Topological Anderson insulating phases in the long-range Su-Schrieffer-Heeger model

Abstract

The long-range Su-Schrieffer-Heeger (SSH) model, in which the second nearest-neighbor hopping is taken into account, exhibits a topological phase diagram that contains winding numbers $w=0,\phantom{\rule{4pt}{0ex}}1$, and 2. In the clean system, the change in winding number stems from the band-touching phenomenon. In the presence of disorder, the renormalization of energy band and Fermi level results in the nonzero density of states in the energy gap. These midgap states cause the crossover phenomenon and the divergence of localization length at a critical disorder strength ${U}_{c}$ in the finite SSH system. In this study, we numerically computed the mean winding number and localization length for the disordered SSH system. We find that the disorder is able to drive phase transitions between different mean winding numbers: $w=0\ensuremath{\rightarrow}1$, $0\ensuremath{\rightarrow}2$, $1\ensuremath{\rightarrow}2$, and $2\ensuremath{\rightarrow}1$ in the weak disorder regime. By investigating the wave function distribution and the self-energy, the nonzero mean winding numbers correspond to the so-called topological Anderson insulating (TAI) phases. The finite size scaling for the mean winding number in the TAI phase is shown. For describing the phase transitions in the thermodynamic limit, we apply the criterion of band gap closure resulting from the broadening of energy band and Fermi level to determine the critical disorder strength. The critical disorder strength for self-consistent Born approximation (SCBA) ${U}_{c}^{\mathrm{SCBA}}$ and that for first Born approximation (FBA) ${U}_{c}^{\mathrm{FBA}}$ are numerically calculated. ${U}_{c}^{\mathrm{SCBA}}$ is found to match with ${U}_{c}$ qualitatively. Nonetheless, SCBA indicates the different roles of band shifts and Fermi level broadening near the topological phase transitions. Band shift/Fermi level broadening is more dominant for the transitions from low-to-high/high-to-low winding number. Interestingly, for the transition from bulk insulator to TAI, ${U}_{c}^{\mathrm{FBA}}$ is quantitatively closer to ${U}_{c}$ than ${U}_{c}^{\mathrm{SCBA}}$ as long as the renormalized band gap is zero within FBA.