Superconductivity in the kagome Hubbard model under the flat-band-preserving disorder

TL;DR

The authors demonstrate that a preserved flat band significantly boosts superconducting robustness against disorder in the kagome-lattice attractive Hubbard model. Using Bogoliubov–de Gennes mean-field theory and exact diagonalization, they compare flat-band-preserving disorder with random hopping disorder, showing that the geometric contribution to the superfluid weight remains substantial under FB-preserving disorder and that the linear $D_s$-$U$ behavior characteristic of flat bands persists, unlike the exponential behavior seen with dispersive-like (random hopping) disorder. The study also links flat-band states to a distinctive plateau and edge discontinuity in the OPDM occupation spectrum, with the plateau and jump more resilient under FB-preserving disorder. Overall, the work highlights the central role of flat-band geometry in stabilizing superconductivity and offers OPDM-based diagnostics for identifying flat-band effects in disordered interacting systems.

Abstract

We investigate the disordered flat-band superconductivity within the attractive Hubbard model on the kagome lattice by contrasting the flat-band-preserving disorder [Phys. Rev. B 98, 235109 (2018)] with the random hopping disorder that breaks the flat-band degeneracy. Through Bogoliubov-de Gennes mean-field calculations, we find that the superfluid weight is much more robust under the flat-band-preserving disorder, while the system eventually undergoes a transition to an insulator as disorder becomes strong enough. The almost linear interaction-dependence of the superfluid weight in the weak coupling limit found with the flat-band-preserving disorder confirms the persistent flat-band signature, whereas the exponential behavior of a dispersive-band character arises with the random hopping counterpart. In addition, in the exact diagonalization of the one-particle density matrix, we identify an occupation spectrum structure attributed to the flat-band states, demonstrating the connection between the resilient flat band and the enhanced robustness of superconductivity.

Figures (7)

• Figure 1: Flat band of the disordered kagome tight-binding model. (a) The structure of the kagome lattice. The dashed lines indicate the $L \times L$ supercells. The case of $L=2$ is exemplified. (b) The band structure for the clean system of the disorder strength $W = 0$. The density of states (DOS) is displayed for (c) the flat-band preserving disorder and (d) the random hopping disorder. The plots of DOS are made for $L = 128$ using the Lorentzian broadening of $0.01$ and averaged over $50$ different disorder realizations.
• Figure 2: Geometry of the clusters with (a) $N=12$ and (b) $N=24$ sites used for the exact diagonalization calculation of the one-particle density matrix and the occupation spectrum.
• Figure 3: Superconductor-insulator transition. The site-averaged pairing amplitude $\overline{\Delta}$, the superfluid weight $D_s$, and the energy gap $E_\mathrm{gap}$ are plotted as a function of disorder strength $W$ and compared between (a) the flat-band-preserving (FBP) disorder and (b) the random hopping (HOP) disorder. The panels (c) and (d) display the single-particle density of states (DOS) for selected values of $W$. The real-space BdG calculations are performed in the system of size $L = 24$ ($1728$ sites) with the interaction strength $U = 1$.
• Figure 4: Site occupancy (left) and local pairing amplitude (right). The sample profiles are collected for selected values of disorder strength $W$ using the BdG calculations with $U = 1$ in the system of $L=24$.
• Figure 5: Superfluid weight as a function of interaction strength $U$. Left and right panels correspond to the flat-band-preserving (FBP) an random hopping (HOP) disorders, respectively. In (c)--(f), the geometric contribution $D_s^\mathrm{geom}$ is compared with the total superfluid weight $D_s$ for the selected values of disorder strength, $W=0.1$ and $W = 0.4$. The dashed lines in (a)--(c) and (e) indicate the superfluid weight for the clean system ($W = 0$). The dashed lines in (d) and (f) are the curve fits to the form of $a\exp(-b/U)$. The momentum-space BdG calculations are performed in the system of $L = 6$ ($108$ sites).