Canted magnetism and ${\mathbb{Z}}_2$ fractionalization in metallic states of the Lieb lattice Hubbard model near quarter filling

Abstract

A recent experiment has examined ultracold, fermionic, spin-1/2 ${}^6$Li atoms in the Lieb lattice at different Hubbard repulsion $U$ and filling fractions $\hat{I}½$ (Lebrat et al. arXiv:2404.17555). At $\hat{I}½=1/2$ and small $U$, they observe an enhanced compressibility on the $p_{x,y}$ sites, pointing to a flat band near the Fermi energy. At $\hat{I}½=1/2$ and large $U$ they observe an insulating ferrimagnet. Both small and large $U$ observations at $\hat{I}½=1/2$ are consistent with theoretical expectations. Surprisingly, near $\hat{I}½=1/4$ and large $U$, they again observe a large $p_{x,y}$ compressibility, pointing to a flat $p_{x,y}$ band of fermions across the Fermi energy. Our Hartree-Fock computations near $\hat{I}½=1/4$ find states with canted magnetism (and related spiral states) at large $U$, which possess nearly flat $p_{x,y}$ bands near the Fermi level. We employ parton theories to describe quantum fluctuations of the magnetic order found in Hartree-Fock. We find a metallic state with ${\mathbb{Z}}_2$ fractionalization possessing gapless, fermionic, spinless `chargons' carrying ${\mathbb{Z}}_2$ gauge charges which have a nearly flat $p_{x,y}$ band near their Fermi level: this fractionalized metal is also consistent with observations. Our DMRG study does not indicate the presence of magnetic order, and so supports a fractionalized ground state. Given the conventional ferrimagnetic insulator at $\hat{I}½=1/2$, the ${\mathbb{Z}}_2$ fractionalized metal at $\hat{I}½=1/4$ represents a remarkable realization of doping-induced fractionalization.