一维化学势调制的p波超导体中的拓扑量子相变

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

本文研究了一维公度势和非公度势调制下的p波超导量子线系统的拓扑相变.在公度势调制下,通过计算Z2拓扑不变量确定系统的相图,指出系统的拓扑相变强烈地依赖于调制参数α和相移δ.在非公度势调制下,以α=(√5-1)/2,δ=0为例,计算系统的低能激发谱、Z2拓扑不变量以及逆参与率等,发现p波配对强度△∈(0,0.33)时,系统存在拓扑非平庸超导相,拓扑平庸超导相和拓扑平庸局域相的转变.而当p波配对强度△>0.33时,系统存在拓扑非平庸超导相和拓扑平庸局域相的转变.

We consider a one-dimensional p-wave superconducting quantum wire with the modulated chemical potential,which is described by H=∑i[(-tc^+i+1+△cici+1+h.c.)+Vini],Vi=Vcos(2πiα+δ)/1-bcos(2πiα+δ) and can be solved by the Bogoliubov-de Gennes method.When b=0,αis 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 Z2 topological invariant of the bulk system.We find the phase transition strongly depends on the strength of potential V and the phase shiftδ.For some certain special parametersαandδ,the critical strength of the phase transition is infinity.For the incommensurate case,i.e.α=(√5-1)/2,the phase diagram is identified by analyzing the low-energy spectrum,the amplitudes of the lowest excitation states,the Z2 topological invariant and the inverse participation ratio(IPR)which characterizes the localization of the wave functions.Three phases emerge in such case forδ=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 Z2 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.