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This paper investigates the topological phase transitions and localization properties in a 1D p-wave superconductor under Fibonacci quasi-periodic potential modulation. By calculating the Z2 topological invariant, we numerically determine the topological phase diagram of the system. We find that, under Fibonacci quasi-periodic modulation, the system can transition from a topologically trivial phase to a topological Anderson superconductor phase. Moreover, under certain parameters, the system undergoes multiple topological Anderson superconductor phases transitions, accompanied by the emergence of zero-energy modes. However, in the case of strong disorder, the topological Anderson superconductor phase is destroyed, indicating that the topological Anderson superconductor phase can only be induced within a finite range of parameters. Furthermore, by calculating and analyzing the fractal dimension and the mean inverse participation ratio (MIPR) order parameter, we analyze the localization properties of the system. The results show that regardless of the increase in disorder strength, the fractal dimension values of most eigenstates always remain within the range (0;1). Subsequently, the variations in the fractal dimensions of all eigenstates for different system sizes were studied. The results show that the fractal dimension values of most eigenstates are away from 0 and 1. These results indicate that the wavefunction in the bulk of the topological Anderson superconductor phase induced by Fibonacci quasi-periodic potential are critical state wavefunction, with the system overall being in a critical phase. The stability of the critical phase is confirmed by scale behavior of MIPR, as shown in Fig.(a).
It differs from the traditional topological Anderson superconductor phase induced by random disorder or AA-type quasi-periodic disorder. The results provide new insights and references for the study of topological phase transitions and localization transitions in 1D p-wave superconductors. -
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