Interplay of local and global quantum geometry in the stability of flat-band superfluids

Abstract

Quantum geometry strongly impacts physical properties in flat-band systems. We consider its role in bosonic condensation and superfluidity on flat bands, and show that the superfluid weight has an important contribution proportional to the condensate quantum metric. Based on this result, we uncover conditions under which flat-band superfluidity is unlikely. For instance, we find that stable flat-band superfluidity in a two-dimensional system requires at least three bands within Bogoliubov theory. Because the quantum geometry at the condensation momentum plays a disproportionately large role, a large integrated quantum metric is not sufficient for flat-band superfluidity, but how the quantum metric is distributed in the Brillouin zone is crucial.

Figures (2)

• Figure 1: (a) Kagome lattice and (b) corresponding band structure. In the kagome lattice, condensation occurs at the $K$ point (indicated by the red star), where the two dispersive bands above the flat band are degenerate. (c) Contributions $D_1$ (blue squares), $D_2^{\rm ndeg}$ (red crosses) and their sum $D_1+D_2^{\rm ndeg}$ (green diamonds) as a function of the on-site interaction $U$. In the kagome lattice, the superfluid weight is diagonal and proportional to the identity matrix, so we plot only the $xx$ component. The predictions relating these contributions to the condensate quantum metric, obtained from Eqs. \ref{['eq.d1']} and \ref{['eq.arbitrary']} (dashed blue, red and green lines) agree perfectly with our numerical results at all interactions. The more general relationship Eq. \ref{['eq.d2']}, valid also when the dispersive bands are not degenerate at $\boldsymbol{k_c}$, captures only the low-interaction behavior (gray dashed line). The dashed lines are plotted using the numerically obtained values for $n_0$ at each interaction. (d) Contributions $D_1+D_2^{\rm ndeg}$ (green diamonds), $D_3'$ (orange triangles), and total superfluid weight $D=D_1+D_2^{\rm ndeg}+D_3'$ (black dots) as a function of the interaction strength. Inset: $D_1+D_2^{\rm ndeg}$ (green diamonds) and $D_3'$ (orange triangles) at low $U$. The gray dashed line is obtained from the low-$U$ prediction Eq. \ref{['eq.d2']} for $D_1+D_2^{\rm ndeg}$, and agrees well with the numerical data. In the kagome lattice, $D_3'$ is positive at very small $U$, and changes sign at approximately $U\approx 0.13t$.
• Figure 2: (a) Kagome-III lattice and (b) corresponding single-particle band structure. The lowest band at $\epsilon_0=-t$ is degenerate. (c) Smallest possible value of $S=\sum_{\alpha}|\langle\alpha|n_{\boldsymbol{k}}\rangle|^4$ obtained from numerical minimization at each $\boldsymbol{k}$ when $|n_{\boldsymbol{k}}\rangle$ is a flat-band Bloch function. When considering condensation in a flat-band eigenstate, the minima of this sum correspond exactly to the minima of $E_{\rm MF} = n_0\epsilon_0-\mu n_0 + (Un_0^2/2)S$. The smallest possible value of $S=1/3$ is obtained whenever there exists a flat-band Bloch state such that $|\langle\alpha|n_{\boldsymbol{k}}\rangle|^2=1/3$ for all orbitals. (d) Left: White indicates all points where the result of the numerical minimization fulfills $|S-1/3|<10^{-10}$. Right: White indicates the region where $|(\cos^2(k_1)+\cos^2(k_2)-\cos^2(k_3))/2\cos(k_1)\cos(k_2)|<1$, where constructing a uniform flat-band Bloch state should be possible. The results of the numerical minimization are in good agreement with this analytical result.