Order-by-disorder degeneracy lifting of interacting bosons on the dice lattice

Motivated by recent experimental progress in the realization of synthetic gauge fields in systems of ultracold atoms, we consider interacting bosons on the dice lattice with half flux per plaquette. All bands of the noninteracting spectrum of this system were previously found to have the remarkable property of being completely dispersionless. We show that degeneracies remain when interactions are treated at the level of mean-field theory, and the ground state exhibits vortex lattice configurations already established in the simpler $XY$ model in the same geometry. We argue that including quantum and thermal fluctuations will select a unique vortex lattice up to overall symmetries based on the order-by-disorder mechanism. We verify the stability of the selected state by analyzing the condensate depletion. The latter is shown to exhibit an unusual nonmonotonic behavior as a function of the interaction parameters which can be understood as a consequence of the dispersionless nature of the noninteracting spectrum. Finally, we comment on the role of domain walls which have interactions mediated through fluctuations.

Figure 1

Dice lattice under an effective magnetic field using the gauge of Ref. [31]. There are two types of sites: hub sites with a coordination number of 6 and rim sites with a coordination number of 3. The links and sites outlined in orange comprise the half-flux-per-plaquette magnetic unit cell. Particles acquire a phase of $\pi$ when hopping across crossed links and no phase when hopping across uncrossed links. The lattice vectors ${\mathit{v}}_1$ and ${\mathit{v}}_2$ can be chosen to be orthogonal, as in the figure where ${\mathit{v}}_1=(1,0)$ and ${\mathit{v}}_2=(0,\sqrt{3})$. For convenience, we have set the lattice constant to unity. The lattice can thus be viewed as a rectangular Bravais lattice with a sixfold basis. The basis vectors are ${\mathit{b}}_1=\mathbf{0},{\mathit{b}}_2=(1/2,1/2\sqrt{3}),{\mathit{b}}_3=(1/2,-1/2\sqrt{3}),{\mathit{b}}_4=(1/2,\sqrt{3}/2),{\mathit{b}}_5={\mathit{b}}_2+{\mathit{b}}_4$, and ${\mathit{b}}_6={\mathit{b}}_3+{\mathit{b}}_4$. In the absence of a gauge field the smallest unit cell is half the size and we can take ${\mathit{v}}_1$ and ${\mathit{b}}_4$ for the lattice vectors and ${\mathit{b}}_{1-3}$ for the three basis vectors.

Figure 2

The four small unit-cell periodic mean-field ground states. The single, double, and triple arrows represent gauge-invariant phase differences ${\Phi }_s,{\Phi }_m$, and ${\Phi }_l$ across links, respectively, and the black (white) disks represent positive (negative) plaquette vorticities. The dashed and dotted lines signify locations of possible domain-wall insertions [subfigure (a)] or domain walls themselves (all other subfigures). The dashed orange lines represent type I domain walls, while the blue dotted lines represent type II domain walls.

Figure 3

The twelve Bogoliubov modes about ground state (b) from Fig. 2 at $U_{*}=U_{\Delta }$, $U_{*}/t=0.5$, and $n_{*}=6$. As the interaction strengths $U_{*,\Delta }$ decrease, the bands flatten and the gaps between them approach $\sqrt{6}t$. At $U_{*,\Delta }=0$ we recover the dispersionless degenerate single-particle spectrum.

Figure 4

Left: quantum free-energy difference per condensed particle with respect to state (b) from Fig. 2 with $n_{*}=6\text{and}{U}_{*}=U_{\Delta }$. Right: total free-energy difference at the same $n_{*}$ and $U_{*}/U_{\Delta }$, for finite temperature and $U_{*}/t=0.5$.

Figure 5

Quantum, thermal, and total depletion per condensed particle for a system consisting of $20\times 20$ unit cells at $n_{*}=6,U_{*}=U_{\Delta }$, and $T=t/10$.