Topological quantum phase transitions in one-dimensional p-wave superconductors with modulated chemical potentials
Authors: Wu Jing-Nan , Xu Zhi-Hao , Lu Zhan-Peng , Zhang Yun-Bo ,   Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China,   State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University,  Taiyuan 030006, China
Published: December 2020

We consider a one-dimensional <i>p</i>-wave superconducting quantum wire with the modulated chemical potential, which is described by <inline-formula><tex-math id="M9">\begin{document}$\hat{H}= \displaystyle\sum\nolimits_{i}\left[ \left( -t\hat{c}_{i}^{\dagger }\hat{c}_{i+1}+\Delta \hat{c}_{i}\hat{c}_{i+1}+ h.c.\right) +V_{i}\hat{n}_{i}\right]$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M9.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$V_{i}=V\dfrac{\cos \left( 2{\text{π}} i\alpha + \delta \right) }{1-b\cos \left( 2{\text{π}} i\alpha+\delta \right) }$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M10.png"/></alternatives></inline-formula> and can be solved by the Bogoliubov-de Gennes method. When <inline-formula><tex-math id="M11">\begin{document}$b=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M11.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$\alpha$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M12.png"/></alternatives></inline-formula> is a rational number, the system undergoes a transition from topologically nontrivial phase to topologically trivial phase which is accompanied by the disappearance of the Majorana fermions and the changing of the <inline-formula><tex-math id="M13">\begin{document}$Z_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M13.png"/></alternatives></inline-formula> topological invariant of the bulk system. We find the phase transition strongly depends on the strength of potential <i>V</i> and the phase shift <inline-formula><tex-math id="M14">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M14.png"/></alternatives></inline-formula>. For some certain special parameters <inline-formula><tex-math id="M15">\begin{document}$\alpha$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M15.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$\delta$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M16.png"/></alternatives></inline-formula>, the critical strength of the phase transition is infinity. For the incommensurate case, i.e. <inline-formula><tex-math id="M17">\begin{document}$\alpha=(\sqrt{5}-1)/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M17.png"/></alternatives></inline-formula>, the phase diagram is identified by analyzing the low-energy spectrum, the amplitudes of the lowest excitation states, the <inline-formula><tex-math id="M18">\begin{document}$Z_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M18.png"/></alternatives></inline-formula> topological invariant and the inverse participation ratio (IPR) which characterizes the localization of the wave functions. Three phases emerge in such case for <inline-formula><tex-math id="M19">\begin{document}$\delta=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M19.png"/></alternatives></inline-formula>, topologically nontrivial superconductor, topologically trivial superconductor and topologically trivial Anderson insulator. For a topologically nontrivial superconductor, it displays zero-energy Majorana fermions with a <inline-formula><tex-math id="M20">\begin{document}$Z_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20191868_M20.png"/></alternatives></inline-formula> topological invariant. By calculating the IPR, we find the lowest excitation states of the topologically trivial superconductor and topologically trivial Anderson insulator show different scaling features. For a topologically trivial superconductor, the IPR of the lowest excitation state tends to zero with the increase of the size, while it keeps a finite value for different sizes in the trivial Anderson localization phase.