Quantum phases of Bose-Bose mixtures on a triangular lattice

We investigate the zero-temperature quantum phases of a Bose-Bose mixture on a triangular lattice using the bosonic dynamical mean-field theory (BDMFT). We consider the case of total filling where geometric frustration arises for asymmetric hopping. We map out a rich ground-state phase diagram including $xy$-ferromagnetic, spin-density wave, superfluid, and supersolid phases. In particular, we identify a stripe spin-density wave phase for highly asymmetric hopping. On top of the spin-density wave, we find that the system generically shows weak charge (particle) density wave order.

Figure 1

(a) Ground-state phase diagram at unit filling $\rho =1$, with $V_a=V_b=48U$, obtained from calculations on a $6\times 6$ triangular lattice. We observe four major different phases, which are the homogeneous superfluid (HSF), $xy$-ferromagnet, SDW, and supersolid (SS). In the SDW region, the area where the SDW has a 3-sublattice structure is marked in yellow, while the area featuring the stripe SDW is marked red. The insets (a.1) and (a.2) show the corresponding $z$-magnetization distribution $\langle S_i^z\rangle$. The small blue area indicates a coexistence region of HSF and stripe SDW phases. (b) The triangular lattice considered here with sublattice indices $A,B,C$ marked near the corresponding lattice sites. ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are the two lattice vectors. (c) $t_b$ dependence of the 3-sublattice SDW order parameter at fixed $9t_a/U=0.6$.

Figure 2

(a–c) $z$-magnetization distribution $\langle S_i^z\rangle$ for different Mott-insulating phases with quantum magnetic order. (a) $xy$-ferromagnet for $9t_a/U=0.6$ and $9t_b/U=0.14$. (b) 3-sublattice SDW for $9t_a/U=0.7$ and $9t_b/U=0.1$. (c) Stripe SDW for $9t_a/U=0.7$ and $9t_b/U=0.05$. In all three cases, we set $V_{a,b}/U=48$. (d) Schematic figure where nearest neighbor, next-nearest neighbor, and next-next-nearest neighbor sites in a triangular lattice are indicated by the purple solid line, red dashed line, and blue dash-dotted line, respectively.

Figure 3

(a) In the stripe SDW phase region, we show the CDW order (blue solid line) and SDW order (black dashed line) as a function of $t_a$ when $9t_b/U=0.01$ is kept fixed. (b) The total density (${\rho }_i$) (upper panel) and the $z$-magnetization ($\langle S_i^z\rangle$) distribution (lower panel) for $V_{a,b}/U=48,9t_a/U=0.7$, and $9t_b/U=0.01$.