Fermionization of bosons in a flat band

Strongly interacting bosons that live in a lattice with degeneracy in its lowest-energy band experience frustration that can prevent the formation of a Bose-Einstein condensate. Such systems form an ideal playground to investigate spin-liquid behavior. We use the variational principle and the Chern-Simons technique of fermionization of hard-core bosons on the kagome lattice to find that below lattice filling fraction $\nu =\frac{1}{3}$ the system favors a topologically ordered chiral spin-liquid state that is gapped in bulk, spontaneously breaks time-reversal symmetry, and supports massless chiral bosonic edge mode. We construct the many-body variational wave function of the state and show that the corresponding energy coincides with the energy of the flat band. This result proves that the ground state of the system cannot stabilize a Bose condensate below $\nu =\frac{1}{3}$. The fermionization and variational scheme we outline apply to any non-Bravais lattice. We distinguish between the roles played by the Chern-Simons gauge field in lattices with a flat band and those exhibiting a moatlike dispersion (which is degenerate along a closed contour in the reciprocal space). We also suggest experimental probes to differentiate the proposed ground state from a condensate.

Figure 1

(a) The electronic structure of kagome lattice has three bands, with the lowest band ($E_1$) being flat. The red and blue dots denote the population of $E_1$ and $E_2$ bands. (b) The real-space depiction of the $\nu =\frac{1}{9}$ state. The shaded region denotes occupancy by a boson. The wave functions do not overlap. (c) The real-space depiction of the $\nu =\frac{1}{3}$ chiral spin-liquid state obtained from fermionization.

Figure 2

(a) A unit cell of the kagome lattice. (b) The unit cell redrawn with a shift. This explicitly shows that the cell area includes three atoms. The dashed lines are the internal bonds chosen to not include any atoms in the triangular loop. All the particles, and hence the flux, is contained within the hexagon. (c) Flux attachment within the unit cell of the kagome lattice. $A_i$ denotes the phase accumulated by $\mathbf{A}\left(\mathbf{r}\right)$ while traversing the direction ${\mathbf{a}}_i$ such that $A_1=A_2+A_3$.

Figure 3

Extended scheme for the flux attachment. Note that ${\varphi }_T$ grows to account for the area law, where as ${\varphi }_C$ is the same in every unit cell. The flux through the hexagon and the unit cell is $8{\varphi }_T$ and through any triangle is ${\varphi }_T-{\varphi }_C$. In our MFA ${\varphi }_T={\varphi }_C$.

Figure 4

Comparison of the kagome energy spectrum in a CS field for two different flux distributions (a) and (b). The flat band is preserved in (b) where the flux through the triangle is zero. In this case, the spectrum is unique only up to $\nu =\frac{1}{3}$.

Figure 5

Ground-state energy of noninteracting fermions in kagome lattice subject to the CS flux at mean field level. Energy begins to rise after $\nu =\frac{1}{3}$ filling.

Figure 6

(a) The kagome lattice has three atoms per unit cell marked $a,b,c$. For each atom, we consider the first-neighbor hoppings ($J_1$) shown by the dashed lines. ${\mathbf{a}}_{1,2}$ are the lattice translation vectors. (b) One possible choice of the unit cell. All atoms and bonds wholly belong to the chosen unit cell. The dashed lines are the bonds that extended to the neighboring unit cells and the solid lines are the bonds within this unit cell.