Ferromagnetism beyond Lieb's theorem

Natanael C. Costa; Tiago Mendes-Santos; Thereza Paiva; Raimundo R. dos Santos; Richard T. Scalettar

doi:10.1103/PhysRevB.94.155107

paper

Ferromagnetism beyond Lieb's theorem

arXiv:1610.03566 · doi:10.1103/PhysRevB.94.155107

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 \emph{uniform} on-site Hubbard interaction $U$ is turned on, Lieb proved rigorously that at half filling ( $ρ = 1$ ) the ground state has a non-zero spin. In this paper we consider a `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. 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 study both the homogeneous (H) case, $U_{d} = U_{p}$ , originally considered by Lieb, and the inhomogeneous (IH) case, $U_{d} \neq = U_{p}$ . 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 $U_{p} \neq = U_{d}$ 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 $ρ = 1/3$ , strong antiferromagnetic correlations set in, caused by the presence of one fermion on each $d$ site.

10 pages, 14 figures

Ferromagnetism beyond Lieb's theorem

Abstract

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