Nontrivial flat bands and quantum Hall crossovers in square-octagon lattice materials

Abstract

Coexistence of nontrivial topology and flat electronic bands in low-energy lattices provides a fertile platform for correlated quantum states. The square-octagon lattice hosts Dirac nodes and flat bands at half-filling, yet the influence of intrinsic spin-orbit coupling (SOC) and staggered magnetic flux on its topological and flat-band properties remains largely unexplored. Here, we examine this lattice using tight-binding models that include SOC and magnetic flux, uncovering a quantum spin Hall phase with spin Chern number $C_s=1$, crossovers to quantum anomalous Hall phases with $C=1$ and $C=2$, and higher-order topological insulator phases carrying quantized quadrupolar corner charges. The initially dispersionless flat bands evolve into quasi-flat, topologically nontrivial bands with uniform quantum geometry and large flatness ratios, conducive to fractional Chern insulator states. We further identify realistic material candidates, including octagraphene, transition-metal dichalcogenides, synthetic $\mathrm{MoSi_2N_4}$, and magnetic $α$-MnO$_2$, as potential candidates for realizing tunable topological phases intertwined with flat-band physics, opening new opportunities for correlated topological matter.

Figures (6)

• Figure 1: Coexistence of flat bands and Dirac cones in a square-octagon lattice. (a) Square-octagon lattice with four sublattices (A-D) per unit cell, shown in distinct colors. $t_1$, $\lambda$, and $t_2$ represent the nearest-neighbor, next-nearest-neighbor, and intercell hopping parameters, respectively. Arrows indicate the magnetic flux directions. (b) Energy dispersion for $t_2 = 2t_1$ and $\lambda = t_1$. Van Hove singularities (VHSs), flat bands (FBs), and a Dirac point (DP) are marked. Colors denote the dominant sublattice contributions to each band. (c) Schematic of a compact localized state associated with the FB at $E = t_1$ (gray). Filled and hollow sites correspond to nonzero and zero wavefunction amplitudes, respectively. The $\pm$ signs indicate phase alternation leading to destructive interference at the hollow sites, which localizes the state.
• Figure 2: Emergence of quantum Hall phases and nontrivial flat bands in square-octagonal lattice. (a) Band structure with SOC strength $\lambda_{SOC}= 0.1$ and intercell hopping $t_2=1.2t_1$. A nontrivial band gap opens across the full Brillouin zone (BZ) at half-filling, giving rise to a quantum spin Hall phase. Occupied and unoccupied bands are shown in orange and violet, respectively. (b) Evolution of Wannier charge centers for two spin channels, exhibiting zero total Chern number but a finite spin Chern number. (c) Edge-state spectrum along the (010) direction, showing spin-polarized edge modes. (d) Band structure in the presence of SOC ($\lambda_{\mathrm{SOC}}=0.1$) and magnetic flux $\Phi=0.3\Phi_0$. (e) Corresponding WCC evolution for two spin channels and (f) associated chiral edge states, indicating a quantum anomalous Hall phase with Chern number $C=2$ (QAH II). (g) Evolution of the spin Chern number with SOC strength $\lambda_{\mathrm{SOC}}$ at 1/2 and 1/4 fillings. (h) Evolution of Chern number as a function of magnetic flux $\Phi$ at the same fillings.
• Figure 3: Nontrivial flatness and quantum geometry. (a-c) Evolution of the flat band and Dirac cone at the $M$ point: (a) without SOC, (b) with SOC, and (c) with both SOC and magnetic flux. The gapless Dirac point in the absence of SOC evolves into a QSH phase with SOC and a QAH phase under magnetic flux. Color maps indicate (b) the spin Berry curvature $\Omega_{xy}^{\sigma}$ and (c) the Berry curvature $\Omega_{xy}$ associated with band inversion. Flat bands acquire finite dispersion near the inversion point, forming quasi-flat topological bands. (d) Flatness ratio of flat band (at half-filling) as a function of magnetic flux $\Phi$ for $t_2=1.2t_1$ and $\lambda_{\mathrm{SOC}}=0.1$. Green regions denote the QAH phase with $C=\pm2$, where the topological flat band retains a high flatness ratio. (e,f) Contour maps of (e) Berry curvature $\Omega_{xy}$ and (f) the difference between the trace of the quantum metric and the absolute value of Berry curvature, $\mathrm{Tr}[Q_{xy}] - |\Omega_{xy}|$, for the topological band with flatness ratio $\sim22$ and Chern numbers $C=2$ and $C=-2$.
• Figure 4: Higher-order topological insulator with quadrupolar charge. (a) Edge band structure for magnetic flux $\Phi = 0.5\Phi_0$, intercell hopping integral $t_2=3.2t_1$ and spin–orbit coupling $\lambda_{\mathrm{SOC}} = 0.1$, showing spin-polarized floating edge states. (b,c) Spatial distribution of corner-state wavefunctions $|\psi|^2$ at $E = t_1$ for up (red) and down (blue) spin channels. Fourfold-degenerate spin-polarized corner states appear at $E = \pm t_1$. Insets display the energy spectrum, highlighting red and blue states within the nontrivial bulk gap.
• Figure 5: Topological phase diagram and spin-polarized topological flat bands. (a) Phase diagram with intercell hopping $t_2$ (vertical axis) and magnetic flux $\Phi$ (horizontal axis). The intrinsic SOC strength is fixed at $\lambda_{SOC}=0.1$ with $\lambda=t_1$ and $t_1=1$. Distinct phases are identified by their Chern number $C$ or spin Chern number $C_s$: quantum anomalous Hall (QAH, QAH I, QAH I$^{\prime}$, QAH II), quantum spin Hall (QSH), higher-order topological insulator (HOTI), and normal insulator (NI). (b) Flatness ratio as a function of $t_2$ and $\Phi$ at half filling. (c-h) Representative band structures of the corresponding phases and the evolution of flat bands in the square-octagon lattice under SOC and magnetic flux. Red and blue denote spin-up and spin-down projections, while black lines represent spin-degenerate bands.