Survival probability of an edge Majorana in a one-dimensional $p$-wave superconducting chain under sudden quenching of parameters

We consider the temporal evolution of a zero-energy edge Majorana of a spinless $p$-wave superconducting chain following a sudden change of a parameter of the Hamiltonian. Starting from one of the topological phases that has an edge Majorana, the system is suddenly driven to the other topological phase or to the (topologically) trivial phases and to the quantum critical points (QCPs) separating these phases. The survival probability of the initial edge Majorana as a function of time is studied following the quench. Interestingly when the chain is quenched to the QCP, we find a nearly perfect oscillation of the survival probability, indicating that the Majorana travels back and forth between two ends, with a time period that scales with the system size. We also generalize to the situation when there is a next-nearest-neighbor hopping in a superconducting chain and there results in a pair of edge Majorana at each end of the chain in the topological phase. We show that the frequency of oscillation of the survival probability gets doubled in this case. We also perform an instantaneous quenching of the Hamiltonian (with two Majorana modes at each end of the chain) to an another Hamiltonian which has only one Majorana mode in equilibrium; the MSP shows oscillations as a function of time with a noticeable decay in the amplitude. On the other hand for a quenching which is reverse to the previous one, the MSP decays rapidly and stays close to zero with fluctuations in amplitude.

Figure 1

Phase diagram of the 1D $p-$ wave superconducting system [see Eq. (1)]. Phases I and II are topologically nontrivial, while phase III is topologically trivial. Quenching paths A and B are shown in the phase diagram.

Figure 2

Two isolated Majorana states are localized in two edges of a 100-site open Majorana chain in phase I ($\xi =0.1$ and $\eta =0.0$). $j$ labels as Majorana sites $1,2,\cdots ,200$. One can see that here probability is nonzero only if $j$ is odd (even) for the left (right) end of Majorana chain.

Figure 3

Energy spectrum of the Majorana chain of system size $N=100$ as a function of $\xi$ with $w=1$ and $\eta =0$ using (a) open boundary condition (OBC) and (b) periodic boundary condition (PBC), respectively. Note that two zero-energy Majorana modes are present in case (a) but not in case (b). We mention that the energy is scaled by a factor $1/4$ in Eq. (5) in comparison to Eq. (2).

Figure 4

Majorana survival probability of a zero-energy Majorana for different quench schemes along the path A and probability of Majorana after quench. (a) MSP decays rapidly and does not revive significantly for quenching from phase I ($\xi =0.1$) to phase II ($\xi =-0.1$). (b) Quenching from phase I ($\xi =0.1$) to the QCP ($\xi =0.0$), MSP shows nearly perfect collapse and revival with time $t$ and scales linearly with the system size $N$. (c) Probability of the end Majorana at time $t=100$ after quench at the QCP with the Majorana site $j$. This shows that at this instant the probability of Majorana is maximum around the center of the chain.

Figure 5

The probabilities of an end mode ($\xi =1.0$ and $\eta =0.9$) after a quench (a) at a point ($\xi =1.0$ and $\eta =1.1$) of phase III and (b) at a QCP ($\xi =1.0$ and $\eta =1.0$) along the path B show same behavior as of the path A.

Figure 6

System is suddenly quenched within the same phase (I) from ($\xi =0.2$ and $\eta =0.0$) to ($\xi =0.1$ and $\eta =0.0$) and MSP becomes (a) rapidly fluctuating function of time $t$ with a mean value of nearly $0.8$ and (b) probability of the end Majorana at $t=100$, with the Majorana sites $j$ after quench.

Figure 7

Two zero-energy Majorana modes of a system Hamiltonian defined in Eq. (11) exist in two edges of a 100-site open Majorana chain in phase I ($\xi =0.1$ and $\eta =0.0$). Similarly to Fig. 2, here as well $j$ labels Majorana sites $1,2,\cdots ,200$. The probabilities are nonzero only if $j$ is odd (even) for the left (right) end of Majorana chain.

Figure 8

$P_m\left(t\right)$ of a right-end zero-energy Majorana mode after quenching along different paths. (a) MSP decays rapidly and stays minimum with some noisy fluctuation of small amplitude when the system [see Eq. (9)] is quenched from phase I ($\xi =0.1$) to phase II ($\xi =-0.1$) along path A. (b) For quenching to QCP $P_m\left(t\right)$ is nearly perfect oscillatory function of time $t$ with a interesting fact that its time period becomes half of the earlier case (see Fig. 4) and scales linearly with the system size $N$. (c) Time variation of MSP with $\xi =1.0$ and $\eta =0.9$ following a quench to a point ($\xi =1.0$ and $\eta =1.1$) and (d) the QCP ($\xi =1.0$ and $\eta =1.0$) along path B shows the same behavior as of the path A($\eta =0$).

Figure 9

Energy spectrum of the Majorana chain [with only next-nearest-neighbor hopping and interaction as defined in Eq. (9)] as a function of $\xi$ with $\eta =0$ and $N=100$ sites for open boundary condition. In this case also a energy scale difference exists (see caption of Fig. 3).

Figure 10

System changes suddenly from a phase with two Majorana fermions to a phase which contains only one Majorana mode at each end of the chain at time $t=0$. We have set here the parameter values $\xi =0.1$ and $\eta =0.0$. (a) MSP of a left end ($c_3$) Majorana mode shows collapse and revival with time, but at each revival there are fluctuations and the peaks of revivals decrease rapidly. (b) On the other hand when the system is quenched reversely the Majorana ($c_1$) decoheres rapidly with time (with no prominent revival) and fluctuates around zero mean.

Figure 11

Initially, the Hamiltonian $H_2$ is fixed at the parameter values $\xi =0.1$ and $\eta =0.0$ and then it is quenched to the critical point ($\xi =0.0$ and $\eta =0.0$) of the Hamiltonian $H_1$ at $t=0$. (a) MSP of a left-end ($c_3$) Majorana mode shows collapse and revival with time quite similar to Fig. 10; the amplitude of the revival decays rapidly if the initial value of $\Delta$ is larger, i.e., the Hamiltonian $H_2$ is deep in the topological phase away from the QCP. (b) There is a complete loss of coherence when the system is quenched reversely.