Kibble-Zurek mechanism with a single particle: Dynamics of the localization-delocalization transition in the Aubry-André model

The Aubry-André one-dimensional lattice model describes a particle hopping in a pseudorandom potential. Depending on its strength $\lambda$, all eigenstates are either localized $\left(\lambda >1\right)$ or delocalized $\left(\lambda <1\right)$. Near the transition, the localization length diverges like $\xi \sim {(\lambda -1)}^{-\nu }$ with $\nu =1$. We show that when the particle is initially prepared in a localized ground state and the potential strength is slowly ramped down across the transition, then—in analogy with the Kibble-Zurek mechanism—it enters the delocalized phase having finite localization length $\widehat{\xi }\sim {\tau }_Q^{\nu /(1+z\nu )}$. Here ${\tau }_Q$ is the ramp/quench time and $z$ is a dynamical exponent. At $\lambda =1$ we determine $z\simeq 2.37$ from the power-law scaling of an energy gap with a lattice size $L$. Even though for infinite $L$ the model is gapless, we show that the gap that is relevant for excitation during the ramp remains finite. Close to the critical point it scales like ${\xi }^{-z}$ with the value of $z$ determined by the finite-size scaling. It is the gap between the ground state and the lowest of those excited states that overlaps with the ground state enough to be accessible for excitation. We propose an experiment with a noninteracting BEC to test our prediction. Our hypothesis is further supported by considering a generalized version of the Aubry-André model possessing an energy-dependent mobility edge.

Figure 1

In (a), eigenstates of the AA model are calculated for a lattice of size $L=144$. Color indicates a measure of the localization length of the eigenstate: light green denotes the localization length $\approx L/\sqrt{12}$ (delocalized) and dark green denotes the localization length of $O\left(1\right)$ (localized). The abrupt color change at critical ${\lambda }_c=1$ separates the localized and delocalized phases. We initialize the quench in the ground state deep in the localized phase and ramp the potential strength $\lambda$ at a finite rate across ${\lambda }_c$. In (b), the top panel shows the initial ground state in the localized phase. The bottom panels show snapshots of states during a slow ramp taken in the critical region. They preserve a finite localization length that grows with a power of the quench time ${\tau }_Q$. This scenario could be realized experimentally with a noninteracting Bose-Einstein condensate in a pseudorandom optical lattice potential as in Ref. [8].

Figure 2

In (a), localization length $\xi$ as a function of distance from the critical point $\epsilon$. Here $\xi$ was calculated as the dispersion of the localized ground state on a lattice of $L=987$ sites. The linear fit $\xi \sim {\epsilon }^{-\nu }$ yields $\nu =0.97\pm 0.01$, in good agreement with exact $\nu =1$. In (b), a gap ${\mathrm{\Delta }}_c$ between the ground state and the first excited state at the critical $\lambda =1$ as a function of the lattice size $L$. Fitting ${\mathrm{\Delta }}_c\sim L^{-z}$ yields a dynamical exponent $z=2.37\pm 0.03$. All results were averaged over 100 random $\varphi$.

Figure 3

Linear ramp across the critical point. Here time 0 corresponds to the critical Hamiltonian with $\lambda =1$. The instantaneous transition rate, $\left|\dot{\epsilon }/\epsilon \right|=1/\left|t\right|$, diverges at the critical point and, at the same time, the relevant energy gap closes like ${\sim \left|\epsilon \right|}^{z\nu }$. Consequently, in the neighborhood of the critical point, between $-\widehat{t}$ and $\widehat{t}$, the evolution is not adiabatic.

