Fate of Majorana fermions and Chern numbers after a quantum quench

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.

Figure 1

Survival probability of the Majorana state of the one-dimensional fully polarized $p$-wave Kitaev model for different transitions across the phase diagram: (a) transition within the same topological phase, I $({\epsilon }_F=0.5,\Delta =0.6)\to ({\epsilon }_F=1.0,\Delta =0.6)$; (b) transition from the topological phase I to the trivial phase, III $({\epsilon }_F=0.5,\Delta =0.6)\to ({\epsilon }_F=2.2,\Delta =0.6)$; (c) transition from the topological phase I with positive $\Delta$ to the topological phase II with negative $\Delta ({\epsilon }_F=0.5,\Delta =0.6)\to ({\epsilon }_F=0.5,\Delta =-0.6)$; (d) transition within the same topological phase, I, to the quantum critical point $({\epsilon }_F=0,\Delta =0.1)\to ({\epsilon }_F=0,\Delta =0)$, where the system is gapless. The system has 100 sites. Time is in units of $1/\tilde{t}$ throughout.

Figure 2

Topological phases and their Chern $\left(C\right)$ numbers as a function of chemical potential and magnetization (expressed in units of $\tilde{t}$). At zero magnetization and when $-4\le {\epsilon }_F\le 4$, the system is in a $Z_2$ phase. At finite magnetization, the system is in a $Z$ phase characterized by the Chern number. The transitions occur at three sets of momenta, $\mathbf{k}=(0,0),(\pi ,0)$, and $(\pi ,\pi )$. The three dashed lines I, II, and III correspond to the slow quenches considered in Sec. 7.

Figure 3

Time evolution of real space $|u_{\uparrow }{|}^2$ for $(M_z=2,{\epsilon }_F=-5)\to (M_z=0,{\epsilon }_F=-5)$ (along line I of Fig. 2), $C=1\to C=0$ (trivial) (left column) for $t=0$, 50, and 62, shown in (a), (b), and (c), respectively and $|u_{\uparrow }{|}^2$ for $(M_z=3.5,{\epsilon }_F=0)\to (M_z=4.5,{\epsilon }_F=0),C=-2\to C=0$ (trivial) (right column) for $t=0$, 11, and 29, shown in (d), (e), and (f), respectively, with strong spin-orbit coupling. Note that in these transitions, there are no edge states in the final states. In the various panels, the horizontal axis is the $y$ direction and the vertical direction is the $x$ direction. The system size is $31\times 41$.

Figure 4

Time evolution of real space $|u_{\uparrow }{|}^2$ for $(Mz=2,{\epsilon }_F=-1)\to (M_z=4,{\epsilon }_F=-1),C=-2\to C=-1$ for $t=0,t=10,t=22$, shown in (a), (b), and (c), respectively, with strong spin-orbit coupling for the same system size as in Fig. 3. Note that there are edge states in the final state.

Figure 5

Survival probability of the Majorana state of the two-dimensional triplet superconductor with strong spin-orbit coupling for the transitions considered in Figs. 3 and 4: (a) transition $(M_z=2,{\epsilon }_F=-5)\to (M_z=0,{\epsilon }_F=-5),C=1\to C=0$ (trivial); (b) transition $(M_z=3.5,{\epsilon }_F=0)\to (M_z=4.5,{\epsilon }_F=0),C=-2\to C=0$ (trivial); and (c) transition $(M_z=2,{\epsilon }_F=-1)\to (M_z=4,{\epsilon }_F=-1),C=-2\to C=-1$.

Figure 6

Time evolution of real space $|u_{\uparrow }{|}^2$ for weak spin-orbit coupling ($\alpha =1,{\epsilon }_F=-1,d=1$ and $M_z=1.2\to M_z=0.5,C=2\to C=0$ for $t=0,t=11,t=21$, shown in (a), (b), (c), respectively.

Figure 7

Survival probability of the Majorana state (highest value) and second excited state (lowest value) of the two-dimensional triplet superconductor for weak spin-orbit coupling: $\alpha =1,{\epsilon }_F=-1,d=1$ and initial state $M_z=1.2$ and new parameter set $M_z=0.5$ (corresponds to a transition from $C=2$ to 0).

Figure 8

Early time survival probability of the Majorana state of the two-dimensional triplet superconductor for the quenches (a), (b), and (c) of Fig. 5. Different system sizes along the transverse, $y$, direction for $k_x=0$ are superimposed. The fast decay of the survival probability is independent of system size.

Figure 9

Scaling of the survival probability of the Majorana state of the two-dimensional triplet superconductor for the quenches (a), (b), and (c) of Fig. 5 and different system sizes along the transverse, $y$, direction for $k_x=0$. The first revival time scales with the system transverse direction, due to the propagation of the mode to the center of the system.

Figure 10

Survival probability of the states with momenta $k_x=0,\pi /2$ in (a) and $k_x=\pi$ in (b), of the two-dimensional triplet superconductor for a quench $(M_z=2,{\epsilon }_F=-3)\to (M_z=0,{\epsilon }_F=-3),C=1$ to $C=0$ ($Z_2$ phase). In the initial state, only the $k_x=0$ state is a Majorana.

Figure 11

Survival probability of the Majorana state of the two-dimensional triplet superconductor for a quench $C=1$ to quantum critical points with $C=0$ (trivial and with edge states). $(M_z=1,{\epsilon }_F=-4)\to (M_z=1,{\epsilon }_F=-5)$ with regions of vanishing survival probability, and $(M_z=1,{\epsilon }_F=-4)\to (M_z=1,{\epsilon }_F=-3)$ for which there is a finite survival probability for the time interval shown.

Figure 12

Survival probability of the Majorana state of the two-dimensional triplet superconductor for the case of weak spin-orbit coupling with ${\epsilon }_F=-1,d=1$ and $(\alpha =1,M_z=1.2)\to (\alpha =1,M_z=0.5)$ corresponding to $C=2\to C=0$.

Figure 13

Comparison of time evolution of (a) survival probability and (b) Chern number for the case of strong spin-orbit coupling for $(M_z=2,{\epsilon }_F=-3)\to (M_z=0,{\epsilon }_F=-3)$ corresponding to $C=1\to C=0$. The Chern number remains stable at the initial state value until the Majorana mode reaches the middle point of the system. Beyond this instant, the Chern number fluctuates.

Figure 14

Time evolution of $P\left(t\right)$ for three different cuts in the phase diagram, I, II and III, from bottom to top in the figure. corresponding to transitions $C=1\to C=0$ (trivial phase), $C=1\to C=0$ ($Z_2$ phase), and $C=-1\to C=-2$, respectively. In all cases, the transition occurs at half the time.

Figure 15

Time evolution of $P\left(t\right)$ for cuts I [(a) and (b) in upper row] and II [(c) and (d) in lower row] for different rates in a strip geometry.

Figure 16

Time evolution of $P\left(t\right)$ for cuts I [(a) and (b) in upper row] and II [(c) and (d) in lower row] for different rates with periodic boundary conditions along the $y$ direction (no edge states).

Figure 17

Defect production after quenches I and III and overlap to the lowest level excited state (LL). As the system size grows the defect production decreases.