Survival probability in a quenched Majorana chain with an impurity

We investigate the dynamics of a one-dimensional $p$-wave superconductor with next-nearest-neighbor hopping and superconducting interaction derived from a three-spin interacting Ising model in transverse field by mapping to Majorana fermions. The next-nearest-neighbor hopping term leads to a new topological phase containing two zero-energy Majorana modes at each end of an open chain, compared to a nearest-neighbor $p$-wave superconducting chain. We study the Majorana survival probability (MSP) of a particular Majorana edge state when the initial Hamiltonian $\left(H_i\right)$ is changed to the quantum critical as well as off-critical final Hamiltonian $\left(H_f\right)$, which additionally contains an impurity term $\left(H_{\mathrm{imp}}\right)$ that breaks the time-reversal invariance. For the off-critical quenching inside the new topological phase with $H_f=H_i+H_{\mathrm{imp}}$, and small impurity strength $\left({\lambda }_d\right)$, we observe a perfect oscillation of the MSP as a function of time with a single frequency (determined by the impurity strength ${\lambda }_d)$ that can be analyzed from an equivalent two-level problem. On the other hand, the MSP shows a beating like structure with time for quenching to the phase boundary separating the topological phase (with two edge Majoranas at each edge) and the nontopological phase where the additional frequency is given by inverse of the system size. We attribute this behavior of the MSP to the modification of the energy levels of the final Hamiltonian due to the application of the impurity term.

Figure 1

Phase diagram of the model with Hamiltonian in Eq. (1) for $h=1$. The phase boundary ${\lambda }_2=1+{\lambda }_1$ separates upper topological phase (with two zero energy Majorana edge states at each end) from one Majorana $\left(n=1\right)$ topological phase. The other two phase boundaries are ${\lambda }_2=1-{\lambda }_1$ $(a\text{−}e\text{−}d$ line) and ${\lambda }_2=-1$ $(b\text{−}d$ line). The paramagnetic region corresponds to a nontopological phase with $n=0$ Majorana modes. Both the topological phases with $n=2$ Majorana modes, i.e., lower and upper, are characterized by the presence of $a_1$ and $a_2$ isolated modes at the left end and $b_{N-1},b_N$ at the right end of the chain.

Figure 2

(a) The MSP of an initial zero-energy edge Majorana $\left(a_1\right)$ shows perfect oscillation with time for smaller values of ${\lambda }_d$, whereas it decays nearly equal to zero value and does not revive significantly for higher values of ${\lambda }_d$. The rate of decay increases as the value of ${\lambda }_d$ is increased. In this case, the initial Hamiltonian is considered to be in the $n=2$ phase with ${\lambda }_1=0.2$ and ${\lambda }_2=2.0$, and the final Hamiltonian has an additional impurity term $H_{\mathrm{imp}}={\lambda }_d{a}_1{a}_2$ over the initial Hamiltonian. (b) indicates that the energy corresponding $a_1$ edge Majorana increases linearly with ${\lambda }_d$ followed by a saturation as ${\lambda }_d$ becomes large. (c) shows the behavior of MSP for $a_1$ with ${\lambda }_d=0.2$ for different interaction range of impurity term $H_{\mathrm{imp}}={\lambda }_d{a}_1{a}_q$ where $q$ can take even or odd values. (d) indicates that the energy corresponding the $a_1$ Majorana decays exponentially with $q$. On the other hand, inset shows that the prefactor of the exponential fall is different for even and odd $q$. Here, $N=200$.

Figure 3

The MSP as a function of time when the system is quenched from upper $n=2$ phase $\left({\lambda }_1=1.8,{\lambda }_2=3.0\right)$ to $n=1$ phase $\left({\lambda }_1=2.2,{\lambda }_2=3.0\right)$ or vice versa. (a) The MSP of $a_1$ Majorana mode decays initially and then becomes rapidly oscillating function of time with different mean values depending upon the values of ${\lambda }_d$. (b) On the other hand, the MSP of $a_2$ Majorana mode decays rapidly and does not revive significantly for this quenching scheme. (c) shows the MSP of $a_1$ mode as a function of time $t$ for smaller value of ${\lambda }_d$ when the sudden quenching is carried out in the reverse path, i.e., from $n=1$ phase to $n=2$ phase. We consider $N=200$ for the above three cases. (d) This plot indicates that the time period of the collapse and revival of the MSP here is independent of system size confirming the off-criticality of the associated dynamics. Here, ${\lambda }_d=0.2$. For all the cases we consider ${\lambda }_d{a}_1{a}_2$ as the impurity Hamiltonian.

Figure 4

Variation of the MSP of a zero-energy Majorana mode $\left(a_1\right)$ as a function of time when the Hamiltonian is quenched from lower $n=2$ phase $\left({\lambda }_1=1.0,{\lambda }_2=-1.1\right)$ to the QCP $\left({\lambda }_1=1.0,{\lambda }_2=-1.0\right)$ lying on $b\text{−}d$ line. (a) Beating pattern in the MSP as a function of time when the impurity term of small magnitude is applied along with the final Hamiltonian. It can be observed that the time period of the envelope of the beating structure varies as $T_{r2}\propto 1/{\lambda }_d$. (b) On the other hand, for larger values of ${\lambda }_d$ one can not find any beating pattern in the MSP. Here, $N=100$ for the above two plots. (c) depicts that the interior frequency of oscillation in the beating pattern is linearly dependent on $N$. Here, ${\lambda }_d=0.1$. (d) Variation of the energy difference between two consecutive energy levels of low lying states as a function of ${\lambda }_d$. For all the above cases we consider ${\lambda }_d{a}_1{a}_2$ as the impurity term.

Figure 5

$P_m\left(t\right)$ for $a_1$ Majorana mode as a function of time $t$ for the quenching from upper $n=2$ phase $\left({\lambda }_1=0.2,{\lambda }_2=2\right)$ to $(n=2)\text{−}(n=1)$ phase boundary $({\lambda }_1=1$ and ${\lambda }_2=2)$ for different values of ${\lambda }_d$ with impurity term $H_{\mathrm{imp}}={\lambda }_d{a}_1{a}_2$ in the final Hamiltonian. In this case the MSP does not exhibit any beating pattern and also the revival structure destroys with increasing value of ${\lambda }_d$.

Figure 6

Time evolution of the MSP of a zero-energy Majorana mode $\left(a_1\right)$ becomes independent of the impurity strength ${\lambda }_d$ when the Hamiltonian is quenched from $n=1$ phase $\left({\lambda }_1=1.0,{\lambda }_2=0.2\right)$ to the QCP $\left({\lambda }_1=1.0,{\lambda }_2=0.0\right)$ lying on the $a\text{−}d$ phase boundary. Here, $N=200$.