Superfluidity of Bosons in Kagome Lattices with Frustration

In this Letter we consider spinless bosons in a kagome lattice with nearest-neighbor hopping and on-site interaction, and the sign of hopping is inverted by insetting a $\pi$ flux in each triangle of the kagome lattice so that the lowest single particle band is perfectly flat. We show that in the high-density limit, despite the infinite degeneracy of the single particle ground states, interaction will select out the Bloch state at the $K$ point of the Brillouin zone for boson condensation at the lowest temperature. As the temperature increases, the single-boson superfluid order can be easily destroyed, while an exotic triple-boson paired superfluid order will remain. We establish that this trion superfluid exists in a broad temperature regime until the temperature is increased to the same order of hopping and then the system turns into normal phases. Finally, we show that time-of-flight measurement of the momentum distribution and its noise correlation can be used to distinguish these three phases.

Figure 1

(a) Kagome lattice, partitioned into $A$, $B$, $C$ sublattices. (b) Band structure with inverted hopping sign, where $\Gamma =(0,0)$, $K=(2\pi /3,0)$, and $M=(\pi /2,\pi /2\sqrt{3})$. (c) Mean-field energy landscape for different $U\rho /t$. Energy shifted by $E_0=-2t+U\rho /3$.

Figure 2

(a), (b) Phase configurations of the condensates at the $\Gamma$ point (a) and the $K$ point (b), (c), (d) random 3-color arrangement. Phase ${\theta }_i$ is denoted by colors: $\mathrm{\text{red circle}}=0$, $\mathrm{\text{green square}}=2\pi i/3$, $\mathrm{\text{blue triangle}}=-2\pi i/3$. “$\pm$” mark out the vorticity around each triangle. (a) ${\psi }_{\Gamma }$ is a “vorticity ferromagnetic” state and (b) ${\psi }_K$ is a “vorticity antiferromagnetic” state.

Figure 3

(a) Bogoliubov spectra for $\Gamma$-point (left) and $K$-point (right) condensates at $U\rho /t=0.3$. Momentum points are defined as $A=(\pi /3,0)$ and $B=(\pi /3,\pi /3\sqrt{3})$. (b) Distribution of the zero-point energy (ZPE) $\Lambda$ of the degenerate mean-field configurations. Two dashed lines denote the ZPE for $K$- and $\Gamma$-point condensates. The vertical axis is the probability for the particular ZPE. (c) $J/t$ vs $U\rho /t$ plot.

Figure 4

(a) Phase diagram. It contains three phases: the conventional superfluid phase with boson condensation at the $K$ point (KSF), trion superfluid (trion SF) phase, and the normal phase. (b) The long-range correlation for $⟨{\widehat{b}}_i^{†}{\widehat{b}}_j⟩$ and $⟨{\widehat{b}}_i^{†3}{\widehat{b}}_j^3⟩$ as a function of temperature $k_{B}T/t$. (c) TOF intensity at the $\Gamma$ and the $K$ point as a function of temperature $k_{B}T/t$. (b) and (c) are calculated at $U\rho /t=1$.

Figure 5

(a)–(c) TOF image of momentum distribution for three phases. (a) KSF, (b) trion SF, (c) normal. The thin lines of the honeycomb trace out the Brillouin zone boundary. For (b) and (c), the image is averaged over many times of TOF experiments. (d) Profile of $C(\mathit{k})$ simulated in the trion SF phase by analyzing the noise correlation among 100 TOF images.