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.
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.
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 (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 (λ_{d}), we observe a perfect oscillation of the MSP as a function of time with a single frequency (determined by the impurity strength λ_{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.
In the context of slow quenching dynamics of a p-wave superconducting chain, it has been shown that a Majorana edge state cannot be adiabatically transported from one topological phase to the other separated by a quantum critical line. On the other hand, the inclusion of a phase factor in the hopping term, which breaks the effective time-reversal invariance, results in an extended gapless region between two topological phases. We show that for a finite chain with an open boundary condition, there exists a nonzero probability that a Majorana edge state can be adiabatically transported from one topological phase to the other across this gapless region following a slow quench of the superconducting term; this happens for an optimum transit time, which is proportional to the system size and diverges for a thermodynamically large chain. We attribute this phenomenon to the mixing of the Majorana only with low-lying inverted bulk states. Remarkably, the Majorana state always persists with the same probability even after the quenching is stopped. For a periodic chain, on the other hand, we find a Kibble-Zurek scaling of the defect density with a renormalized rate of quenching.
The fermionic and Majorana edge mode dynamics of various topological systems are compared, after a sudden global quench of the Hamiltonian parameters takes place. Attention is focused on the regimes where the survival probability of an edge state has oscillations either due to critical or off-critical quenches. The nature of the wave functions and the overlaps between the eigenstates of different points in parameter space determine the various types of behaviors, and the distinction due to the Majorana nature of the excitations plays a lesser role. Performing a sequence of quenches, it is shown that the edge states, including Majorana modes, may be switched off and on. Also, the generation of Majoranas due to quenching from a trivial phase is discussed.
In the sequence of quenches to either nontopological phases or other topological phases, we study the stability of Majorana fermions at the edges of a two-dimensional topological superconductor with spin-orbit coupling and in the presence of a Zeeman term. Both instantaneous and slow quenches are considered. In the case of instantaneous quenches, the Majorana modes generally decay, but for a finite system there is a revival time that scales to infinity as the system size grows. Exceptions to this decaying behavior are found in some cases due to the presence of edge states with the same momentum in the final state. Quenches to a topological Z(2) phase reveal some robustness of the Majorana fermions in the sense that even though the survival probability of the Majorana state is small, it does not vanish. If the pairing is not aligned with the spin-orbit Rashba coupling, it is found that the Majorana fermions are fairly robust with a finite survival probability. It is also shown that the Chern number remains invariant after the quench, until the propagation of the mode along the transverse direction reaches the middle point, beyond which the Chern number fluctuates between increasing values. The effect of varying the rate of change in slow quenches is also analyzed. It is found that the defect production is nonuniversal and does not follow the Kibble-Zurek scaling with the quench rate, as obtained before for other systems with topological edge states.
We investigate the evolutions of edge Majorana fermions (MFs), and we show that they can be used to characterize different topological phases and study topological phase transitions. For some limiting cases of the evolution process for the Kitaev chain model and the Su-Schrieffer-Heeger (SSH) model, we give analytical expressions of the survival probabilities (SPs) of the edge MFs. For a general case, we consider a dimerized Kitaev model as well as the Kitaev chain with disorder chemical potential, and we numerically calculate the SPs of two edge correlation MFs. Our results show that both of them are equal to zero, one of them is equal to zero at some times, or neither of them is equal to zero, which corresponds to the SSH-like topological, topological superconductor, and trivial phases, respectively.
A one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with p-wave superconductivity. In the weak-coupling BCS regime, it exhibits a zero-energy Majorana mode at each end of the chain. Here, we consider a variation of the model, which represents a superconductor with longer-ranged kinetic energy and pairing amplitudes, as is likely to occur in more realistic systems. It possesses a richer zero-temperature phase diagram and has several quantum phase transitions. From an exact solution of the model, we find that these phases can be classified according to the number of Majorana zero modes of an open chain: zero, one, or two at each end. The model possesses a multicritical point where phases with zero, one, and two Majorana end modes meet. The number of Majorana modes at each end of the chain is identical to the topological winding number of the Anderson pseudospin vector that describes the BCS Hamiltonian. The topological classification of the phases requires a unitary time-reversal symmetry to be present. When this symmetry is broken, only the number of Majorana end modes modulo 2 can be used to distinguish two phases. In one of the regimes, the wave functions of the two phase-shifted Majorana zero modes decay exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to that in the asymptotic connected spin-spin correlation of the XY model in a transverse field, to which our model is dual.