Flat-band Ferromagnetism of SU$(N)$ Hubbard Model on the Kagome Lattices

TL;DR

This work studies flat-band ferromagnetism of the SU($N$) Hubbard model on the kagome lattice, where a dispersionless flat band at energy $\epsilon_0=-2t$ enables localized hexagon states and a percolation-based description. By mapping the quantum problem to a classical $N$-state Pauli-correlated site-percolation model on a triangular lattice, with cluster degeneracy $d_{\text{SU}(N)}(|C|)=\frac{(N+|C|-1)!}{|C|!(N-1)!}$ and configuration weight $W(q)=\prod_i e^{\mu|C_i|} d_{\text{SU}(N)}(|C_i|)$, the authors perform sign-problem-free Metropolis Monte Carlo simulations. They observe a first-order para-ferro transition between densities $p_-$ and $p_+$ that rise with $N$ (e.g., SU($3$): $p_-\approx0.55$, $p_+\approx0.69$; SU($4$): $p_-\approx0.58$, $p_+\approx0.73$; SU($10$): $p_-\approx0.68$, $p_+\approx0.85$), indicating stronger entropic repulsion for larger SU symmetry. The phase diagram comprises paramagnetic, phase-separated, and unsaturated ferromagnetic regions, and the mapping provides a general, sign-problem-free framework for SU($N$) lattice ferromagnetism on frustrated geometries. The results have potential relevance for cold-atom realizations with SU($N$) symmetry and underscore the role of lattice geometry in correlation-driven order.

Abstract

The kagome lattice, a well known example of the geometrically frustrated system, hosts a dispersionless flat band that offers a unique platform for studying correlation-driven quantum phenomena. At appropriate particle concentrations, the existence of a flat band allows a representation of percolation with nontrivial weights. In this work, we investigate the paramagnetic-ferromagnetic transition in the repulsive SU($N$) Hubbard model on the kagome lattice within this percolation framework. In this representation, the model can be rigorously mapped to a classical $N$-state site-percolation problem on a triangular lattice, with the SU($N$) symmetry reflected in the nontrivial weights. By large-scale Monte Carlo simulations for SU($3$), SU($4$), and SU($10$) symmetries, we demonstrate that the critical particle concentration for ferromagnetism exceeds the standard percolation threshold and increases with $N$, indicating a strengthening of the effective entropic repulsion.

Figures (5)

• Figure 1: (a) Localized hexagon state on the kagome lattice, where the signs $+$ and $-$ label the wavefunction amplitude on each site around the hexagon. (b) A sketch of kagome lattice with lattice vectors $\mathbf{a}_1$ and $\mathbf{a}_2$, and the basis sites $A$, $B$, and $C$ in each unit cell (enclosed by dashed lines). (c) Single-particle dispersion of kagome lattice along high-symmetric lines in the Brillouin zone.
• Figure 2: An illustration of the mapping from the Hubbard model on a kagome lattice to the percolation problem on an effective triangular lattice. The red lines depict the localized states on the boundary of a double and triple plaquette, with alternating $+$ and $-$ signs labeling the amplitude of the wavefunction on each site. The filled circles depict the linked clusters of the $2$-color percolation model defined on a triangular lattice, which is formed by the centers of the hexagons (trapping cells).
• Figure 3: Particle concentration $p$ as a function of fugacity $z$ in the grand canonical ensemble for various system sizes $|\Lambda|=L\times L$ for the Hubbard model with (a) SU$(3)$, (b) SU$(4)$, and (c) SU($10$) symmetries.
• Figure 4: Finite-size scaling of the normalized macroscopic magnetic moment, $\langle S^2 \rangle / S^2_{\max}$, for the Hubbard model on a two-dimensional kagome lattice. Results are shown for SU($3$) [(a)-(c)], SU($4$) [(d)-(f)], and SU($10$) [(g)-(i)] symmetries, with maximum system sizes of $L=240$, $200$, and $110$, respectively. The corresponding particle concentrations $p$ are: (a) $0.50$, (b) $0.62$, (c) $0.72$ for SU($3$); (d) $0.51$, (e) $0.65$, (f) $0.76$ for SU($4$); and (g) $0.52$, (h) $0.76$, (i) $0.87$ for SU($10$).
• Figure 5: Snapshots of typical configurations for Pauli-correlated percolation for small deviations from critical concentration, by Monte Carlo simulation for SU($3$) Hubbard model. Snapshots for concentrations at (a) $p_1=0.48$ (paramagnetic), (b) $p_2=0.62$ (phase-separated), and (c) $p_3=0.73$ (ferromagnetic). Lattice extension $L=200$. Empty sites are white, and the largest cluster is magenta.