard Model

Recently, the frustrated $XY$ model for spins $1/2$ on the honeycomb lattice has attracted a lot of attention in relation with the possibility to realize a chiral spin liquid state. This model is relevant to the physics of some quantum magnets. Using the flexibility of ultracold atom setups, we propose an alternative way to realize this model through the Mott regime of the bosonic Kane-Mele-Hubbard model. The phase diagram of this model is derived using bosonic dynamical mean-field theory. Focusing on the Mott phase, we investigate its magnetic properties as a function of frustration. We do find an emergent chiral spin state in the intermediate frustration regime. Using exact diagonalization we study more closely the physics of the effective frustrated $XY$ model and the properties of the chiral spin state. This gapped phase displays a chiral order, breaking time-reversal and parity symmetry, but is not topologically ordered (the Chern number is zero).

Figure 1

(a) Honeycomb lattice and its first Brillouin zone. (b) Phase diagram of the BKMH model obtained using B-DMFT containing Mott insulator, uniform superfluid, and chiral superfluid phases with different regimes of the MI phase marked in italic. The central gray region corresponds to the states with no coplanar order. Parameters $U_{\uparrow \downarrow }/U=0.5$, $\mu /U_{\uparrow \downarrow }=0.5$, lattice of 96 sites. “Pentagons” mark parameter values that we further explore in Figs. 2.

Figure 2

Results of the B-DMFT for different values of ${(t_2/t_1)}^2=J_2/J_1$ for $h_z/U={10}^{-3}$, $U_{\uparrow \downarrow }/U=0.5$, $t_1/U=0.025$ on a lattice of 24 sites. (a)–(d) Different spin configurations. The color palette gives $⟨S_{{\mathit{r}}_i}^z⟩$, while arrows depict ordering in the $xy$ plane. (a) Uniform state with ferromagnetic ordering; (b) chiral spin state with no coplanar order; (c) a configuration of spiral states, in which each pseudospin is aligned with only one of its three first neighbors and antialigned with two of its six second neighbors; (d) a 120° configuration. (e) Absolute value of $|⟨S_{{\mathit{r}}_i}^z⟩|$. For each ratio $(t_2/t_1{)}^2$ we plot the result for all 24 sites and compare it to the classical solution. Pentagons mark results presented in (a)–(d). Note that for finite values of $h_z$ the border between the 120° Mott state and CSF is slightly shifted in favor of the Mott state.

Figure 3

Results of the ED. (a) Phase diagram of the frustrated $XY$ model. (b)–(d) Variation of the observables with the dimensionless parameter $J_2/J_1$ for different values of $h_z$, with $J_2^{\prime }=0.01J_1$, on a lattice of $6\times 2$ unit cells. (b) Difference of the average Ising magnetization on two sublattices $m$. (c) Scalar spin chirality $\chi$. (d) Pseudospin density wave structure factor $S_{\mathrm{PSDW}}(\mathbf{\Gamma })$. (e) Schematic representation of the perturbation term $H_{J_2^{\prime }}$.

Figure 4

ED calculations of the low energy spectra as a function of $J_2/J_1$ showing (using red circles) the formation of a quasidegenerate twofold ground-state manifold (a) on a lattice of $4\times 3$ unit cells for various $S_{\mathrm{Tot}}^z$; (b) on a lattice of $4\times 4$ unit cells in the $S_{\mathrm{Tot}}^z=0$ sector only. (c) Low energy spectrum as a function of the twist angle ${\theta }_1$ for $J_2/J_1=0.3$ and ${\theta }_2=0$ on a lattice of $4\times 3$ unit cells. (d) Berry curvature calculated using the non-Abelian formalism resulting in a vanishing Chern number shown for $J_2/J_1=0.3$, $h_z/J_1=J_2^{\prime }/J_1=0.02$ on a lattice of $4\times 3$ unit cells.