Acta Physica Sinica, Volume. 69, Issue 7, 070302-1(2020)
Topological quantum phase transitions in one-dimensional p-wave superconductors with modulated chemical potentials
Jing-Nan Wu1,2, Zhi-Hao Xu1,2、*, Zhan-Peng Lu1,2, and Yun-Bo Zhang1
Author Affiliations
1Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, Chinashow less
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.