Topological quantum phase transitions in one-dimensional p-wave superconductors with modulated chemical potentials

We consider a one-dimensional p -wave superconducting quantum wire with the modulated chemical potential, which is described by \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}, \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}and can be solved by the Bogoliubov-de Gennes method. When \begin{document}$b=0$\end{document}, \begin{document}$\alpha$\end{document}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 \begin{document}$Z_2$\end{document}topological invariant of the bulk system. We find the phase transition strongly depends on the strength of potential V and the phase shift \begin{document}$\delta$\end{document}. For some certain special parameters \begin{document}$\alpha$\end{document}and \begin{document}$\delta$\end{document}, the critical strength of the phase transition is infinity. For the incommensurate case, i.e. \begin{document}$\alpha=(\sqrt{5}-1)/2$\end{document}, the phase diagram is identified by analyzing the low-energy spectrum, the amplitudes of the lowest excitation states, the \begin{document}$Z_2$\end{document}topological invariant and the inverse participation ratio (IPR) which characterizes the localization of the wave functions. Three phases emerge in such case for \begin{document}$\delta=0$\end{document}, topologically nontrivial superconductor, topologically trivial superconductor and topologically trivial Anderson insulator. For a topologically nontrivial superconductor, it displays zero-energy Majorana fermions with a \begin{document}$Z_2$\end{document}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.