Higher-Order Topological Mott Insulators

We propose a new correlated topological state, which we call a higher-order topological Mott insulator (HOTMI). This state exhibits a striking bulk-boundary correspondence due to electron correlations. Namely, the topological properties in the bulk, characterized by the ${\mathbb{Z}}_3$ spin-Berry phase, result in gapless corner modes emerging only in spin excitations (i.e., the single-particle excitations remain gapped around the corner). We demonstrate the emergence of the HOTMI in a Hubbard model on the kagome lattice, and elucidate how strong correlations change gapless corner modes at the noninteracting case.

Figure 1

(a) Sketch of the kagome lattice with $N_{\mathrm{UC}}=10$ and the boundary condition. Blue and red bonds represent nearest-neighbor hoppings corresponding to the Hamiltonian $H_{\mathrm{kin}}^{△}$ and $H_{\mathrm{kin}}^{▽}$, respectively. (b) $⟨{\mathit{S}}_i^2⟩-⟨{\mathit{S}}_i^2{⟩}_0$ computed by the Lanczos method for $N_{\mathrm{UC}}=6$, $t_{△}/t_{▽}=0.4$. In panel (b), the calculated value is represented by the radius of the spheres which is obtained under the OBC. The spin operator can be rewritten as ${\mathit{S}}_i^2=3(n_{i\uparrow }+n_{i\downarrow }-2n_{i\uparrow }{n}_{i\downarrow })/4$.

Figure 2

Numerical results for the spin model (2). (a) $S_{\mathrm{tot}}^z$ of the ground state as a function of $\varphi$ and $J^{(3)}/J^{(2)}$. (b) Spin-Berry phase ${\gamma }_{-}$ of the ground state for $J^{(3)}/J^{(2)}=0.1$ as a function of $\varphi$. The gray region in panel (b) indicates that $S_{\mathrm{tot}}^z$ of the ground state is not $N_{\mathrm{UC}}/2=4.5$. The system is periodic and its size is $N_{\mathrm{UC}}=3\times 3$ for panels (a) and (b).

Figure 3

Numerical results for the spin model (2). (a) Energy spectra $E/J^{(2)}$ as functions of $\varphi$. (b) Energy spectra $E/J^{(2)}$ at $\tan \varphi =0.5$ for each value of $p$, where $p$ is the number of removed sites. Panels (a) and (b) are obtained for $N_{\mathrm{UC}}=10$ and $J^{(3)}/J^{(2)}=0.1$ under the OBC. Only the four lowest energies in the subspace specified by $S_{\mathrm{tot}}^z$ are shown here. The black lines in panel (a) express the energy for $S_{\mathrm{tot}}^z<-1$ or $6<S_{\mathrm{tot}}^z$. The dashed line in panel (a) represents $\tan \varphi =0.5$.

Figure 4

Numerical results for the Hubbard model (1). (a) $S_{\mathrm{tot}}^z$ of the ground state as a function of $\varphi$ and $t/U$. (b) Charge gap ${\mathrm{\Delta }}_c$ and spin gap ${\mathrm{\Delta }}_s$ as functions of $U/t$. (c) Spin-Berry phase ${\gamma }_{-}$ as functions of $U/t$. (d) Charge gap ${\mathrm{\Delta }}_{\mathrm{c}}$ and spin gap ${\mathrm{\Delta }}_{\mathrm{s}}$ as functions of $U/t$. Panels (a), (b), and (c) are obtained for $N_{\mathrm{UC}}=2\times 2$ under the PBC. Panel (d) is obtained for $N_{\mathrm{UC}}=6$ under the OBC. We set $t_{△}/t_{▽}=0.4$ in panels (b), (c), and (d).