Figure 4

In (a) we show localized eigenstates at $-\widehat{t}$: the adiabatic ground state (black), the relevant excited state with the most probable transition (red), and the first excited state (blue). The relevant state overlaps with the ground state. The first excited state, even though it has the smallest gap, has no overlap with the ground state and hence is irrelevant for excitation. Consequently, in (b) we plot the relevant gap ${\mathrm{\Delta }}_r$ as a function of the distance from the critical point $\epsilon$. The four plots correspond to different choices of preselecting the relevant states with $O_e$ greater than ${10}^{-5},{10}^{-6},{10}^{-7},{10}^{-8}$. These log-log plots have a common steplike structure with an overall linear dependence on $\epsilon$. The fit to the linear region yields ${\mathrm{\Delta }}_r\sim {\epsilon }^{z\nu }$ with $z\nu =2.4\pm 0.1$ close to the expected $z\nu \simeq 2.37$ obtained from the finite-size scaling at criticality; see Fig. 2. We focus on the linear region as finite-size effects, shown in Fig. 2, appear below $\mathrm{\Delta }\sim {10}^{-5}$. Here ${\mathrm{\Delta }}_r$ was averaged over 100 random $\varphi$ on a lattice of $L=987$ sites.

Figure 5

In (a), the width of the wave packet at the critical point as a function of the quench time ${\tau }_Q$. The fit gives $\widehat{\xi }\sim {\tau }_Q^{0.299\pm 0.001}$; cf. Eq. (7). In (b), a similar plot for the energy dispersion at the critical point. The linear fit yields $\widehat{\mathrm{\Delta }}\sim {\tau }_Q^{-0.688\pm 0.004}$; see Eq. (10). Here the lattice size $L=987$, and averaging is done over 10 random values of $\varphi$.

Figure 6

The top (bottom) row shows the width (dispersion) as a function of scaled time. In the left panels, the two quantities are not scaled. In the right panels, they are scaled and collapse to their respective scaling functions. The collapse demonstrates the KZ scaling hypothesis.

Figure 7

Eigenstates of the GAA model are calculated taking $\alpha =0.5$ for a lattice size $L=144$. Color indicates a measure of the localization length of the eigenstate: light green denotes localization length $\approx L/\sqrt{12}$ (completely delocalized) and dark green denotes the localization length of $O\left(1\right)$ (completely localized). The abrupt color change separates the localized and delocalized phases defining the mobility edge (red line), which is consistent with the analytic prediction. We initialize the quench in the ground state deep in the localized phase and ramp the potential strength $\lambda$ at a finite rate across the mobility edge.

Figure 8

Correlation and dynamical exponents for the generalized Aubry-André model: In (a), the localization length $\xi$ as a function of the distance from the critical point $\epsilon$. Here $\xi$ was calculated as the dispersion of a localized ground state on a lattice of $L=987$ sites. The fit $\xi \sim {\epsilon }^{-\nu }$ yields $\nu =0.982$ similar to $\nu =0.97$ obtained for the AA model; see Fig. 2. In (b), a gap ${\mathrm{\Delta }}_c$ between the ground state and the first excited state at the critical point ${\lambda }_c\approx 1.638$ as a function of the lattice size $L$. Fitting ${\mathrm{\Delta }}_c\sim L^{-z}$ yields a dynamical exponent $z=2.378$ that is again similar to the AA model (see Fig. 2). Results were averaged over 100 random $\varphi$. In (c), we plot the relevant gap ${\mathrm{\Delta }}_r$ as a function of the distance from the critical point $\epsilon$ similar to Fig. 4 for the AA model. Once again the fit to the linear region yields ${\mathrm{\Delta }}_r\sim {\epsilon }^{z\nu }$ with $z\nu =2.46$ close to the expected $z\nu \simeq 2.378$ obtained from the finite-size scaling at criticality; see Fig. 2. We focus on the linear region as finite-size effects, shown in Fig. 2, appear below $\mathrm{\Delta }\sim {10}^{-5}$. Here ${\mathrm{\Delta }}_r$ was averaged over 100 random $\varphi$ on a lattice of $L=987$ sites.

Figure 9

In (a), the width of the wave packet at the critical point ${\lambda }_c$ as a function of the quench time ${\tau }_Q$. The fit gives $\widehat{\xi }\sim {{\tau }_Q}^{0.3092}$, which yields $z=2.27$ for $\nu =0.982$. In (b), the scaled widths as a function of the scaled times for different ${\tau }_Q$'s collapse to a common scaling function. The collapse demonstrates the KZ scaling hypothesis.