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.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.

arXiv:2505.16206Physical review. B./Physical review. BAug 26, 20254 citations

Quantum Monte Carlo study of spin chains

Wataru Yokozeki, Kazuo Shiiki, Mitsuru Fukuchi

Physica A Statistical Mechanics and its ApplicationsJan 1, 19960 citations

Ferromagnetism beyond Lieb's theorem

Natanael C. Costa, Tiago Mendes-Santos, Thereza Paiva +2 more

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.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.

arXiv:1610.03566Physical review. B./Physical review. BOct 6, 201635 citations

Spin and charge density waves in the Lieb lattice

José Daniel Gouveia, R. G. Dias

arXiv:1505.01656Journal of Magnetism and Magnetic MaterialsJan 2, 201615 citations

Quantum Magnetism at High Resolution

Sergey Shinkevich

Jan 1, 20130 citations

{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

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.

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.