Superpotentials, flat bands and the role of Quantum Geometry for the superfluid stiffness

TL;DR

The study demonstrates that imposing a periodic superpotential on a 2D superconductor can create flat electronic minibands and substantially raise the critical temperature $T_c$ via increased density of states. Unlike conventional flat-band systems, the finite kinetic energy from filled lower minibands ensures a robust superfluid stiffness $D_s$, aided by a positive geometric contribution from the quantum metric. Including phase fluctuations through BCS+RPA preserves gauge invariance and shows stiffness remains appreciable even when the top miniband is nearly flat, making the superconducting state coherent under moderate disorder. This moiré-inspired, tunable superlattice approach provides a general route to enhance superconductivity beyond twisted van der Waals materials and underscores the central role of quantum geometry in sustaining phase coherence in engineered flat-band systems.

Abstract

Enhancing superconductivity through material design is a central goal in quantum materials research. Moire engineering, where twisting stacked layers creates long-wavelength modulations and flat bands, has shown how electronic correlations can be amplified and eventually used to raise the superconducting critical temperature Tc. Yet this approach is largely confined to van der Waals materials and offers limited tunability. Here we explore a moire-inspired alternative: imposing artificial superpotentials on otherwise homogeneous systems to engineer flat electronic minibands. Whether such superlattice potentials can truly enhance superconductivity and sustain a finite superfluid stiffness remains, however, an open question. Our calculations show that a periodic superpotential imposed to a 2D system can indeed enhance superconductivity by reconstructing the electronic bands and creating regions of large density of states, leading to a substantial increase of Tc. In contrast to conventional flat band systems, where the superfluid stiffness arises solely from quantum geometry through the quantum metric, a modulated system inherits kinetic energy from the filled minibands below the Fermi level. This inherited component coexists with a positive quantum geometric contribution, yielding a finite and robust stiffness even when the upper band becomes nearly flat. The resulting superconducting state remains coherent and resilient against weak to moderate disorder. Our findings demonstrate that engineered superpotentials offer a tunable route to enhance superconductivity beyond twist based moire systems, unifying flat band amplification of pairing with preserved phase stiffness. They further highlight the central role of quantum geometry in shaping collective electronic phenomena and point to superlattice design as a promising platform for next-generation superconductors.

Figures (6)

• Figure 1: Realization and electronic properties of a device with periodidic gating. (a) Schematic of a 2D superconducting electron gas modulated by a periodic gate potential. A representative platform for realizing such a system is the LaAlO$_3$/SrTiO$_3$ interface. (b) Color surface plot of the modulated charge density for a checkerboard lattice of $4\times 4$ plaquettes. (c) Band structure for a diagonal cut across the reduced Brillouin zone. (d) Color surface plot of the modulated order parameter. Parameters: $U/t=1.5$, $V_0/t=4 (c)$, $V_0/t=2$ (b,d), charge concentration $p=0.5$.
• Figure 2: Critical temperature in the presence of a superpotential. Comparison of $T_c$ between homogeneous system ($V_0/t=0$, blue) and in the presence of a superpotential with $V_0/t=2$ (green) and $V_0/t=4$ (red). Inset: Ratio between $T_c$ with a finite superpotential and $T_c$ for the homogeneous system. The solid red line in (e) is a fit to $T_c/T_c^{hom}=\alpha |U| e^{\beta t/|U|}$ with $\alpha=0.0436$ and $\beta=6.663$ as described in the text. Charge concentration: $p=0.5$.
• Figure 3: Current flow through the periodically modulated superconductor. Illustration of the gauge invariant SC current distribution (white arrows) for a vector potential applied along the x-direction. The local gaps are shown as colored background. $U/t=2$, $V_0/t=4$, Charge concentration $n_{el}=0.5$.
• Figure 4: Impact of superpotential and gauge invariance on the superfluid stiffness. Interaction dependence of the stiffness $D_{s,tot}$for superpotential values $V_0/t=0$ (blue) and $V_0/t=4$ (red, squares) separated into the diamganetic (mostly intraband) ($D_{s}^{dia}$) and paramagnetic (mostly interband) ($D_{s}^{para}$) contribution. Also shown is the geometric contribution for $r\ne m$ (black) which positively contributes to the stiffness. Lines in the foreground (full symbols) and in the background (open symbols) refer to the RPA (BCS) result, respectively. For $|U|/t=0$ the charge stiffness $D_ c$ is shown by full circles while $D_s$ vanishes in this limit. charge concentration: $p=0.5$.
• Figure 5: Intra- and interband contributions to the superfluid stiffness. Color surface plot of ${\cal D}_{r,m}$, see Eq. (\ref{['eq:defdrm']}), divided into intraband (a) and interband (b) contributions. charge concentration $p=0.5$; $|U|/t=2$.