Multifractal wave functions of simple quantum maps

We study numerically multifractal properties of two models of one-dimensional quantum maps: a map with pseudointegrable dynamics and intermediate spectral statistics and a map with an Anderson-like transition recently implemented with cold atoms. Using extensive numerical simulations, we compute the multifractal exponents of quantum wave functions and study their properties, with the help of two different numerical methods used for classical multifractal systems (box-counting and wavelet methods). We compare the results of the two methods over a wide range of values. We show that the wave functions of the Anderson map display a multifractal behavior similar to eigenfunctions of the three-dimensional Anderson transition but of a weaker type. Wave functions of the intermediate map share some common properties with eigenfunctions at the Anderson transition (two sets of multifractal exponents, with similar asymptotic behavior), but other properties are markedly different (large linear regime for multifractal exponents even for strong multifractality, different distributions of moments of wave functions, and absence of symmetry of the exponents). Our results thus indicate that the intermediate map presents original properties, different from certain characteristics of the Anderson transition derived from the nonlinear sigma model. We also discuss the importance of finite-size effects.

Figure 1

(Color online) Top panel: instance of an eigenvector of the intermediate map with random phases for $N=2^{14}$ and $\gamma =1/3$. Bottom panel: instance of an iterate of the Anderson map with random phases for $k=1.81$, $N=2^{11}$, and $t={10}^8$; $|\psi \left(0\right)⟩=|i⟩$, with $i=1024$.

Figure 2

(Color online) Top panels: average moments ${\text{log}}_2⟨P_{q,k}⟩$ as a function of $k$, the logarithm of the box sizes. Bottom panels: partition function ${\text{log}}_2⟨\mathcal{Z}\left(q,A\right)⟩$ as a function of the logarithm of the scale $A$. The left panels correspond to eigenvectors of the intermediate map with random phases, $N=2^{14}$, and $\gamma =1/3$. Here, the average is taken over $98304$ vectors (respectively, $32768$ vectors for the partition function). The right panels correspond to iterates of the Anderson map for $k=1.81$, $N=2^{11}$, $|\psi \left(0\right)⟩=|i⟩$, with $i=1024$, and $t={10}^8$, where the average is taken over 1302 vectors. The values chosen for $q$ are $q=-2$ (red diamonds), $q=2$ (blue circles), and $q=6$ (green squares). The gray shaded regions show the fitting interval.

Figure 3

(Color online) Top panel: multifractal exponents $D_q$ for eigenvectors of the intermediate map with random phases, $N=2^{14}$, and $\gamma =1/3$. Empty (filled) red circles correspond to the method of moments (wavelets). The (light blue and light red) shaded regions indicate standard error in the least-squares fitting. The multifractal analysis was done over $98304$ vectors with box sizes ranging from 16 to 1024 and scales ranging from $2^{-12}\text{to}{2}^{-5}$. Bottom panel: typical multifractal exponents $D_q^{\text{typ}}$ for the same data. In both panels, the gray solid line is the linear approximation $1-q/3$.

Figure 4

(Color online) Multifractal exponents $D_q$ and $D_q^{\text{typ}}$ for iterates of the Anderson map with $k=1.81$ and $N=2^{11}$. Same convention as in Fig. 3. The multifractal analysis was done over 1302 realizations of size $N=2^{11}$ with box sizes ranging from 4 to 512 and scales ranging from $2^{-9}\text{to}{2}^{-4}$.

Figure 5

(Color online) Top: multifractal exponents $D_q$ and $D_q^{\text{typ}}$ for the Anderson map with $k=1.81$, $N=2^{11}$, $|\psi \left(0\right)⟩=|i⟩$, with $i=1024$, and $t={10}^8$. Same convention as in Fig. 3. The multifractal analysis was done over 1302 realizations with box sizes ranging from 4 to 512. Bottom: multifractal exponents $D_q$ and $D_q^{\text{typ}}$ for iterates of the intermediate map with $b=3$, $N=2^{12}$, $|\psi \left(0\right)⟩=|i⟩$, with $i=2048$, and $t={10}^8$. The multifractal analysis was done over 2000 realizations with box sizes ranging from 4 to 512. In both cases we used the box-counting method.

Figure 6

(Color online) Top panel: multifractal exponents $D_q$ for eigenvectors of the intermediate map with (circles) and without (triangles) random phases. Bottom panel: typical multifractal exponents $D_q^{\text{typ}}$ for eigenvectors of the intermediate map with (squares) and without (diamonds) random phases. In both cases we used the box-counting method. Conventions and parameter values are the same as in Fig. 3 except that for nonrandom phases, a single realization of size $N=2^{14}$ was considered. In both panels, the gray solid line is the linear approximation $1-q/3$.

