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arXiv:2505.16206 · Physical review. B./Physical review. B

Finite temperatures and flat bands: The Hubbard model on three-dimensional Lieb lattices

L. Oliveira-Lima, Julián Faúndez, Natanael C. Costa, Raimundo R. dos Santos

Published Aug 26, 2025• 4 citations
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Abstract

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 ${T}_{c}$, 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 ${T}_{c}$ 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 ${T}_{c}$ 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 ${U}_{d}$; this behavior was not confirmed by our DQMC simulations.

DOI
10.1103/qm8r-btz4

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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{T}_{c}Tc​, as a function of the uniform onsite coupling UUU, 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{T}_{c}Tc​ at large interaction strength U/tU/tU/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 ddd sites, which increases Tc{T}_{c}Tc​ more significantly. At mean-field level, we have obtained that setting U=0U=0U=0 on ppp sites and U=Ud\ensuremath≠0U={U}_{d}\ensuremath{\ne}0U=Ud​\ensuremath=0 on ddd sites leads to a quantum critical point at some Ud{U}_{d}Ud​; this behavior was not confirmed by our DQMC simulations.
We explore the magnetic properties of a two-dimensional Hubbard model on an inhomogeneous square lattice, which provides a platform for tuning the bandwidth of the flat band. In its limit, this inhomogeneous square lattice turns into a Lieb lattice, and it exhibits abundant properties due to the flat band structure at the Fermi level. By using the determinant quantum Monte Carlo simulation, we calculate the spin susceptibility, double occupancy, magnetization, spin structure factor, and effective pairing interaction of the system. It is found that the antiferromagnetic correlation is suppressed by the inhomogeneous strength and that the ferromagnetic correlation is enhanced. Both the antiferromagnetic correlation and ferromagnetic correlation are enhanced as the interaction increases. It is also found that the effective ddd-wave pairing interaction is suppressed by the increasing inhomogeneity. In addition, we also study the thermodynamic properties of the inhomogeneous square lattice, and the calculation of specific heat provides good support for our point. Our intensive numerical results provide a rich magnetic phase diagram over both the inhomogeneity and interaction.
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 UUU 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 ``CuO2{\mathrm{CuO}}_{2}CuO2​ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``ddd orbitals'' occupy the vertices of the squares, while ``ppp orbitals'' lie halfway between two ddd orbitals; both ddd and ppp 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 UUU and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, Ud=Up{U}_{d}={U}_{p}Ud​=Up​, 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 UUU. 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 Up=0,Ud=U{U}_{p}=0,{U}_{d}=UUp​=0,Ud​=U and Up=U,Ud=0{U}_{p}=U,{U}_{d}=0Up​=U,Ud​=0. We found that the different environments of ddd and ppp sites lead to a ferromagnetic insulator when Ud=0{U}_{d}=0Ud​=0; by contrast, Up=0{U}_{p}=0Up​=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 ddd site.
We describe the enhanced magnetic transition temperatures Tc{T}_{c}Tc​ of two-component fermions in three-dimensional layered Lieb lattices, which are created in cold-atom experiments. We determine the phase diagram at half filling using the dynamical mean-field theory. The dominant mechanism of enhanced Tc{T}_{c}Tc​ gradually changes from the ($\ensuremath{\delta}$-functional) flat-band to the (logarithmic) Van Hove singularity as the interlayer hopping increases. We elucidate that the interaction induces an effective flat-band singularity from a dispersive flat (or narrow) band. We offer a general analytical framework for investigating the singularity effects, where a singularity is treated as one parameter in the density of states. This framework provides a unified description of the singularity-induced phase transitions, such as magnetism and superconductivity, where the weight of the singularity characterizes physical quantities. This treatment of the flat-band provides the transition temperature and magnetization as a universal form (i.e., including the Lambert function). We also elucidate a specific feature of the magnetic crossover in magnetization at finite temperatures.
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 UUU. 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 UUU, 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.
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 SSS 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.
In itinerant systems, electron-electron interactions may lead to the formation of local magnetic moments and their effective exchange coupling, which in turn gives rise to long-range magnetic order. Therefore, when moment formation is weakened, such as in the single-band Hubbard model on a square lattice with the on-site repulsion being randomly switched off on a fraction xxx of sites, magnetic order is suppressed beyond some critical xc{x}_{c}xc​, which was found to lie below the classical percolation threshold xc(perc,sq){x}_{c}^{\text{(perc,sq)}}xc(perc,sq)​. Here we study dilute magnetism in flat band systems, namely, in the Hubbard model on a ``Lieb'' lattice. Interestingly, we show that magnetic order persists to xxx almost twice as large as the classical percolation threshold for the lattice, thus emphasizing the central role of electron itinerancy to the magnetic response. The analysis of the orbital-resolved order parameters reveals that the contribution of the fourfold coordinated ``ddd'' sites to magnetism is dramatically affected by dilution, while the localized ``ppp'' states of the flat band provide the dominant contribution to long-range correlations. We also examine the transport properties, which suggest the existence of an insulator-to-metal transition in the same range of the critical magnetic dilution.