Nonequilibrium quantum relaxation across a localization-delocalization transition

We consider the one-dimensional $XX$ model in a quasiperiodic transverse field described by the Harper potential, which is equivalent to a tight-binding model of spinless fermions with a quasiperiodic chemical potential. For weak transverse field (chemical potential), $h<h_c$, the excitations (fermions) are delocalized, but become localized for $h>h_c$. We study the nonequilibrium relaxation of the system by applying two protocols: a sudden change of $h$ (quench dynamics) and a slow change of $h$ in time (adiabatic dynamics). For a quench into the delocalized (localized) phase, the entanglement entropy grows linearly (saturates) and the order parameter decreases exponentially (has a finite limiting value). For a critical quench the entropy increases algebraically with time, whereas the order parameter decreases with a stretched exponential. The density of defects after an adiabatic field change through the critical point is shown to scale with a power of the rate of field change and a scaling relation for the exponent is derived.

Figure 1

Dynamical entanglement entropy after a quench from $h_0=0$ to different values of $h$ (upper panel). Saturation values of the entanglement entropy and the limiting value of the width of the wave packet (diffusion) in the localized phase show a power-law divergence close to the transition point (lower panel).

Figure 2

Prefactor of the linear part of the dynamical entanglement entropy (left axis) and the relaxation time (right axis) after a quench from $h_0=0$ to different values of $h$.

Figure 3

Bulk magnetization after a quench from $h_0=0$ to different values of $h$. In the inset quench to the critical region is shown in agreement with the stretched-exponential form in Eq. (17) (the straight lines have a slope $\mu =0.47$).

Figure 4

Time-dependent width of the wave packet at different amplitudes of the transverse field.

Figure 5

Excitation probability as a function of the time scale, $\tau$, after an adiabatic process from $h=-\infty$ to $h=0$ (upper panel) and to $h=\infty$ (lower panel) calculated in finite systems of sizes $L=2F_n$ with $n=13,14,\cdots ,18$.

Figure 6

Normalized partial excitation probabilities as a function of ${(N/L)}^2$ for different sizes: $L=2F_n$ with $n=13,14,\cdots ,18$ at $\tau =100$ (upper panel) the same at $L=2F_{18}$ for different values of $\tau$ (lower panel) both in log-log scale.