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研究了一维Fibonacci准晶势调制下的p波超导体下的拓扑相变和局域化性质. 在Fibonacci准晶势调制下, 通过计算$Z_2$拓扑不变量确定了系统的拓扑相图. 分析相图指出在Fibonacci准晶势调制下, 系统可以由拓扑平庸超导相进入拓扑安德森超导相. 进一步分析发现, 在某些参数下, 系统会发生多次拓扑安德森超导相转变并伴随零能态的出现. 此外, 还研究了系统的局域化性质, 通过分析分形维度、平均逆参与率序参量, 发现Fibonacci准晶势诱导的拓扑安德森超导相, 其体态的波函数表现出多重分形行为, 这与随机无序诱导出来的传统拓扑安德森超导相完全不同. 该研究结果为一维p波超导体中拓扑相变和局域化转变的研究提供了一些新的理解和参考.
The topological phase transitions and localization properties in a 1D p-wave superconductor under Fibonacci quasi-periodic potential modulation are investigated in this work. By calculating the Z2 topological invariant, the topological phase diagram of the system is determined numerically. It is found that the system can transition from a topologically trivial phase to a topological Anderson superconductor phase through the Fibonacci quasi-periodic modulation. Moreover, under certain parameters, the system undergoes multiple topological Anderson superconductor phase 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 be induced only within a finite range of parameters. Furthermore, by calculating and analyzing the fractal dimension and the mean inverse participation ratio (MIPR) order parameter, the localization properties of the system are analyzed. The results show that regardless of how the disorder intensity increases, the fractal dimension values of most eigenstates always remain within a range of 0–1. Subsequently, the variations in the fractal dimensions of all eigenstates for different system sizes are 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 is a 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 into and references for studying 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$\varDelta_{\mathrm{g}}$和$Z_{2}$拓扑不变量Q随无序强度V的变化. (b)插图对应的是无序强度$V=0.2$时, L能级对应的波函数分布. 这里, $\varDelta=0.4$和$L=2000$
Fig. 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 $\varDelta_{\mathrm{g}}$ 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 panel (b). Other parameters: $\varDelta=0.4$ 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(${\varGamma}_n$)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(${\varGamma}_n$)的大小. (c)当$V=2$且$M=1$时, 不同系统尺寸下, 系统分形维度(${\varGamma}_n$)的值. (d)当$V=2$且$M=-3$时, 不同系统尺寸下, 系统分形维度(${\varGamma}_n$)的值. 其他参数取值为$\varDelta=0.4$
Fig. 2. The fractal dimension ${\varGamma}_n$ 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 ${\varGamma}_n$ for different system sizes for $V=2$ and $M=1$. (d) The fractal dimensions ${\varGamma}_n$ for different system sizes for $V=2$ and $M=-3$. Other parameters: $\varDelta=0.4$.
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,\;3 $ 情况下的标度行为. 其他参数$\varDelta=0.4$
Fig. 3. The scaling behavior of MIPR in (a) $M=1$ and (b) $M=-3$ with $V=1,\; 2,\;3$. Other parameters: $\varDelta=0.4$.
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[1] Nakajima S, Takei N, Sakuma K, Kuno Y, Marra P, Takahashi Y 2021 Nat. Phys. 17 844
Google Scholar
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Google Scholar
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Google Scholar
[4] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[5] Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004
Google Scholar
[6] Xiao T, Xie D, Dong Z, Chen T, Yi W, Yan B 2021 Sci. Bull. 66 2175
Google Scholar
[7] Szameit A, Rechtsman M C 2024 Nat. Phys. 20 905
Google Scholar
[8] König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L, Qi X L, Zhang S C 2007 Science 318 766
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[9] Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438
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