Edge mode dynamics of quenched topological wires

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.

Figure 1

Phases of the Kitaev model, Eq. (10). At each lattice site, $j$, the two dots represent the two Majorana operators, ${\gamma }_{j,1}$ and ${\gamma }_{j,2}$ (real and imaginary parts of $c_j$), defined after Eq. (17). The lines represent the links between Majorana operators at given points in parameter space. Trivial phase with parameters: $\mathrm{\Delta }=0,\left|\mu \right|>2t$ and topological phase with $\mu =0,\mathrm{\Delta }=t$. In the trivial phase, the two Majoranas at each site are linked in the Hamiltonian and they constitute usual fermionic modes. In the topological phase, the Majoranas are linked at nearest-neighbor sites and the first and last Majorana operators are decoupled, and therefore they have zero energy.

Figure 2

Phases of the Shockley model, Eq. (21). The symbols have the same meaning as in Fig. 1. Trivial phase with $t_2=0$ and topological phase with $t_1=0$. In the trivial phase, the Majorana fermions are coupled to form fermionic modes at the same site. In the topological phase, the links are between nearest-neighbor sites and there are four Majorana operators decoupled at the edges. However, these give origin to zero-energy fermionic modes, one at each end of the system.

Figure 3

Phases of the SSH-Kitaev model, Eq. (26). The symbols have the same meaning as in Fig. 1. When $\mathrm{\Delta }=0$ the model reduces to the SSH model, and for negative $\eta$ the model is topologically nontrivial with edge states represented by the decoupled Majorana operators. As in the Shockley model, since at each end site there are two decoupled Majoranas, these combine to form edge fermionic modes. This constitutes phase SSH2 with $\eta =-1,\mathrm{\Delta }=0$ and two edge modes. If superconductivity is present, and there is no dimerization $\eta =0$, the model reduces to the Kitaev model. The phase K1 with $\eta =0,\mathrm{\Delta }=t$ has two decoupled Majorana operators, one at each end, and therefore there is one Majorana mode at each edge. The model interpolates between Majorana modes and fermionic modes as the parameters change. There is also a trivial phase with no zero-energy modes denoted SSH0, which is similar to the trivial phase of the Shockley model.

Figure 4

Phase diagram of the Kitaev model. Regions I and II are topologically nontrivial, and region III is trivial. The points separate regions where critical quenches do or do not lead to oscillations in the survival probability of a Majorana state of region I.

Figure 5

Zero-energy mode wave functions of the Kitaev model for different points approaching the critical point at $\mu =0$. In the last panel we show the dependence of the decay length of the wave functions as a function of $\mathrm{\Delta }$ and $1/\mathrm{\Delta }$. The results are for a system size of $N=200$. At the point $\mathrm{\Delta }=1$ the wave function is strictly local at one edge of the system. Each Majorana state is perfectly localized with a decay length $\xi =0$. For $\mathrm{\Delta }=0.5$ the decay length is also very small and the two Majorana modes (in black, left edge, and in red, right edge) are decoupled. For smaller values of $\mathrm{\Delta }$, the two Majorana modes are coupled and each is peaked at both ends of the chain. For very small values of $\mathrm{\Delta }$ the modes get extended as one tends to the gapless regime at $\mathrm{\Delta }=0$.

Figure 6

In the first two panels the survival probability, $P\left(T\right)$, of a Majorana mode in the Kitaev model is shown as one approaches critical points. In the third panel the crossover to period doubling is shown as one approaches the critical region. In panels (a), (b), and (c) $N=100$. In (a) $\mu =0$, in (b) $\delta =0.5$, and in (c) $\mu =0$. In panel (d) the linear dependence of the point of crossover, ${\mathrm{\Delta }}_d$, on $1/N$ is shown taking $\mu =0$.

Figure 7

(a) At finite chemical potential, $\mu =0.5$, the critical quench to the line $\mathrm{\Delta }=0$ does not lead to oscillations of the survival probability of the Majorana mode. Close to the intersection of the two critical lines near $\mu =2t,\mathrm{\Delta }=0$ there are no oscillations as well as shown in (b) for $\mu =1.9$.

Figure 8

Overlaps for the Kitaev model as a function of energy. (a) The critical point is $\mu =0,\mathrm{\Delta }=0$ and in (b) it is $\mu =2,\mathrm{\Delta }=0.5$. In the third panel, (c), the even-odd effect in the overlaps is shown considering $\mu =0$.

