Sudden quenches in a quasiperiodic Ising model

We present here the nonequilibrium dynamics of the recently studied quasiperiodic Ising model. The zero temperature phase diagram of this model mainly consists of three phases, where each of these three phases can have extended, localized, or critically delocalized low-energy excited states. We explore the nature of excitations in these different phases by studying the evolution of entanglement entropy after performing sudden quenches of different strengths to different phases. Our results on nonequilibrium dynamics of entanglement entropy are concurrent with the nature of excitations predicted in Chandran and Laumann [Phys. Rev. X 7, 031061 (2017)].

Figure 1

Phase diagram of QPTIM consisting of FM, PM, and QPFM ground states. Depending on the strength of the quasiperiodic modulation, the low-energy excitations can exhibit localized, extended, or critically delocalized behavior, also shown in the figure. For more details, see Ref. [12].

Figure 2

Evolution of entanglement entropy in a log-log scale after quenching to different regions in the phase diagram starting from the same initial point $J_0=2$ and $A_J^0=0$ for a system of size $N=512$. The black continuous line in (a) corresponds to $S_l\left(t\right)\sim t$ confirming our claim of linear increase in $t$ when quenched to the extended regime. Panel (b) shows quenches to different localized phases. Panel (c) corresponds to a different initial Hamiltonian $H_0$ (with $J_0=2,A_J^0=3$) but showing behavior similar to (a) and (b). Here $E$, $CD$, and $L$ correspond to the extended, critically delocalized, and localized nature of excitations in the final Hamiltonian. Quench to CD phase is discussed in details in Fig. 3.

Figure 3

Quenches of different strengths to critically delocalized phase. The parameters of the initial and final Hamiltonian are shown in the label in the format $(J_0,A_J^0)\mathrm{to}(J,A_J)$. For comparison, we have also plotted the black solid line to depict the general linear increase in time when quenched to extended phase. Clearly, the increase in entanglement entropy seems to be with an exponent smaller than unity.

Figure 4

Finite-size effects in the evolution of entanglement entropy. Clearly, the evolution for $N=512$ is similar to that of $N=1024$ but different from $N=256$, indicating that $N=512$ is sufficiently large system size.

Figure 5

Quench to $J=1.55$ and $A_J=1.5$ from two different initial points written in ($J_0,A_j^0$) format in the label. The final point has extended low-lying excited states but have localized high-energy excited states. A strong quench from $J_0=100,A_J^0=1.5$ is expected to explore the high-energy localized states which, in principle, should be reflected in the evolution of entanglement entropy. On the contrary, we see that such a strong quench is still resulting to a linear increase in $S_l\left(t\right)$, similar to quench from $J_0=1.7,A_J^0=1.5$ referred to as a weak quench.