Energy-dependent dynamical quantum phase transitions in quasicrystals

Recently the role of the mobility edge in localization transitions has been extensively studied for one-dimensional tight-binding quasiperiodic models. In this work we study the mobility edge in a family of quasiperiodic systems evolving far from equilibrium, such as quench dynamics. We report numerical simulations of the Loschmidt echo based on a polynomial expansion-based technique with a moderate computational cost. Remarkably, we obtain an identical energy dependence on the equilibrium and dynamical quantum phase transitions of quasiperiodic models. The self-dual energy-independent localization model under quench dynamics exhibits energy-independent dynamical quantum phase transitions. On the other hand, self-dual energy-dependent localization models undergo energy-dependent dynamical quantum phase transitions. The results provide insights into energy-dependent dynamical localization transitions in quasiperiodic systems relevant to experiments.

Figure 1

Phase diagrams of the AA lattice in the energy-potential plane. Phase diagrams are obtained by calculating the LE when an initial ground state with (a) ${\lambda }_i=0$, (b) ${\lambda }_i=0.1$, and (c) ${\lambda }_i=0.2$ is quenched into a time-evolved state with ${\lambda }_f={10}^4$ and when an initial localized state with ${\lambda }_i={10}^4$ is quenched into a time-dependent state with (c) ${\lambda }_f=0$, (d) ${\lambda }_f=0.1$, and (f) ${\lambda }_f=0.2$. Numerical calculations are carried out for the system of size $L=1024$ with period boundary conditions using an exact diagonalization method.

Figure 2

Phase diagrams of the GAA lattice with $\beta =0.5$ in the energy-potential plane. Phase diagrams are obtained by calculating the LE when an initial ground state with (a) ${\lambda }_i=0$, (b) ${\lambda }_i=0.1$, and (c) ${\lambda }_i=0.2$ is quenched into a time-evolved state with ${\lambda }_f=2$ and when an initial eigenstate with ${\lambda }_i=2$ is quenched into a time-dependent state with (c) ${\lambda }_f=0$, (d) ${\lambda }_f=0.1$, and (f) ${\lambda }_f=0.2$. Numerical calculations are carried out for the system of size $L=512$ with period boundary conditions using an exact diagonalization method.

Figure 3

LE of the GAA lattice when a plane wave is quenched into the time-evolved state with strong potential modulation ${\lambda }_f\to \infty$ in the thermodynamic limit. Inset: LE as a function of rescaled evolving time for $\beta =0.9$ in log-log scale. The oscillatory decay of LE is very well fitted with a line, $y=ax$, where $a$ is a fitting parameter.

Figure 4

Phase diagram of the MAA model in the $E-{\lambda }_f\tau$ plane for (a) $\kappa =2$, and (b) $\kappa =3$ with ${\lambda }_f=3$. The phase diagram is obtained by computing the KPM estimates of the LE of the system of size $N=131072$ with $M=1024$ Chebyshev moments.

Figure 5

Phase diagrams of the AAF lattice in the $E\text{−}{\lambda }_f\tau$ plane. Phase diagrams are obtained by computing the KPM estimates of the LE when an initial ground state $({\lambda }_i=0)$ is quenched into a time-evolved state for (a) $\gamma =1$, (b) $\gamma =5$, and (c) $\gamma \to \infty$ with ${\lambda }_f=2$ and (d) $\gamma =1$, (e) $\gamma =5$, and (f) $\gamma \to \infty$ with ${\lambda }_f=\infty$.

Figure 6

LE of the AAF model when a plane wave quenched into the time-evolved state of the Hamiltonian with a strongly potential modulation ${\lambda }_f\to \infty$ in the thermodynamic limit.

Figure 7

Phase diagram of the AA model in the $E\text{−}\lambda$ plane. The diagram is obtained by computing the IPR of the system of size $L=1024$. The red, blue, and magenta dashed lines at $\lambda =0, \lambda =0.1$, and $\lambda =0.2$ correspond to the data presented in Fig. 1 for the quench dynamics. In insets we enlarge the region around $E\approx \pm 0.69$ to show that the system displays no eigenenergy for $\lambda >0$.

Figure 8

Phase diagram of the GAA model in the $E\text{−}\lambda$ plane. The diagram is obtained by computing the IPR of the system of size $L=1024$. The red, blue, and magenta dashed lines at $\lambda =0, \lambda =0.1$, and $\lambda =0.2$ correspond to the data presented in Fig. 2 for the quench dynamics. The green dashed line represents the mobility edge separating extended and localized regimes. In insets we show that the system displays no eigenenergy for $\lambda >0$ around $E\approx \pm 0.69$.