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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 $Z_2$ 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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图 1 (a)以拓扑不变量 Q大小为背景颜色填充的 V- M参数平面的拓扑相图. 颜色条代表 $Z_{2}$拓扑不变量 Q的大小. 绿色虚线代表拓扑相变点解析解, 由( 20)式确定. (b)—(d) 当常数势强度 $M=1$, $-3$和 $-4$时, 能隙 $\Delta_g$和 $Z_{2}$拓扑不变量 Q随着无序强度 V的变化. (b)插图中对应的无序强度 $V=0.2$时, L能级对应的波函数分布. 这里, $\Delta=0.4$和 $L=2000$
Figure 1. The $Z_{2}$ topological invariant Q as a function of the disorder strength V and constant potential M. The colorbar shows the value of the $Z_{2}$ topological invariant Q. The green dashed line represents the analytical solution of the topological phase transition point, determined by Eq. ( 20). Energy gap $\Delta_g$ and the $Z_{2}$ topological invariant Q as a function of V for (b) $M=1$, (c) $M=-3$ and (d) $M=-4$. The wave function distributions corresponding to the L energy levels at a disorder strength of $V=0.2$ in the inset of (b). Other parameters: $\Delta=0.4$ and $L=2000$.
图 2 当系统尺寸 $L=500$, M分别为(a) $M=1$和(b) $M=-3$时, 分形维度( ${\Gamma}_n$)随着本征能量 E和无序强度 V的变化. 图中的颜色条代表分形维度( ${\Gamma}_n$)的大小. (c)当 $V=2$且 $M=1$时, 不同系统尺寸下, 系统分形维度( ${\Gamma}_n$)的值. (d)当 $V=2$且 $M=-3$时, 不同系统尺寸下, 系统分形维度( ${\Gamma}_n$)的值. 其他参数: $\Delta=0.4$
Figure 2. The fractal dimension ${\Gamma}_n$ of different eigenstates as a function of the corresponding E and the modulation strength V with $L=500$ for (a) $M=1$ and (b) $M=-3$. (c) The Fractal dimensions ${\Gamma}_n$ for different system sizes for $V=2$ and $M=1$. (d) The Fractal dimensions ${\Gamma}_n$ for different system sizes for $V=2$ and $M=-3$. Other parameters: $\Delta=0.4$.
图 3 当 M分别为(a) $M=1$和(b) $M=-3$时, MNPR在无序强度为 $V=1$, $2$ 和 $3$情况下的标度行为. 其他参数: $\Delta=0.4$
Figure 3. The scaling behavior of MIPR in (a) $M=1$ and (b) $M=-3$ with $V=1, 2$ and $3$. Other parameters: $\Delta=0.4$.
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