Segregated quantum phases of dipolar bosonic mixtures in two-dimensional optical lattices

We identify the quantum phases in a binary mixture of dipolar bosons in two-dimensional optical lattices. Our study is motivated by the recent experimental realization of binary dipolar condensate mixtures of Er-Dy [Phys. Rev. Lett. 121, 213601 (2018)]. We model the system by using the extended two-species Bose-Hubbard model and calculate the ground-state phase diagrams by using mean-field theory. For selected cases we also obtain analytical phase boundaries by using the site-decoupled mean-field theory. For comparison we also examine the phase diagram of two-species Bose-Hubbard model. Our results show that the quantum phases with the long-range intraspecies interaction phase separate with no phase ordering. The introduction of the long-range interspecies interaction modifies the quantum phases of the system. It leads to the emergence of phase-separated quantum phases with phase ordering. The transition from the phase-separated quantum phases without phase ordering to phase ordered ones breaks the inversion symmetry.

Figure 1

Phase diagram of TBHM by varying the interspecies interaction strength $U_{12}$. Blue solid lines represent numerically obtained phase boundaries for the mean field Hamiltonian. Filled dots marks phase boundaries between compressible and incompressible phases, obtained analytically by perturbative analysis of the mean-field Hamiltonian. The odd occupancy Mott lobes appear for nonzero $U_{12}$ and enlarges with increasing $U_{12}$.

Figure 2

Phase diagram of TBHM at different temperatures for $U_{12}=0.4U$. Maroon-colored line represents the phase boundaries between the incompressible lobes and the superfluid phase, while the green line forms the boundary of region of the normal fluid (NF) phase.

Figure 3

Phase diagram of eTBHM at different interspecies interaction strength $U_{12}$ and for interspecies NN interaction $V_{12}=V_{21}=0$, $V_1=V_2=0.05U$. Maroon-colored line forms the boundary of the region comprised of incompressible phases (MI, DW). The filled black dots mark the phase boundaries obtained analytically by perturbative analysis of the mean field Hamiltonian. Around the density wave phase, a supersolid phase exists and the boundary between supersolid and superfluid phases is represented by green lines. The supersolid region around the density wave region enlarges with increasing $U_{12}$. In DW($n,0$) phase both species have DW($n,0$) pattern and in MI(1,1) phase both species have uniform unit occupancy.

Figure 4

Phase diagram of eTBHM at the different interspecies interaction strength $U_{12}$ and for interspecies NN interaction $V_{12}=V_{21}=0.05$, $V_1=V_2=0.05U$. The incompressible (MI, cDW) and compressible phase (SS, SF) regions are separated by maroon-colored lines. In correlated density wave phase the two species occupy lattice sites randomly in such a way such that total density $\rho ={\rho }^1+{\rho }^2$ have density wave pattern. And, around this phase, the supersolid phase exists and its boundary is marked by green lines. For $U_{12}=1.2$ the density wave and superfluid phases are phase separated.

Figure 5

Phase separation with a side-by-side pattern of species occupancies obtained with periodic boundary conditions along both the $x$ and $y$ axes. The density distribution of the species over lattice sites is shown in panels (a)–(c) for the cDW(2,1) phase, in panels (d)–(f) for the supersolid phase, and in panels (g)–(i) for the superfluid phase.

Figure 6

Phase separation with side-by-side pattern of species superfluid order parameter, obtained with periodic boundary conditions along both $x$ and $y$ axes. The superfluid order parameter at the lattice sites is shown in panels (a)–(c) for the supersolid phase and in panels (d)–(f) for the superfluid phase.