Flat-Band Ferromagnetism as a Pauli-Correlated Percolation Problem

We investigate the location and nature of the para-ferro transition of interacting electrons in dispersionless bands using the example of the Hubbard model on the Tasaki lattice. This case can be analyzed as a geometric site-percolation problem where different configurations appear with nontrivial weights. We provide a complete exact solution for the one-dimensional case and develop a numerical algorithm for the two-dimensional case. In two dimensions the paramagnetic phase persists beyond the uncorrelated percolation point, and the grand-canonical transition is via a first-order jump to an unsaturated ferromagnetic phase.

Figure 1

Left: Two-dimensional Tasaki lattice. A trapping cell contains five sites (dashed red lines). The green circles and lines show the 1D variant of the lattice (sawtooth chain). Right: Snapshots of configurations for standard and Pauli-correlated percolation for small deviations from critical concentration. Panels (a) and (b) show snapshots (lattice extension $\mathcal{L}=200$) of configurations for standard percolation for concentrations $p_1=0.574$ and $p_2=0.6$ ($p_c=0.592746\dots$), while panels (c), (d), (e), and (f) show snapshots for Pauli-correlated percolation for $p_3=0.62$ (paramagnetic), $p_4=0.65$, $p_5=0.7$ (phase-separated), and $p_6=0.78$ (ferromagnetic). Pink color denotes the largest cluster.

Figure 2

Left: $n(l)$ (top) and $g(l)$ (bottom) at $p=0.99$ for PCP (solid line) and standard percolation (dotted line) in 1D. Right: Deviation of $⟨{\mathit{S}}^2⟩/{\mathit{S}}_{\max }^2$ from saturation for large $p$ for PCP and standard percolation in 2D.

Figure 3

Density of electrons $p$ vs chemical potential $\mu$, controlling the filling of the flat band in 2D. Inset: Histograms of density for a “finite-size” critical value $\mu ={\mu }_c$.

Figure 4

Left: Finite-size scaling of the square of the magnetic moment $⟨{\mathit{S}}^2⟩/{\mathit{S}}_{\max }^2$ for the 2D Tasaki lattice (up to $\mathcal{L}=270$). Right: Normalized number of clusters $n(l)$ for $\mathcal{L}=200$.