Superconductivity and geometric superfluid weight of a tunable flat band system

TL;DR

This work analyzes superconductivity and the geometric contribution to the superfluid weight in the tunable α-$\mathcal{T}_3$ lattice with on-site asymmetries that create an isolated quasi-flat band. Using a mean-field treatment of the attractive Hubbard model and linear-response theory, it decomposes the superfluid weight into conventional and geometric parts, highlighting the role of the quantum metric, which is enhanced by increasing $α$. At quasi-flat-band filling, pairing grows with a power-law-like dependence on the interaction $U$ due to a divergent density of states, and the geometric component can dominate near half-filling, especially as $α$ grows. Finite-temperature analysis shows the Berezinskii-Kosterlitz-Thouless transition temperature $T_{BKT}$ is strongly enhanced by $α$ via the amplified quantum geometry, positioning the α-$\mathcal{T}_3$ lattice as a tunable platform for geometry-driven superconductivity in quantum materials.

Abstract

We study superconductivity and superfluid weight of the two-dimensional $α$-$\mathcal{T}_3$ lattice with on-site asymmetries, hosting an isolated quasi-flat band with tunable bandwidth via a parameter $α$. Within a mean-field approximation of the attractive Hubbard model, we obtain the superconducting order parameters on the three inequivalent sublattices and show their strong dependence on $α$, interaction strength, and electron filling. At quasi-flat band filling, a superconducting gap opens and grows power-law fast with interaction strength, instead of the usual slow exponential growth, due to diverging density of states. We calculate the superfluid weight from linear response theory and study its band dispersion and geometric contributions. While the conventional part proportional to band derivatives is suppressed in the quasi-flat band regime, the contribution dominated by the quantum metric grows linearly for small interaction strength. We further demonstrate how tuning $α$ enhances the quantum metric and thus the geometric superfluid weight especially near half-filling, while increasing on-site asymmetries increases the conventional contribution by broadening the quasi-flat band. We obtain the Berezinskii-Kosterlitz-Thouless transition temperature and demonstrate its strong dependence and enhancement with the parameter $α$. Our results establish a tunable flat band system, the $α$-$\mathcal{T}_3$ lattice model, as a candidate for tunable quantum geometry and superfluid weight and as a prototype of related behavior in tunable quantum materials.

Figures (10)

• Figure 1: The $\alpha$-$\mathcal{T}_3$ lattice connects sublattices A and B with electron hopping energy $-t\cos\phi_\alpha$, and sublattices B and C with hopping energy $-t\sin\phi_\alpha$, where $\alpha=\tan\phi_\alpha$ and $\alpha\in(0,1]$. For small $\alpha$, the hopping amplitude between A and B sites dominates. As $\alpha\rightarrow0$, the C sites become disconnected.
• Figure 2: Energy bands and density of states of the $\alpha$-$\mathcal{T}_3$ lattice with $\alpha=0.6$ and $\varepsilon=0.5$ along a high-symmetry path in the Brillouin zone. Dotted lines correspond to $\varepsilon=0$, where the energy spectrum becomes $\alpha$-independent.
• Figure 3: Bandwidth of the quasi-flat band $\varepsilon_1({\bf k})$ ($i=1$ in Eq. \ref{['bands']}) as a function of $\alpha$ for several values of A-B site asymmetry $\varepsilon$.
• Figure 4: (a) Zero temperature superconducting order parameters $\Delta_i/U$ as a function of $U$ for $\alpha=0.6$. The inset shows the behavior at small $U$. In (b), we set $U=0.5$ and plot as a function of $\alpha$. In both cases $\varepsilon=0.5$ and $n_e=1$ (half-filling).
• Figure 5: Superfluid weight as a function of $U$ for $\alpha=0.6$ at half-filling ($n_e=1$). The inset shows the small $U$ regime, where the geometric contribution grows linearly with $U$. Dashed line shows the contribution from the quantum metric in Eq. \ref{['Dgeomg']}. Here we take $\varepsilon=0.5$.