Ferromagnetism and antiferromagnetism of a correlated topological insulator with a flat band

In this paper, based on the mean-field approach and random-phase approximation, we studied the magnetic properties of the spinfull Haldane model on honeycomb lattice of topological flat band with onsite repulsive Coulomb interaction. We found that the antiferromagnetic (AF) order is more stable than the ferromagnetic (FM) order at, or near, half filling. Away from half filling, the phase diagram becomes complex: at large doping, the FM order is more stable than the AF order due to the flatness of band structure. In particular, we found that at quarter filling, the system becomes a Chern number $Q=1$ topological insulator induced by the FM order.

Figure 1

The illustration of the honeycomb lattice. The red and black dots denote the bipartite lattice sites of the honeycomb lattice. The green lines, dashed blue lines, and dashed magenta lines represent the NN hopping $t$, the NNN hopping $t^{\prime },$ and the NNNN hopping $t^{\prime \prime }$, respectively.

Figure 2

Free electronic dispersion ($U/t=0$) at $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi$. The energy scale is $t$. (a) illustrates the total energy bands, and we can see that the energy bands of electrons with up spin and those of electrons with down spin are degenerate due to the spin-rotation symmetry; (b) shows the detailed structure of the flat bands, and one can see that the bandwidth is much smaller than the band gap.

Figure 3

The zigzag edge states along the $x$ direction for free electrons at $t^{\prime }=0.6t,$$t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$ The energy scale is $t$. The black dashed line is the position of the chemical potential.

Figure 4

The electronic dispersions of the AF order at $U/t=7.0$ $(M_{\text{AF}}\ne 0),$ $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi$ for half filling. The energy scale is $t$. We can see that the AF order parameter changes the flat bands into the dispersive energy bands.

Figure 5

The electronic dispersions of the FM order at $U/t=7.0$ $(M_F\ne 0),$ $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi$ for half filling. The energy scale is $t$. We can see that the FM order parameter splits the two degenerate flat bands.

Figure 6

The global phase diagram in the TFB limit. For the half filling case $d=0,$ there are three phases: $Q=2$ TI, $Q=2$ AF-TI, and normal AF insulator. For the quarter-filling case $d=0.5,$ there are two phases: normal metal and $Q=1$ TI induced by the FM order ($Q=1$ FM-TI). For other cases, there are three phases for the region of $0<d<0.5$: normal metal, FM metal, and AF metal, while there are only two phases for the region of $0.5<d<1$: normal metal and FM metal.

Figure 7

The per-site ground-state energies for the AF order (blue line), the FM order (red line), and no spin order (black line), respectively. The energy scale is $t$. The parameters are set to $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$$\varphi =0.4\pi$ at half filling.

Figure 8

The phase diagram at half filling. There are four phases: $Q=2$ TI, FM insulator, $Q=2$ AF-TI, and normal AF insulator. The dashed blue line represents the FM phase transition ${(U/t)}_{\text{FM}}$, the black line represents the AF phase transition ${(U/t)}_{\text{AF}},$ the red line represents a topological quantum phase transition. The purple dashed line is the case with the largest flatness ratio at $U/t=0$, $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t$, $\varphi =0.4\pi$.

Figure 9

The staggered magnetization $M_{\text{AF}}$ for the half-filling case with the parameters $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$ ${(U/t)}_{\text{AF}}$ is the magnetic phase transition.

Figure 10

The energy gap of electrons for the half-filling case with the parameters $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$ The energy scale is $t$. There are two quantum phase transitions (the black dashed lines): one is the magnetic phase transition at ${(U/t)}_{\text{AF}}$, the other is a topological quantum phase transition with gap closing.

Figure 11

The spin wave's dispersions of the AF order for the TFB case of $t^{\prime }=0.6t$, $t^{\prime \prime }=-0.58t$, $\varphi =0.4\pi$ with $U/t=7.0$ (black line) and $U/t=9.0$ (red line), respectively. The spin wave's dispersions of the AF order away from the flat-band case of $t^{\prime }=0.6t$, $t^{\prime \prime }=-0.54t$, $\varphi =0.4\pi$ with $U/t=7.0$ (blue line) and $U/t=9.0$ (pink line), respectively. The energy scale is $t$.

Figure 12

The magnetization $M_F$ at the quarter filling ($d=0.5$) with the parameters $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi$. There are two phases: the normal metal (the green region) and the $Q=1$ FM-TI. In the FM order, the ferromagnetization saturates suddenly.

Figure 13

The electronic energy gap at the quarter filling ($d=0.5$) with the parameters $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi$. The energy scale is $t$. There have two phases: the normal metal (the green region) and the $Q=1$ FM-TI.

Figure 14

The electronic dispersions at the quarter filling ($d=0.5$) with the parameters $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ $\varphi =0.4\pi$ for $U/t=5.0.$ The energy scale is $t$. The chemical potential ${\mu }_{\text{eff}}$ (the green plane) locates at the middle of the energy gap. The ground state is a $Q=1$ FM-TI with filled lowest topological flat band (the blue plane).

Figure 15

The zigzag edge states along the $x$ direction at $U/t=5.0$ with $t^{\prime }=0.6t$, $t^{\prime \prime }=-0.58t$, $\varphi =0.4\pi$ for the quarter-filling case $d=0.5$. The energy scale is $t$. The black dashed line is the position of the chemical potential.

Figure 16

The spin wave's dispersions of the FM order for the case of $t^{\prime }=0.6t$, $t^{\prime \prime }=-0.58t$, $\varphi =0.4\pi$ with $U/t=3.0$ and $6.0$ at quarter filling $d=0.5$. The energy scale is $t$.

Figure 17

The magnetization ($M_F$ or $M_{\text{AF}}$) for the doping case $d=0.1$ at $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$ There we have three phases: normal metal (the green region), FM metal, and AF metal.

Figure 18

The per-site ground-state energies of the AF order and the FM order for the case of $d=0.1$ at $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$ The energy scale is $t$. The red line is the per-site ground-state energy of the FM order and the black line is the per-site ground-state energy of the AF order. The blue spots $A$ and $B$ denote the quantum phase transitions between the AF order and the FM order.

Figure 19

The electronic dispersions of the FM order and the AF order at $d=0.1$ for $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$ The energy scale is $t$. ${\mu }_{\text{eff}}$ (the green plane) locates in the spectra. (a) illustrates the FM case and (b) illustrates the AF case.

Figure 20

The spin susceptibility of AF metal for the doping case $d=0.1$ at $U/t=6.0,$$t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$

Figure 21

The spin susceptibility of the FM-metal for the doping case $d=0.1$ at $U/t=1.5,$ $t^{\prime }=0.6t,$ $t^{\prime \prime }=-0.58t,$ and $\varphi =0.4\pi .$