Flat bands with nontrivial topology in three dimensions

We construct a simple model for electrons in a three-dimensional crystal where a combination of short-range hopping and spin-orbit coupling results in nearly flat bands characterized by a nontrivial ${\mathbb{Z}}_2$ topological index. The flat band is separated from other bands by a band gap $\Delta$ that is much larger than the bandwidth $W$. When the flat band is partially filled we show that the system remains nonmagnetic for a significant range of repulsive interactions. In this regime we conjecture that the true many-body ground state may become a three-dimensional fractional topological insulator.

Figure 1

(a) Lieb lattice showing a three-site basis in the unit cell and basis vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$. The second- and third-neighbor hoppings are indicated by red/gray and green/light gray dotted lines, respectively. (b) Tight-binding dispersion for ${\mathcal{H}}_{\mathbf{k}}$ after tuning parameters to achieve a maximal figure of merit $\Delta /W$. For clarity, the bulk energy bands are plotted over the shifted Brillouin zone $0\le k_x\le \pi /a$, $0\le k_y\le \pi /a$, where $a$ represents the length of the nearest-neighbor bond. (c) Band structure for a strip of $N_y=10$ unit cells with open-boundary conditions along the $y$ direction and infinite along $x$ with the same parameters as in (b). (d) Semilog plot showing the evolution of the gap as one moves between a known topological insulating phase and the final set of finely tuned parameters.

Figure 2

(a) Perovskite lattice showing four sites in a unit cell along with basis vectors. The second- and third-neighbor hoppings are indicated by red/gray and green/light gray dotted lines, respectively. (b) High symmetry points in the Brillouin zone. (c) Band structure inside the bulk along the path of high symmetry for ${\mathcal{H}}_{\mathbf{k}}$ with a finely tuned parameter set. (d) Semilog plot showing the evolution of the gap as one moves between a known topological insulating phase and the final set of finely tuned parameters.

Figure 3

(a) Phase diagram for Lieb lattice ($W\approx 0.034$) and (b) perovskite lattice ($W\approx 0.072$). Critical values $U_c$,$U_c^{\prime }$ for the onset of magnetic order reflect the charge distribution on the basis sites. The latter is approximately uniform for the Lieb lattice, implying $U_c\approx U_c^{\prime }$. On the perovskite lattice most of the charge density is on site 1, causing $U_c\ll U_c^{\prime }$. (c) Band structure for the Lieb lattice with $U/W=U^{\prime }/W=15$, $m^{\left(1\right)}\approx 0.35$. and $m^{\left(2\right)}=m^{\left(3\right)}\approx 0.32$, and (d) the perovskite lattice with $U/W=5$, $U^{\prime }/W=25$, $m^{\left(1\right)}\approx 0.62$, $m^{\left(2\right)}=m^{\left(3\right)}\approx 0.11$, and $m^{\left(4\right)}\approx 0.05$.