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

Wu Jing-Nan^{1,2}, Xu Zhi-Hao^{1,2,*}, Lu Zhan-Peng^{1,2}, and Zhang Yun-Bo^{1}

Author Affiliations

^{1}Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China^{2}State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China

## Abstract

We consider a one-dimensional *p*-wave superconducting quantum wire with the modulated chemical potential, which is described by $\hat{H}= \displaystyle\sum\nolimits_{i}\left[ \left( -t\hat{c}_{i}^{\dagger }\hat{c}_{i+1}+\Delta \hat{c}_{i}\hat{c}_{i+1}+ h.c.\right) +V_{i}\hat{n}_{i}\right]$![]()

, $V_{i}=V\dfrac{\cos \left( 2{\text{π}} i\alpha + \delta \right) }{1-b\cos \left( 2{\text{π}} i\alpha+\delta \right) }$![]()

and can be solved by the Bogoliubov-de Gennes method. When $b=0$![]()

, $\alpha$![]()

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 $Z_2$![]()

topological invariant of the bulk system. We find the phase transition strongly depends on the strength of potential *V* and the phase shift $\delta$![]()

. For some certain special parameters $\alpha$![]()

and $\delta$![]()

, the critical strength of the phase transition is infinity. For the incommensurate case, i.e. $\alpha=(\sqrt{5}-1)/2$![]()

, the phase diagram is identified by analyzing the low-energy spectrum, the amplitudes of the lowest excitation states, the $Z_2$![]()

topological invariant and the inverse participation ratio (IPR) which characterizes the localization of the wave functions. Three phases emerge in such case for $\delta=0$![]()

, topologically nontrivial superconductor, topologically trivial superconductor and topologically trivial Anderson insulator. For a topologically nontrivial superconductor, it displays zero-energy Majorana fermions with a $Z_2$![]()

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.

## keywords