Stacking-induced symmetry-protected topological phase transitions

We study symmetry-protected topological (SPT) phase transitions induced by stacking two gapped one-dimensional subsystems in the BDI symmetry class. The topological invariant of the entire system is a sum of three topological invariants: two from each subsystem and an emerging topological invariant from the stacking. We find that any symmetry-preserving stacking of topologically trivial subsystems can drive the entire system into a topologically nontrivial phase for a certain coupling strength. We explain this intriguing SPT phase transition by conditions set by orbital degrees of freedom and time-reversal symmetry. To exemplify the SPT transition, we provide a concrete model which consists of an atomic chain and a spinful nanowire with spin-orbit interaction and $s$-wave superconducting order. The stacking-induced SPT transition drives this heterostructure into a zero-field topological superconducting phase.

Figure 1

Symmetry-preserving stacking of two trivial subsystems induces a nontrivial SPT phase with nonzero winding number ${\mathcal{W}}_{ud}$ evaluated by the loop $z_{\lambda }\left(k\right)$ (blue). The loops deform by tuning the coupling strength $\lambda$. (a) Deformation of the loop in the presence of time-reversal symmetry and ${(-1)}^{min\{N_u,N_d\}}=-1$. (b) Same as (a) for ${(-1)}^{min\{N_u,N_d\}}=+1$. (c) Same as (a) in the absence of time-reversal symmetry.

Figure 2

(a) Two loops of $z_{{\lambda }_0+\epsilon }\left(k\right)$ and $z_{{\lambda }_0-\epsilon }\left(k\right)$ yield $\delta {\mathcal{W}}_{ud}$: The sum of contour integrals along the loops shown in (b) and (c) equals $\delta {\mathcal{W}}_{ud}$.

Figure 3

Stacking of two topologically trivial subsystems yields a zero-field TSC phase. (a) Spinful (top) and spin-polarized (bottom) quantum wires are stacked with arbitrary coupling strength (dashed lines). (b) The spinful wire exhibits an $s$-wave superconducting gap along with spin-orbit coupling. (c) The chemical potential gaps out the spin-polarized wire. The lower branches $E_{u,d}<0$ depict the hole band.

Figure 4

(a) Phase diagram of the zero-field TSC (green) in terms of the coupling angle $\chi$ and strength $\lambda$. (b) ${\mathcal{W}}_{ud}={\int }_0^{2\pi }dkarg\left\{z_{\lambda }\left(k\right)\right\}/\left(2\pi \right)$ counts the number of times the loop $z_{\lambda }\left(k\right)$ winds around the origin. The loops depict the cases (i)–(iii) in panel (a). (c) Energy spectra of the heterostructure with a finite size (200 unit cells). The Majorana bound states appear at zero energy (red). (d) Electron bands for $\lambda =0$ (black) and $\lambda \ne 0$ (red). We obtain all figures with $t_u=-1,t_d=-0.7,{\mu }_u=-1.05,{\mu }_d=-0.8,l_{\text{so}}=0.5,\mathrm{\Delta }=0.3$.