Effect of spin–orbit coupling in one-dimensional quasicrystals with power-law hopping

Abstract

In the one-dimensional quasiperiodic Aubry–André–Harper Hamiltonian with nearest-neighbor hopping, all single-particle eigenstates undergo a phase transition from ergodic to localized states at a critical value of the quasiperiodic potential $W/t = 2.0$. There is no mobility edge in this system. However, in the presence of power-law hopping having the form $1/r^a$, beyond a finite value of the quasiperiodic potential $(W_c)$ the mobility edge appears for $a > 1$, while, for $0< a\le 1$, a multifractal edge separates the extended and the multifractal states. In both these limits, depending on the quasiperiodicity strength, the lowest $\beta ^s L$ states are delocalized. We have found that, in the presence of the spin–orbit coupling, the critical quasiperiodicity strength is always larger irrespective of the value of the parameter a. Furthermore, we demonstrate that for $0< a\le 1$, in the presence of spin–orbit coupling, there exists multiple multifractal edges, and the energy spectrum splits up into alternative bands of delocalized and multifractal states. Moreover, the location of the multifractal edges are generally given by the fraction $(\beta ^s \pm \beta ^m)$. The qualitative behavior of the energy spectrum remains unaffected for $a > 1$. However, in contrast to the previously reported results, we find that in this limit, similar to the other case, multiple mobility edges can exist with or without the spin–orbit coupling.

Graphic Abstract

Appendices

Appendix A: IPR for weak RSO interaction in the short-range limit

In this section, we have discussed the behavior of mobility edges in the energy spectrum for a short-range $(a>1)$ GAA model having a smaller amplitude of RSO coefficients $\alpha _y,\alpha _z$ (as shown in Fig. 10). We can observe that with weaker RSO coupling, only the first window (where only a single mobility edge exists) just after the critical point gets destroyed, while the rest of such windows remain unaffected. In Fig. 4 of the main text, we have observed that as the strength of the RSO coupling is increased, such windows appear only at a higher quasiperiodicity strength.

Appendix B: Finite size scaling of IPR

In this section, we have presented the finite-size scaling of the average IPR (averaged over states) for both the short-range $(a = 1.5)$ and long-range $(a = 0.5)$ GAA Hamiltonian with the RSO coupling. Figure 11a shows the finite-size scaling of the average IPR as a function of 1/L at $W/t = 3.0$, where the average is taken over the electronic states corresponding to the $\textrm{ERG}^1$ and $\textrm{LOC}^1$ regions for the short-range GAA model, as shown in Fig. 5b. This finite scaling of IPR confirms the presence of both the delocalized and localized states in the energy spectrum in the short-range GAA model. Similarly, we have shown the finite-size scaling of IPR for the long-range GAA model in Fig. 11b. Here, we have presented the scaling for two different regions, $\textrm{ERG}^1$ and $\textrm{MF}^1$, as shown in Fig. 7b at $W=4.0$. It is evident that the IPR value for states belonging to the region $\textrm{ERG}^1$ vanishes as 1/L, indicating that these states are extended. For the states belonging to the region $\textrm{LOC}^1$, the average IPR value approaches unity for all L. It is almost independent of the system size that is $IPR \propto L^0$, indicating that these states are localized. In contrast to these, the average IPR shows a large fluctuation for the states corresponding to the region $\textrm{MF}^1$ and does not scale well with 1/L, indicating that these states are multifractal types.