Ferromagnetism beyond Lieb's theorem

Abstract

The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction $U$ is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``${\mathrm{CuO}}_{2}$ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``$d$ orbitals'' occupy the vertices of the squares, while ``$p$ orbitals'' lie halfway between two $d$ orbitals; both $d$ and $p$ orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of $U$ and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, ${U}_{d}={U}_{p}$, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all $U$. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases ${U}_{p}=0,{U}_{d}=U$ and ${U}_{p}=U,{U}_{d}=0$. We found that the different environments of $d$ and $p$ sites lead to a ferromagnetic insulator when ${U}_{d}=0$; by contrast, ${U}_{p}=0$ leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each $d$ site.