Survival probability in a quenched Majorana chain with an impurity

TL;DR

This paper studies the dynamics of Majorana edge states in a one-dimensional p-wave superconductor with a new topological phase, analyzing how impurities affect their survival probability during quantum quenches.

Contribution

It introduces a model with next-nearest-neighbor hopping leading to a new topological phase and analyzes the impact of impurities on Majorana survival probability during quenches.

Findings

Perfect oscillations of MSP with a single frequency for small impurities

Beating patterns in MSP near phase boundaries due to energy level modifications

Impurity-induced energy level shifts explain MSP behavior

Abstract

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 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 ($H_i$) is changed to the quantum critical as well as off-critical final Hamiltonian ($H_f$) which additionally contains an impurity term ($H_{imp}$) that breaks the time-reversal invariance. For the off-critical quenching inside the new topological phase with $H_f= H_i +H_{imp}$, and small impurity strength ($\lambda_d$), 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 non-topological 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.