We investigate some thermodynamic and magnetic properties of the Hubbard model on two three-dimensional extensions of the Lieb lattice: the perovskite Lieb lattice (PLL) and the layered Lieb lattice (LLL). Using determinant quantum Monte Carlo (DQMC) simulations alongside Hartree-Fock and cluster mean-field theory approaches, we analyze how flat-band degeneracy, connectivity, and lattice anisotropy influence the emergence of magnetic order. Our results show that both geometries support finite-temperature magnetic transitions, namely, ferromagnetic (FM) on the PLL and antiferromagnetic (AFM) on the LLL. Further, we have established that the critical temperature
Tc, as a function of the uniform onsite coupling
U, displays a maximum, which is larger in the AFM case than in the FM one, despite the absence of flat bands in the LLL. We also provide numerical evidence to show that flat bands in the PLL rapidly generate magnetic moments, but a small interorbital coordination suppresses the increase of
Tc at large interaction strength
U/t. By contrast, the LLL benefits from higher connectivity, favoring magnetic order even in the absence of flat bands. The possibilities of anisotropic interlayer hoppings and inhomogeneous onsite interactions were separately explored. We have found that magnetism in the PLL is hardly affected by hopping anisotropy since the main driving mechanism is the preserved flat band; for the LLL, by contrast, spectral weight is removed from
d sites, which increases
Tc more significantly. At mean-field level, we have obtained that setting
U=0 on
p sites and $U={U}_{d}\ensuremath{\ne}0$ on
d sites leads to a quantum critical point at some
Ud; this behavior was not confirmed by our DQMC simulations.
The physics of strongly correlated quantum particles within a flat band was originally explored as a route to itinerant ferromagnetism and, indeed, a celebrated theorem by Lieb rigorously establishes that the ground state of the repulsive Hubbard model on a bipartite lattice with an unequal number of sites in each sublattice must have nonzero spin
S at half filling. Recently, there has been interest in Lieb geometries due to the possibility of topological insulator, nematic, and Bose-Einstein condensed (BEC) phases. In this paper, we extend the understanding of the attractive Hubbard model on the Lieb lattice by using determinant quantum Monte Carlo to study real space charge and pair correlation functions not addressed by the Lieb theorems. Specifically, our results show unusual charge and charge transfer signatures within the flat band, and a reduction in pairing order at $\ensuremath{\rho}=2/3$ and $\ensuremath{\rho}=4/3$, the points at which the flat band is first occupied and then completely filled. We compare our results to the case of flat bands in the Kagome lattice and demonstrate that the behavior observed in the two cases is rather different.
The properties of a phase at finite interactions can be significantly influenced by the underlying dispersion of the noninteracting Hamiltonian. We demonstrate this by studying the repulsive Hubbard model on the two-dimensional Lieb lattice, which has a flat band for vanishing interaction
U. We perform real-space dynamical mean-field theory calculations at different temperatures and dopings using a continuous-time quantum Monte Carlo impurity solver. Studying the frequency dependence of the self-energy, we find that a nonmagnetic metallic region at finite temperature displays non-Fermi-liquid behavior, which is a concomitant of the flat-band singularity. At half-filling, we also find a magnetically ordered region, where the order parameter varies linearly with the interaction strength, and a strongly correlated Mott insulating phase. The double occupancy decreases sharply for small
U, highlighting the flat-band contribution. Away from half-filling, we observe the stripe order, i.e., an inhomogeneous spin and charge density wave of finite wavelength, which turns into a sublattice ordering at higher temperatures.