Generalized effective-potential Landau theory for bosonic quadratic superlattices

We study the properties of the Bose-Hubbard model for square and cubic superlattices. To this end we generalize a recently established effective potential Landau theory for a single component to the case of multiple components and not only find the characteristic incompressible solid phases with fractional filling, but also obtain the underlying quantum phase diagram in the whole parameter region at zero temperature. A comparison of our analytic results with corresponding ones from quantum Monte Carlo simulations demonstrates the high accuracy of the generalized effective-potential Landau theory (GEPLT). Finally, we comment on the advantages and disadvantages of the GEPLT in view of a direct comparison with a corresponding decoupled mean-field theory.

Figure 1

Schematic illustration for a square superlattice in two dimensions. The square drawn with solid lines represents one type of unit cell. The solid curve (green) shows the optical potential in the $x$ direction. Lattice sites $A$ are deeper by $\Delta \mu$ than lattice sites $B$.

Figure 2

The quantum phase diagram of a bosonic 2D square superlattice in the whole parameter region from first-hopping-order GEPLT, and the phase diagram projected in the $t$-$\mu$ plane comparing the quantum Monte Carlo simulation (dotted line), the first-order (dashed line) and the second-order (solid line) analytical results at $\Delta \mu /U=0.5$.

Figure 3

The quantum phase boundaries of a bosonic 3D cubic superlattice at $\Delta \mu /U=0.5$ which is obtained by first-order (dashed line) and second-order (solid line) generalized effective-potential Landau theory, and a quantum Monte Carlo simulation (dotted line) in the thermodynamic limit. Inset: Finite-size scaling of the critical points of DW-I at $t/U=0.04125$.

Figure 4

Maxima of Mott-$n$ and DW-$n$ lobes as functions of $\Delta \mu$ for a 3D cubic superlattice.