Efficient prediction of topological superlattice bands with spin-orbit coupling

TL;DR

The paper develops a symmetry-indicator framework to predict the topology of superlattice minibands in two-dimensional materials with spin-orbit coupling, using only the parent-band data and a weak superlattice potential. By combining degenerate perturbation theory with Fu-Kane and rotation-based Chern indicator formulas, it yields analytic expressions for the Z2 invariant and Chern number in TRS and TRS-broken settings, respectively. The approach works beyond perturbation theory as long as SL-induced gaps remain open and is validated across BHZ models, TI thin films, and TMDs, revealing that topological minibands can arise even from non-topological parents under appropriate SL geometry and periodicity. This provides a fast, transparent design principle for engineering topological flat bands in moiré- and nanopatterned-heterostructures, broadening the material pool for experimental realization and high-throughput screening.

Abstract

We develop a symmetry indicator framework to efficiently predict the topology of superlattice-induced minibands with spin-orbit coupling. Our algorithm requires input only from the parent material before the superlattice is applied. The simplification arises by assuming a perturbatively weak superlattice potential; however, our results extend beyond the perturbative regime as long as the superlattice-induced gaps remain open. We first consider a time-reversal- and inversion-symmetric system subject to a weak superlattice potential and derive a compact formula for the $\mathbb{Z}_2$ invariant of the lowest miniband. We then extend to time-reversal breaking systems and compute the Chern number. We apply our theory to selected transition metal dichalcogenides, HgTe/CdTe quantum wells, and thin films of three-dimensional topological insulators and Dirac semimetals. We find topological superlattice bands can arise even from non-topological materials, broadening the pool of candidates for realizing topological flat bands. Our theory predicts which geometry and periodicity of superlattice will yield topological bands for a given material, providing a clear guiding principle for designing topological superlattice heterostructures.

Figures (4)

• Figure 1: Phase diagrams of the (a) rectangular, (b) square and (c) triangular lattice BHZ model without a superlattice potential. (The parameters $C$, $D_j$ and $A$ do not affect the topological phase diagram.)
• Figure 2: Phase diagram showing the $\mathbb{Z}_2$ index of the conduction band for (a-b) SL$_2$-BHZ$_2$, (c-d) SL$_4$-BHZ$_4$ and (e-f) SL$_6$-BHZ$_6$, as a function of $M/B$ and the ratio of the SL periodicity to the original lattice constant $L = a_{\text{SL}}/a$. Here, BHZ$_n$ represents the $\mathcal{C}_n$ symetric BHZ model. In panels (a-b), $B_x=B$ and $B_y=2B$. We fix $A = -3$eV in panels (c-d) ($A$ does not enter the $\mathbb{Z}_2$ invariant for SL$_{2,6}$.) The left (right) panels show the topological phase diagrams for positive (negative) values of the SL potential harmonics.
• Figure 3: Topological phase diagram of (a-b) SL$_2$-Cd$_3$As$_2$, (c-d) SL$_4$-Cd$_3$As$_2$ and (e-f) SL$_6$-Cd$_3$As$_2$ as a function of $\delta/b$ and the SL periodicity $a_{\text{SL}}$. In panels (a-b), $a_x=a_{\text{SL}}$ and $a_y=2a_{\text{SL}}$. The left (right) panels show the topological phase diagrams for positive (negative) values of the SL potential harmonics.
• Figure 4: Topological phase diagram of the valence miniband at the $\xi=-1$ valley of SL$_3$-TMD as a function of the argument of the form factor $\Lambda_{R_3\kappa, \kappa}$ and the phase of the superlattice potential $\theta$.