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 ``
CuO2 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,
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
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
Up=0,Ud=U and
Up=U,Ud=0. We found that the different environments of
d and
p sites lead to a ferromagnetic insulator when
Ud=0; by contrast,
Up=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.
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.
Using a cluster extension of the dynamical mean-field theory (CDMFT) we map out the magnetic phase diagram of the anisotropic square lattice Hubbard model with nearest-neighbor intrachain
t and interchain ${t}_{\ensuremath{\perp}}$ hopping amplitudes at half filling. A fixed value of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}=\ensuremath{-}{t}_{\ensuremath{\perp}}/2$ removes the nesting property of the Fermi surface and stabilizes a paramagnetic metal phase in the weak-coupling regime. In the isotropic and moderately anisotropic regions, a growing spin entropy in the metal phase is quenched out at a critical interaction strength by the onset of long-range antiferromagnetic (AF) order of preformed local moments. It gives rise to a first-order metal-insulator transition consistent with the Mott-Heisenberg picture. In contrast, a strongly anisotropic regime ${t}_{\ensuremath{\perp}}/t\ensuremath{\lesssim}0.3$ displays a quantum critical behavior related to the continuous transition between an AF metal phase and the AF insulator. Hence, within the present framework of CDMFT, the opening of the charge gap is magnetically driven as advocated in the Slater picture. We also discuss how the lattice-anisotropy-induced evolution of the electronic structure on a metallic side of the phase diagram is tied to the emergence of quantum criticality.
The Lieb lattice possesses three bands and with intrinsic spin orbit coupling
λ, supports topologically non-trivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction
U on all sites. Within a slave rotor mean field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (
Ucrit), creating a correlated flat band insulator. Beyond
Ucrit, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in presence of strong correlation. Furthermore, in the limiting case of
U introduced only on one sublattice where
λ=0, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.
The Hubbard model in the strong-coupling regime is mainly studied by Kondo-lattice theory or 1/d expansion theory, with d the spatial dimensionality. In two dimensions and higher, the ground state within the Hilbert subspace with no order parameter is a normal Fermi liquid except for n=1 and U/W=+infinity, with n the electron density per unit cell, U the on-site repulsion, and W the bandwidth; the cooperation between the Kondo effect, which favors a local singlet on each unit cell, and a resonating-valence-bond effect, which favors a local singlet on each pair of nearest-neighbor unit cells, stabilizes the Fermi liquid, whose ground state is a singlet as a whole, in the strong-coupling regime. In the whole Hilbert space with no restriction, the normal Fermi liquid is unstable at least against a magnetic or superconducting state. This analysis confirms an early Fermi-liquid theory of high-temperature superconductivity, F. J. Ohkawa, Jpn. J. Appl. Phys. 26, L652 (1987). The ground state for n=1 and U/W=+infinity is a Mott insulator. Actual metal-insulator transitions cannot be explained within the Hubbard model. In order to explain them, the electron-phonon interaction, multi-band or multi-orbital effects, and effects of disorder should be considered beyond the Hubbard model. The crossover between local-moment magnetism and itinerant-electron magnetism corresponds to that between a localized spin and a normal Fermi liquid in the Kondo effect and it is simply a Mott metal-insulator crossover.
The Lieb lattice possesses three bands and, with intrinsic spin-orbit coupling $\ensuremath{\lambda}$, supports topologically nontrivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction
U on all sites. Within a slave rotor mean-field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (
Ucrit), creating a correlated flat band insulator. Beyond
Ucrit, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in the presence of strong correlation. Furthermore, in the limiting case of
U introduced only on one sublattice where $\ensuremath{\lambda}=0$, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.