Quantum Melting of Valence-Bond Crystal Insulators and Novel Supersolid Phase at Commensurate Density

Bosonic and fermionic Hubbard models on the checkerboard lattice are studied numerically for infinite on-site repulsion. At particle density $n=1/4$ and strong nearest-neighbor repulsion, insulating Valence-Bond crystals (VBC) of resonating particle pairs are stabilized. Their melting into superfluid or metallic phases under increasing hopping is investigated at $T=0\mathrm{K}$. We identify a novel and unconventional commensurate VBC supersolid region, precursor to the melting of the bosonic crystal. Hardcore bosons (spins) are compared to fermions (electrons), as well as positive to negative (frustrating) hoppings.

Figure 1

(a) Topological defects of fractional charge $-e/2$ (2-particle tetrahedron) and $+e/2$ (empty tetrahedron) and particle-hole excitation with two pairs of NN defects. (b) Artistic view of the mixed-VBC supersolid phase at commensurate density $n=1/4$ showing inhomogeneous charge density (dots), resonating plaquettes (dark), and superfluid density (light).

Figure 2

Phase diagram of the $t>0$ bosonic hardcore Hubbard model as a function of the parameter $t/V$ and the chemical potential $\mu$. Extrapolation to the thermodynamic limit of $T=0\mathrm{K}$ GFQMC results in the canonical ensemble (dots) are compared to ED results. The two lobes correspond to the $n=1/4$ (bottom) and $n=1/2$ (top) VBC insulators (see text). A region of supersolid phase is found at commensurate $n=1/4$ filling (thick segment).

Figure 3

ED results obtained on a 32-site cluster for both fermions (empty symbols) and bosons (filled symbols): (a) charge (circles) and optical (squares) gaps, all in units of $V$, (b) order parameter $M_{+}(K)$, (c) Drude weight and, (d) average number of $-e/2$ charge defects. For fermions, discontinuities due to a level crossing at $t/V\simeq 0.035$ might be a finite size effect.

Figure 4

GFQMC results for (hardcore) bosons with $t>0$. Thermodynamic limit of the charge gap in units of $V$ (a), the order parameters $M_{+}(K)$ and $M_{-}(\Gamma )$ with the Brillouin zone depicted in the inset (b), the stiffness parameter rescaled by the number of sites $N$ (c), and the average number of $-e/2$ charge defects (d). The same scale is used to plot these quantities versus $t/V$ to emphasize the transition point close to 0.085. Size scalings are depicted in the insets.

Figure 5

$\sigma (\omega )$ for (a) the fermions and (b) the bosons at different values of $t/V$ (ED on a 32-site cluster). Insets: relative weight of the low-energy dipolar matrix elements [only visible for fermions on the $\sigma (\omega )$ plot], plotted vs $t/V$ in log-log scale.