Flat-band ferromagnetism in the multilayer Lieb optical lattice

We theoretically study magnetic properties of two-component cold fermions in half-filled multilayer Lieb optical lattices, i.e., two, three, and several layers, using the dynamical mean-field theory. We clarify that the magnetic properties of this system become quite different depending on whether the number of layers is odd or even. In odd-number layers in an odd-number-layer system, finite magnetization emerges even with an infinitesimal interaction. This is a striking feature of the flat-band ferromagnetic state in multilayer systems as a consequence of the Lieb theorem. In contrast, in even-number layers, magnetization develops from zero on a finite interaction. These different magnetic behaviors are triggered by the flat bands in the local density of states and become identical in the limit of the infinite-layer (i.e., three-dimensional) system. We also address how interlayer hopping affects the magnetization process. Further, we point out that layer magnetization, which is a population imbalance between up and down atoms on a layer, can be employed to detect the emergence of the flat-band ferromagnetic state without addressing sublattice magnetization.

Figure 1

(a) Lieb lattice is constructed by sites $H,A$, and $B$. Because this lattice is $\alpha \beta$ bipartite, site $H$ $(A,B)$ belongs to sublattice $\alpha \left(\beta \right)$. The dashed line shows the unit cell. (b) Schematic picture of the multilayer system. $t_z$ represents interlayer hopping. Note that site $H$ does not always belong to sublattice $\alpha$, while the multilayer system is also $\alpha \beta$ bipartite.

Figure 2

DOS for (a) $L=1$, (b) $L=2$, and (c) $L=3$ with $t_z=1.0$. “$1A$” means site $A$ on the first layer. Note that the relation ${\rho }_{mA}\left(\omega \right)={\rho }_{mB}\left(\omega \right)$ $\left(m=1,...,L\right)$ exists for all layers. Additionally, ${\rho }_{1\gamma }\left(\omega \right)={\rho }_{2\gamma }\left(\omega \right)$ for $L=2$ and ${\rho }_{1\gamma }\left(\omega \right)={\rho }_{3\gamma }\left(\omega \right)$ for $L=3$ $\left(\gamma =H,A,B\right)$ exist. $t$ is a unit of energy.

Figure 3

Sublattice magnetization of site $\gamma (=H,A,B)$ on $m(=1,2,3)\mathrm{th}$ layer in the three-layer system $\left(L=3\right)$ with $t_z=1.0$. Filled (blank) symbols denote the first (second) layer. Note that relations $M_{mA}=M_{mB}$ and $M_{1\gamma }=M_{3\gamma }$ exist. $t$ is a unit of energy.

Figure 4

Local DOS $\rho \left(\omega \right)$ of (a) site $H$ [(b) $A]$ on the first layer and (c) site $H$ [(d) $A]$ on the second layer in the three-layer system $\left(L=3\right)$ with $t_z=1.0$. Insets show $\rho \left(\omega \right)$ with $U=1.0$ near the Fermi energy, where vertical ranges are as the same as in the large panel. For comparison, the DOS at $U=0.0$ is extracted from Fig. 2. Note that relations ${\rho }_{mA\sigma }\left(\omega \right)={\rho }_{mB\sigma }\left(\omega \right)$ $\left(m=1,2,3\right)$ and ${\rho }_{1\gamma \sigma }\left(\omega \right)={\rho }_{3\gamma \sigma }\left(\omega \right)$ $\left(\gamma =H,A,B\right)$ exist. $t$ is a unit of energy.

Figure 5

Sublattice magnetizations of site $\gamma (=H,A,B)$ on $m(=1,2)\mathrm{th}$ layer in the two-layer system $\left(L=2\right)$ with $t_z=1.0$. Only the first layer results are shown because of $M_{1\gamma }=-M_{2\gamma }$ and $M_{mA}=M_{mB}$. $t$ is a unit of energy.

Figure 6

Local DOS of (a) site $H$ [(b) $A]$ on the first layer in the two-layer system $\left(L=2\right)$ with $t_z=1.0$. For comparison, the DOS at $U=0.0$ is extracted from Fig. 2. Note that relations ${\rho }_{mA\sigma }\left(\omega \right)={\rho }_{mB\sigma }\left(\omega \right)$ $\left(m=1,2\right)$ and ${\rho }_{1\gamma \sigma }\left(\omega \right)={\rho }_{2\gamma \overline{\sigma }}\left(\omega \right)$ $\left(\gamma =H,A,B\right)$ exist. $t$ is a unit of energy.

Figure 7

Average of magnetization as a function of $U$ with the condition $t_z=1.0$. (a) Odd-layer systems. (b) Even-layer systems. $t$ is a unit of energy.

Figure 8

Magnetization of site $A$ (a) and site $H$ (b) in $L=3,5,7,9$ for $U=0.2$. The dashed line shows $-0.5/(L+1)$, which is equal to $-0.25/l$ in the $L=2l-1$ layered system.

Figure 9

Magnetization as a function of $U$ with the condition $t_z=0.1$ for $L=3$. (a) Sublattice magnetization. (b) Layer magnetization. In the lower figure, results for $t_z=1.0$ are also shown for comparison. Note that relations $M_{mA}=M_{mB}$ $\left(m=1,2,3\right)$, $M_{1\gamma }=M_{3\gamma }$ $\left(\gamma =H,A,B\right)$, and $M_1^{\mathrm{lay}}=M_3^{\mathrm{lay}}$ exist. $t$ is a unit of energy.