Vortexable Chern Bands Sit Exactly on the Topological Floor of Flat-Band Superfluid Stiffness

Technical Summary: Vortexable Chern Bands Sit Exactly on the Topological Floor of Flat-Band Superfluid Stiffness

Problem Statement
The paper addresses the relationship between the quantum geometry of flat Chern bands and the superfluid stiffness of time-reversed flat-band superconductors. While the superfluid weight in topological flat bands is bounded from below by the Chern number (the "topological floor"), the conditions under which this bound is saturated, and how deviations from the bound relate to the concept of "vortexability" (a criterion for ideal fractional Chern insulators), remain to be fully characterized. Specifically, the authors investigate whether vortexable bands—those that share the holomorphic structure of the lowest Landau level—are optimal for superconductivity in the sense of minimizing superfluid stiffness for a fixed Chern number, filling, and pairing amplitude.

Methodology
The authors employ a combination of rigorous analytical derivations and numerical verification across several lattice models:

1. Geometric Optimization: The study compares two distinct optimization problems over the space of orbital embeddings (displacements of orbitals within the unit cell):
   • Minimizing the integrated quantum metric (which determines the superfluid stiffness in the mean-field limit).
   • Minimizing the Ledwith-Vishwanath-Parker (LVP) measure of vortexability, defined as the minimum of the determinant of the quantum geometric tensor over lattice-periodic diffeomorphisms.
2. Gram Identity and Splitting Defect: The authors derive a "Gram identity" showing that the integrated metric is a quadratic form in orbital displacements. They define a "natural embedding" ($\eta_*$) that minimizes the metric over all possible orbital positions. They then restrict this minimization to "site-rigid" embeddings (where orbitals on the same physical site move together). The difference between the global minimum and the site-rigid minimum is a positive-semidefinite "splitting defect" ($\Sigma$).
3. Exact Stiffness Proofs: Leveraging recent proofs regarding "quantum-geometric-nesting" (QGN) models, the authors establish that for isolated, exactly flat bands with on-site, orbital-diagonal attraction and uniform pairing, the exact zero-temperature superfluid stiffness equals the mean-field value determined by the minimal quantum metric, provided the paired ground state is non-degenerate.
4. Numerical Verification: The theoretical identities are tested using self-consistent Bogoliubov-de Gennes (BdG) calculations, exact diagonalization, and exact pair mass calculations on several models: the Qi-Wu-Zhang (QWZ) band, the $\pi$-flux checkerboard band, the theta band, the honeycomb model of Li and Wu, and heterostrained twisted bilayer graphene.

Key Contributions and Results

• The Topological Floor Identity: The paper establishes an exact identity for the determinant of the superfluid stiffness tensor ($D_s$) relative to the topological floor ($D_s^{\text{floor}}$):
  $$\frac{\det D_s(0)}{\det D_s^{\text{floor}}(0)} = 1 + \left( \frac{T_{\text{orb}}}{2\pi|C|} \right)^2 \leq 1 + \left( \frac{T_{\text{LVP}}^{\text{cl}}}{2\pi|C|} \right)^2$$
  Here, $T_{\text{orb}}$ measures the departure from vortexability in the orbital-resolved frame, and $T_{\text{LVP}}^{\text{cl}}$ is the closure of the LVP measure over site-rigid frames.
• Vortexability as a Cost: The results demonstrate that vortexable bands (where $T_{\text{orb}} = 0$) sit exactly on the topological floor of superfluid stiffness. Any enhancement of the stiffness above this floor comes at the cost of reduced vortexability. The inequality is strict only if the splitting defect $\Sigma$ is non-zero.
• The Splitting Defect ($\Sigma$): The converse—that a band sitting on the floor is necessarily vortexable—fails only when internal components of the band share a physical site. In such cases, Cooper pairs can relax the phases of orbitals independently, whereas a vortex coordinate must move all orbitals at a site together. This mismatch creates a non-zero $\Sigma$, allowing the band to sit on the floor ($T_{\text{orb}}=0$) while having $T_{\text{LVP}}^{\text{cl}} > 0$.
• Model-Specific Findings:
  • Theta Band: A lattice model where orbitals are placed on one site versus two sites. When on two sites, the band is vortexable and sits on the floor. When forced onto one site, the stiffness remains on the floor (as it is independent of embedding), but the band is no longer vortexable ($T_{\text{LVP}}^{\text{cl}} > 0$), illustrating the role of $\Sigma$.
  • $\pi$-Flux Band: Sits on the boundary curve where $\Sigma = 0$ but $T_{\text{LVP}} > 0$, confirming the trade-off.
  • Heterostrained Twisted Bilayer Graphene: Numerical analysis of the chiral model with relaxation strain shows a non-zero splitting defect ($\Sigma \neq 0$) due to layer-dependent Fermi velocities, resulting in $T_{\text{orb}} > 0$ and $T_{\text{LVP}} > 0$.
  • Landau Levels: The $n$-th Landau level is shown to lie on the boundary curve with $R = (2n+1)^2$ in mean-field theory.

Significance and Claims
The paper claims to provide a definitive answer to the question posed by Ledwith, Vishwanath, and Parker regarding whether vortexable bands are "ideal" for superconductors. The authors conclude that for time-reversed flat-band superconductors with uniform pairing and a non-degenerate ground state:

1. Vortexable bands are the least stiff flat-band superconductors for a fixed Chern number, filling, and pairing amplitude.
2. The minimal quantum metric (fixing the exact stiffness) and the frame-optimized metric (measuring vortexability) are minima of the same quadratic form over nested spaces.
3. The "cost" of exceeding the topological floor of stiffness is paid in the loss of vortexability.
4. The paper explicitly states that these results concern band geometry and the paired state, not the energetics of competing phases (e.g., fractional Chern insulators vs. superconductivity). It does not claim that small $T_{\text{LVP}}$ proves the stability of fractional Chern insulators, but rather that it is evidence for the geometric conditions required for such states.

The work unifies the geometric criteria for ideal fractional Chern insulators and flat-band superconductivity, showing that the same geometric constraints (vortexability) that enable exact fractional quantum Hall states also minimize the superfluid stiffness in the superconducting channel.