Geometric Origin of Superfluidity in the Lieb-Lattice Flat Band

The ground state and transport properties of the Lieb lattice flat band in the presence of an attractive Hubbard interaction are considered. It is shown that the superfluid weight can be large even for an isolated and strictly flat band. Moreover the superfluid weight is proportional to the interaction strength and to the quantum metric, a band structure quantity derived solely from the flat-band Bloch functions. These predictions are amenable to verification with ultracold gases and may explain the anomalous behavior of the superfluid weight of high-$T_c$ superconductors.

Figure 1

(a) The Lieb lattice and its unit cell (gray box) are shown. The orbitals in the unit cell are labeled by $\alpha =A$, $B$, $C$. The thick lines represent nearest-neighbor hoppings with energy $(1+\delta )J$, while the hopping energy corresponding to the thin lines is $(1-\delta )J$ with $0\le \delta \le 1$ parametrizing the staggered hopping. (b)–(c) The energy dispersion as a function of quasimomentum $\mathbf{k}$ for $\delta =0$ (b) and $\delta =0.3$ (c), respectively. The middle band is strictly flat ${ϵ}_{0\mathbf{k}}=0$ for any value of $\delta$ while the upper and lower band have dispersions ${ϵ}_{\pm ,\mathbf{k}}=\pm 2\mathrm{J}\sqrt{1+{\delta }^2+(1-{\delta }^2)(\cos k_{x}a+\cos k_{y}a)/2}$.

Figure 2

Order parameters ${\mathrm{\Delta }}_A/J$ (left) and ${\mathrm{\Delta }}_B/J$ (right) as a function of $\delta$ obtained with DMFT and mean-field BCS theory at temperatures $k_{B}T=5\times {10}^{-3}$ and ${10}^{-2}J$, filling $\nu =1.5$ and coupling strength $U=0.4J$. At these temperatures, significantly lower than the BCS critical temperature $k_B{T}_{c,\mathrm{BCS}}\approx {\mathrm{\Delta }}_A/2=U/8=5\times {10}^{-2}J$, the BCS results are indistinguishable from the zero temperatures ones.

Figure 3

Diagonal components of the superfluid weight tensor $[D_s{]}_{x,x}=[D_s{]}_{y,y}\approx D_s$ as a function of interaction $U/J$ and filling $\nu$ for $\delta ={10}^{-3}$ and at zero temperature. The superfluid weight for partially filled flat band ($1\le \nu \le 2$) depends strongly on $U$ in contrast to the other bands.

Figure 4

(a)–(b) Conventional superfluid weight $D_{s,\text{conv}}$ (blue area) compared with the geometric one $D_{s,\text{geom}}$ (red area) for $\nu =1.5$ (a) and $\nu =2.5$ (b). Here $T=0$ and $\delta ={10}^{-3}$. Also the Drude weight $D$ obtained from ED is shown. Squares and circles correspond to the 12 sites and 18 sites clusters, respectively. Data for the 24 sites cluster at $\nu =2.5$ (shown in Ref. [45]) do not deviate significantly with respect to 18 sites. (c)–(d) Various superfluid weight contributions for half-filled flat band, small $U\le 0.2J$, $\delta =5\times {10}^{-3}$ (c), $\delta =0.1$ (d).

Figure 5

Brillouin-zone integral of the flat-band quantum metric ${{\mathcal{M}}_{ij}^R|}_{\mathrm{f}.\mathrm{b}.}$ as a function of $\delta$. The diagonal components have a logarithmic singularity at $\delta =0$.