Quantum Geometry and Flat Band Bose-Einstein Condensation

We study the properties of a weakly interacting Bose-Einstein condensate (BEC) in a flat band lattice system by using the multiband Bogoliubov theory and discover fundamental connections to the underlying quantum geometry. In a flat band, the speed of sound and the quantum depletion of the condensate are dictated by the quantum geometry, and a finite quantum distance between the condensed and other states guarantees stability of the BEC. Our results reveal that a suitable quantum geometry allows one to reach the strong quantum correlation regime even with weak interactions.

Figure 1

(a) Kagome lattice geometry. The unit cell is shown as a blue parallelogram and black arrows are the basis vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. Purple lines depict nearest-neighbor (NN) hopping terms of strength $t$. (b) Bloch bands of the kagome lattice with $t=1$ along the path connecting the high-symmetry points shown in the inset. The lowest band is strictly flat. The black dot marks the Dirac point $\mathbf{k}=[4\pi /3,0]$ in which BEC can take place. (c) Speed of sound $c_s$ for the kagome flat band BEC as a function of $U$. Total density was chosen to be $n_{\mathrm{tot}}=3$, i.e., one particle per lattice site. We also show the weak-coupling result of Eq. (5) as a solid line. The energy scale $E_g=3t$ is the energy gap from the flat band to the dispersive bands at ${\mathbf{k}}_c$.

Figure 2

(a) Excitation fraction $n_{\mathrm{ex}}/n_{\mathrm{tot}}$ at $n_{\mathrm{tot}}=3$ as a function of $U$ for the flat band BEC ($t=1$) and dispersive-band BEC ($t=-1$). Purple triangle depicts the analytical result of Eq. (8) integrated over the first BZ. (b) Momentum dependence of $n_{\mathrm{ex}}(\mathbf{k})$ at $U{n}_0/E_g=5.13\times {10}^{-4}$. (c) Quantum distance $D(\mathbf{q})$ as a function of $\mathbf{k}={\mathbf{k}}_c+\mathbf{q}$. (b),(c) The red dot depicts the momentum ${\mathbf{k}}_c=[4\pi /3,0]$ of the flat band BEC. (d) Densities $n_0$ and $n_{\mathrm{ex}}$ as a function of $n_{\mathrm{tot}}$ for the flat band condensation at $U=|t|/1800$. Excitation density $n_{\mathrm{ex}}$ remains constant, as it is determined by the quantum distance.