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  • Received: Dec. 10, 2019

    Accepted: --

    Posted: Nov. 20, 2020

    Published Online: Nov. 20, 2020

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    DOI: 10.7498/aps.69.20191868

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    Jing-Nan Wu, Zhi-Hao Xu, Zhan-Peng Lu, Yun-Bo Zhang. Topological quantum phase transitions in one-dimensional p-wave superconductors with modulated chemical potentials[J]. Acta Physica Sinica, 2020, 69(7): 070302-1

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Acta Physica Sinica, Vol. 69, Issue 7, 070302-1 (2020)

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

Wu Jing-Nan1,2, Xu Zhi-Hao1,2,*, Lu Zhan-Peng1,2, and Zhang Yun-Bo1

Author Affiliations

  • 1Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
  • 2State 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.

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