Anderson transition in a three-dimensional kicked rotor

We investigate Anderson localization in a three-dimensional (3D) kicked rotor. By a finite-size scaling analysis we identify a mobility edge for a certain value of the kicking strength $k=k_c$. For $k>k_c$ dynamical localization does not occur, all eigenstates are delocalized and the spectral correlations are well described by Wigner-Dyson statistics. This can be understood by mapping the kicked rotor problem onto a 3D Anderson model (AM) where a band of metallic states exists for sufficiently weak disorder. Around the critical region $k\approx k_c$ we carry out a detailed study of the level statistics and quantum diffusion. In agreement with the predictions of the one parameter scaling theory (OPT) and with previous numerical simulations, the number variance is linear, level repulsion is still observed, and quantum diffusion is anomalous with $⟨p^2⟩\propto t^{2∕3}$. We note that in the 3D kicked rotor the dynamics is not random but deterministic. In order to estimate the differences between these two situations we have studied a 3D kicked rotor in which the kinetic term of the associated evolution matrix is random. A detailed numerical comparison shows that the differences between the two cases are relatively small. However in the deterministic case only a small set of irrational periods was used. A qualitative analysis of a much larger set suggests that deviations between the random and the deterministic kicked rotor can be important for certain choices of periods. Heuristically it is expected that localization effects will be weaker in a nonrandom potential since destructive interference will be less effective to arrest quantum diffusion. However we have found that certain choices of irrational periods enhance Anderson localization effects.

Figure 1

(Color online) Scaling variable $\eta$ given by Eq. (19) as a function of $k$ for different system sizes $N$. Plots (a) and (b) show results for two different sets of $\left\{{\tau }_i\right\}$ generated by following method 1 (see text) with $\alpha \approx 18.1557570570506071$ (a) and $\alpha \approx 18.5328366352669$ (b). Plot (c) shows the averaged results of (a) and (b). The finite-size scaling analysis suggests that a metal-insulator transition occurs at $k_c=2.27\pm 0.07$ with a critical exponent $\nu =1.61\pm 0.25$. Plots (d), (e), and (f) are the equivalent of plots (a), (b), and (c) but with $\left\{{\tau }_i\right\}$ obtained by method 2 with $\alpha \approx 17.2139263167902$ (d) and $\alpha \approx 13.3195523927742$ (e) instead. In this case we obtain $k_c=2.35\pm 0.1$ with $\nu =1.67\pm 0.27$.

Figure 2

(Color online) The spectral rigidity ${\Delta }_3\left(\Lambda \right)$ and level spacing distribution $P\left(s\right)$ for three different kicking values, above, below, and close to the transition $k_c\approx 2.4$ in the 3D kicked rotor, Eq. (13), with $\left\{{\tau }_i\right\}$ the same as in Fig. 1a. A transition from Poisson to Wigner-Dyson statistics for a system with time reversal invariance (GOE) is clearly observed as the kicking strength crosses the critical value $k\approx k_c$ from below. For $k\approx k_c$ the level statistics have all the signatures of a metal-insulator transition such as (a) an asymptotically linear ${\Delta }_3\left(\Lambda \right)\sim \chi L∕15$ with $\chi \sim 0.29$, (b) level repulsion for $s\to 0$ as in a metal $[{\lim }_{s\to 0}P\left(s\right)=0]$, and (c) exponential decay of $P\left(s\right)$ for $s⪢1$ as in an insulator. In agreement with the OPT, level statistics are scale invariant for $k\approx k_c$.

Figure 3

Classical (a) and quantum (b) diffusion, $⟨\Delta p^2\left(t\right)⟩\equiv ⟨{[p\left(t\right)-p\left(0\right)]}^2⟩$, for three different kicking values, above, below, and close to the transition $k_c\approx 2.4$ in the 3D kicked rotor, Eq. (13), with $\left\{{\tau }_i\right\}$ as in Fig. 1a. Classical diffusion is normal for the three kicking values however interference effects slows down and eventually arrests quantum diffusion for $k<k_c$. In agreement with OPT $⟨p^2⟩\propto t^{2∕3}$ at $k\approx k_c$. Thin solid lines in (b) are the best fits in the window $10<t<1000$.

Figure 4

(Color online) (a) $\eta$ versus the kicking strength for different system sizes $N$ for a 3D kicked rotor in which the phase of the kinetic term of the evolution matrix is random (see text). A transition is again observed but for a slightly bigger $k_c=2.4\pm 0.05$. (b) $\eta$ for a randomized 3D kicked rotor (random phase plot) and for the deterministic kicked rotor of Eq. (13) with $\left\{{\tau }_i\right\}$ chosen from the two methods mentioned in the text. In both cases $N=16$. Small but sizable differences are clearly observed specially in the insulator region. For these $\left\{{\tau }_i\right\}$ correlations in the deterministic 3D kicked rotor seem to slightly weaken localization effects. In order to a meaningful comparison in the deterministic case we carry out a full ensemble average over 1280 sets of $\left\{{\tau }_i\right\}$.

Figure 5

(Color online) Distribution of $\eta$ for a 3D kicked rotor, Eq. (13), with (a) the kinetic term replaced by a random phase, (b) $\left\{{\tau }_i\right\}$ given by method 1, (d) $\left\{{\tau }_i\right\}$ given by method 2, and (c) the same for method 2 but where the integer sequence $[a_1,a_2,\dots ]$ for the continuous fraction form of ${\tau }_i∕\pi$ satisfies $\mid a_i\mid <9$ instead. In all cases 1280 sets of $\left\{{\tau }_i\right\}$ are considered and $N=16$.