Eur. Phys. J. B (2022) 95: 191
https://doi.org/10.1140/epjb/s10051-022-00454-2

Regular Article - Solid State and Materials

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

¹ Department of Physics and Astronomy, National Institute of Technology Rourkela, 769008, Rourkela, Odisha, India
² Center for Nanomaterials, National Institute of Technology Rourkela, 769008, Rourkela, Odisha, India

^b dattas@nitrkl.ac.in

Received: 14 June 2022
Accepted: 8 November 2022
Published online: 18 November 2022

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.