Figure 9

Solitonic-like vs constructive interference behavior of the wave functions in the Kitaev model. In (a) the initial state is far from the critical point (CP), and in (b) one is close to the CP. In (a) the quench takes place from $\mu =0,\mathrm{\Delta }=0.5$ to $\mu =0,\mathrm{\Delta }=0$ and in (b) from $\mu =0,\mathrm{\Delta }=0.1$ to the same CP. The results are for a system size $N=100$. In (a) the sequence of times is $T=2,10,20,26,30,40$ and in (b) $T=2,10,20,25,30,40$.

Figure 10

Top panel: work distribution for the Kitaev model. Lower panel: survival probability and its Fourier analysis for the Kitaev model.

Figure 11

Survival probability of different initial single-particle states, labeled by $n$, where $n=200,201$ are the Majorana zero-energy modes. In the top panel the case of initial $\mathrm{\Delta }=0.01$ is shown, and in the middle panel the case of $\mathrm{\Delta }=0.1$ is shown. In the lower panel, a low-energy zoom is shown of the case with $\mathrm{\Delta }=0.1$.

Figure 12

Overlap of states in trivial phase to the Majorana final state in the topological phase far and close to the critical line, respectively.

Figure 13

Probability to find the original Majorana state, $n_0=m_0$, due to quench ${\xi }_2\to {\xi }_3$ as a function of $T_2$ and $T$, described by Eq. (8). The sequence of quenches is such that at $T_0=0$ there is a quench ${\xi }_0\to {\xi }_1$. At time $T_1$ there is a quench to ${\xi }_2={\xi }_0$. The time $T_1$ is chosen such that the survival probability of the original Majorana vanishes. At time $T_2$ there is a new quench to ${\xi }_3$ in the topological region, close to but different from ${\xi }_0$. The survival probability is finite, and similar results are obtained for other levels.

Figure 14

Survival probability of edge modes of the Shockley model. Top panel: off-critical quench from the topological region ($t_1=1,t_2=2$) to the trivial region ($t_1=1,t_2=0.5$). In the lower panel, critical quenches are considered from the topological region to the transition point ($t_1=t_2$).

Figure 15

Phase diagram of the SSH-Kitaev model for $\mu =0$.

Figure 16

Survival probability in off-critical quenches for the SSH-Kitaev model across a transition line. The parameters are $\mu =0,\eta =0,\mathrm{\Delta }=0.2$ to $\eta =0.7$ and $-0.7$, respectively.

Figure 17

Critical quenches in the SSH-Kitaev model: survival probability and overlaps. The CP on the first two panels is $\eta =0,\mathrm{\Delta }=0$ (SSH2) and on the last two panels the CP is $\eta =0.5,\mathrm{\Delta }=0.5$ (K1).

Figure 18

Survival probabilities of the SSH0-K1 transition in the SSH-Kitaev model (trivial to topological). Originally we have an extended state, and in the final we have an edge state but not very localized. In the top panel, we have $\mu =0$ and $\eta =0.7,\mathrm{\Delta }=0$ to $\eta =0.1,\mathrm{\Delta }=0.15$ or the initial value of $\mathrm{\Delta }=0.1$. In the lower panel, we have also SSH0 to K1 and $\mu =0$, but we start from $\eta =0.2,\mathrm{\Delta }=0$ and change to $\mathrm{\Delta }=0.21,0.25,0.3,0.4,0.5,0.6,0.7,0.8,0.9$.

Figure 19

Survival probability of different states for the transition K1-SSH0 in the SSH-Kitaev model.

Figure 20

Modulated frequencies in the survival probability of Majorana states and finite energy peaks of the overlaps in the SSH-Kitaev model. The transition is of the type SSH2-K1.

Figure 21

Fourier decomposition of survival probability of several quenches in the SSH-Kitaev model. The parameters are as follows: (a) $\mu =0,\eta =0.7,\mathrm{\Delta }=0.15\to \mu =0,\eta =0.1,\mathrm{\Delta }=0.15$ (SSH0 $\to$ K1); (b) $\mu =0,\eta =0.7,\mathrm{\Delta }=0\to \mu =0,\eta =0.1,\mathrm{\Delta }=0.15$ (SSH0 $\to$ K1); (c) $\mu =0,\eta =0.2,\mathrm{\Delta }=0\to \mu =0,\eta =0.2,\mathrm{\Delta }=0.25$ (SSH0 $\to$ K1); and (d) $\mu =0,\eta =-0.2,\mathrm{\Delta }=0.1\to \mu =1,\eta =-0.2,\mathrm{\Delta }=0.1$ (SSH2 $\to$ K1).