Correlated Phases of Bosons in the Flat Lowest Band of the Dice Lattice

We study correlated phases occurring in the flat lowest band of the dice-lattice model at flux density one-half. We discuss how to realize this model, also referred to as the ${\mathcal{T}}_3$ lattice, in cold atomic gases. We construct the projection of the model to the lowest dice band, which yields a Hubbard Hamiltonian with interaction-assisted hopping processes. We solve this model for bosons in two limits. In the limit of large density, we use Gross-Pitaevskii mean-field theory to reveal time-reversal symmetry breaking vortex lattice phases. At low density, we use exact diagonalization to identify three stable phases at fractional filling factors $\nu$ of the lowest band, including a classical crystal at $\nu =1/3$, a supersolid state at $\nu =1/2$, and a Mott insulator at $\nu =1$.

Figure 1

Figure showing the structure of the dice lattice, highlighting a rectangular unit cell and negative hoppings on three bonds (double hatched) that realize the case of flux density $n_{\varphi }=\frac{1}{2}$. The blue contour highlights the extent of a maximally localized single-particle wave function in the flat lowest band. Localized states centered around all sixfold connected sites of the lattice (encircled) form an orthonormal basis for this band and lie on a triangular lattice.

Figure 2

(a) The projective dynamics within the low energy band of the dice lattice at flux $n_{\varphi }=1/2$ realizes an effective model on a triangular lattice with an enlarged magnetic unit cell of two distinct triangular sites $A$ and $B$. (b) The density-density interactions of the dice Hamiltonian (1) give rise to the five distinct processes (i)–(v), including an on-site (i) and nearest-neighbor (ii) interactions, coherent hopping of pairs of particles onto the same (iii) and two distinct neighboring sites (iv), and a stimulated hopping process (v). Hoppings are shown as arrows, density interactions as dotted lines. Processes (iv) and (v) have a nontrivial dependency on the gauge from (vi); see main text. (c) Signs for the processes (iv),(v) in the real gauge of Fig. 1: these processes involve sites on a triangular plaquette, of which there are four types classified by the participating sublattice indices [empty ($A$) or full ($B$) circles]. In addition, one of the three sites carries two creation or annihilation operators (marked by a hexagon).

Figure 3

(a) Energy per particle as a function of the filling factor $\nu =3n$ of the low-lying band for the projected Hamiltonian (2) at $n_{\varphi }=1/2$ with hard-core interactions. Finite size effects between different lattice geometries and the effect of twisted boundary conditions (bandwidth shown as error bars for $3\times 3$ and $3\times 4$ geometries) are small. Inset: Energy measured with respect to the tangent at points $\nu =1/2$ and $\nu =1$ (dashed line in main panel), showing that $\nu =1/2$ is the only stable phase between the crystal at $\nu =1/3$ and Mott state at $\nu =1$ for $u\gg 1$. (b) Second difference of the energy per particle. Maxima indicate potential incompressible phases, including the states at $\nu =1/3$, $1/2$, and 1 as well as an additional weaker candidate at $\nu =2/3$ that is overridden by phase separation.

Figure 4

Data for $N=8$ atoms at $\nu =1/2$ on a lattice of $4\times 2$ unit cells. (a) Two-point correlation function $⟨{\widehat{n}}_r{\widehat{n}}_0⟩$ of the ground state $|0⟩$ for $\overrightarrow{\theta }=(0,\pi )$. The reference site “0” is visually highlighted. Axes carry units $a=|{\overrightarrow{v}}_1|$, $h=\sqrt{3/4}|{\overrightarrow{v}}_1|$. (b) Density $⟨{\widehat{n}}_r⟩$ of the symmetry-broken crystal state $|S⟩$ (see main text). (c) Fluctuations $⟨{\widehat{n}}_r^2⟩-⟨n_r{⟩}^2$ (blue circles) and condensate wave function ${\overrightarrow{v}}_0$ of $|S⟩$ (red arrows). (d) Scaling of the condensate fraction ${\lambda }_0/N$ against $N^{-1}$, for superposition states $|S⟩$ and pinned states with pinning centers of strength $V_p$ on crystal sites.