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

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 (CMFT) 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 on-site coupling, $U$, displays a maximum, which is smaller 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 on-site interactions were separateley 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 $U=U_d\neq0$ on $d$ sites leads to a quantum critical point at some $U_d$; this behavior was not confirmed by our DQMC simulations.

Figures (12)

• Figure 1: 3D extensions of the Lieb lattice: (a) as a perovskite lattice (PLL) and (b) as stacked layers (LLL). The unit cell in the LLL configuration is identical to that of the 2D case, consisting of $d$, $p^x$, and $p^y$ sites in the $xy$ plane. In the PLL model, an additional $p$ site is introduced along the $z$-axis, with $d$, $p^x$, $p^y$, and $p^z$ sites forming the unit cell. Solid and dashed lines denote hopping amplitudes in the $xy$-plane, $t_{xy}$, and along the $z$-direction, $t_z$, respectively.
• Figure 2: Site-resolved non-interacting DOS for tight-binding fermions on (a) PLL and (b) LLL. The energy $\hbar\omega$ is measured relative to the $\epsilon_F$, assuming half filling.
• Figure 3: Internal energy and specific heat as a function of the temperature (linear-log scale) for various values of $U/t$ and fixed lattice size $\tilde{L}=(4,4,4)$, shown for PLL (left panels) and LLL (right panels), respectively. Panels (a) and (d): Symbols represent internal energy data from DQMC simulations, while lines show exponential fits using the function $f_{\text{fit}}$ (see text). Panels (b) and (e): Specific heat as a function of temperature for fixed $U/t=4.0$, with symbols indicating numerical differentiation of the DQMC internal energy data in panels (a) and (d), and lines showing differentiation of the fitted function $f_{\text{fit}}$. Panels (c) and (f): Specific heat as a function of temperature, derived from the full differentiation procedure of $f_{\text{fit}}$ for various values of $U/t$.
• Figure 4: (a) Linear-log plot of the local moment at $d$-sites of the PLL as a function of temperature, for different values of $U/t$. (b) Same as (a), but for $p$-sites of the PLL. (c) Local moment on $p$ (empty symbols) and $d$ (filled symbols) sites of the PLL as a function of $U/t$, at fixed temperature $T/t = 0.1$. (d)–(f): Corresponding data for the LLL. All results correspond to lattice size $\tilde{L} = (4,4,4)$.
• Figure 5: Global magnetic structure factor as functions of inverse temperature for (a) the PLL and (b) the LLL. DQMC data are for different lattice sizes, $\tilde{L}$, and fixed $U/t = 5.0$; solid lines are guides to the eye.