Figure 7

(Color online) Difference $D_q-D_q^{\text{lin}}$ for eigenvectors of the intermediate map with random phases for $\gamma =1/3$ (blue solid curve), $\gamma =1/7$ (green dotted curve), and $\gamma =1/11$ (red dashed-dotted curve). Blue dashed curve shows the same difference for the intermediate map without random phases for $\gamma =1/3$. Inset: separation points between $D_q$ and $D_q^{\text{lin}}$ (green dots), determined by $\left(D_q-D_q^{\text{lin}}\right)/\left(D_q+D_q^{\text{lin}}\right)=0.01$, for eigenvectors of the intermediate map with random phases for different values of $\gamma =1/b$. Other parameters as in Fig. 6.

Figure 8

(Color online) Difference $D_q-D_q^{\text{lin}}$ for iterates of the Anderson map, with $D_q^{\text{lin}}$ defined by $D_q^{\text{lin}}=1+q{D}_q^{\prime }\left(0\right)$. Same parameters as in Fig. 4.

Figure 9

(Color online) Top panel: exponents ${\tau }_q$ (red solid lower curve) and ${\tau }_q^{\text{typ}}$ (blue solid upper curve) as a function of $q$ for eigenvectors of the intermediate map with random phases, $N=2^{14}$, and $\gamma =1/3$. Bottom panel: same but for $\gamma =1/11$. In both panels, the black dashed lines are linear fits of ${\tau }_q^{\text{typ}}$ at large $\left|q\right|$. The arrows indicate the critical $q$ value determined by $x_q=1$ (see Sec. 5D), equal to 2.51 for $\gamma =1/3$ and to 4.30 for $\gamma =1/11$. The slopes of the black dashed lines are, respectively, given by ${\alpha }_{-}\approx 2.28$ and ${\alpha }_{+}\approx 0.32$ for $\gamma =1/3$ and by ${\alpha }_{-}\approx 1.57$ and ${\alpha }_{+}\approx 0.60$ for $\gamma =1/11$.

Figure 10

(Color online) Top panel: difference $D_q-D_q^{\text{typ}}$ between average and typical exponents for eigenvectors of the intermediate map with random phases for $\gamma =1/3$ (red solid curve) and $\gamma =1/11$ (blue dashed curve). Bottom panel: same figure for iterates of the Anderson map. Same parameters as in Fig. 4.

Figure 11

(Color online) Separation points $q_{\pm 0.01}$ between $D_q$ and $D_q^{\text{typ}}$ (green dots), determined by $\left(D_q-D_q^{\text{typ}}\right)/\left(D_q+D_q^{\text{typ}}\right)=\pm 0.01$, for eigenvectors of the intermediate map with random phases for different values of $\gamma =1/b$. The blue solid lines are the linear fits $q_{-0.01}\approx -0.10b-1.10$ and $q_{+0.01}\approx 0.15b+0.86$. Same parameters as in Fig. 4.

Figure 12

(Color online) Left panel: singularity spectra $f\left(\alpha \right)$ and $f^{\text{typ}}\left(\alpha \right)$ for eigenvectors of the intermediate map with random phases for $\gamma =1/3$. The inset shows the linear behavior of $\sqrt{b}/\left({\alpha }_{-}-{\alpha }_{+}\right)\simeq 0.307b-0.037$ as a function of $b$ (in $\gamma =1/b$). The values of ${\alpha }_{\pm }$ have been extracted from linear fits of ${\tau }_q^{\text{typ}}$ at large $\left|q\right|$. Right panel: singularity spectra $f\left(\alpha \right)$ and $f^{\text{typ}}\left(\alpha \right)$ for iterates of the Anderson map. In both panels, the singularity spectra are given in the range $\alpha \left(q=-16\right)\le \alpha \le \alpha \left(q=16\right)$.

Figure 13

(Color online) Probability distribution of the logarithm of the moment $P_q=P\left(q,0\right)$ for $q=1.25$ of eigenvectors of the random intermediate map for $\gamma =1/3$. Same parameters as in Fig. 3. The solid line shows a linear fit (in a logarithmic scale) whose slope yields the tail exponent $-x_q$.

Figure 14

(Color online) Tail exponents $x_q$ for eigenvectors of the random intermediate map for $\gamma =1/3$ with $N=2^{12}$ (blue filled circles) and $N=2^{14}$ (blue empty circles) and for $\gamma =1/11$ with $N=2^{14}$ (red squares). Same parameters as in Fig. 3. Inset: tail exponent for $\gamma =1/11$ for larger values of $q$.

Figure 15

(Color online) Tail exponents $x_q$ for iterates of the Anderson map. Same parameters as in Fig. 4. The inset shows that $1/x_q$ is well fitted by a parabola.

Figure 16

(Color online) Relation between ${\tau }_{q{x}_q}$ and $x_q{\tau }_q^{\text{typ}}$ for eigenvectors of the intermediate map for $\gamma =1/3$. Parameter values are the same as in Fig. 3. Blue/green (black/gray) circles correspond to values of $q$ larger/smaller than 2.5 (the values closer to $q=2.5$ are on the left for both series of points). Dashed line is formula (6).

Figure 17

(Color online) Relation between ${\tau }_{q{x}_q}$ and $x_q{\tau }_q^{\text{typ}}$ for eigenvectors of the intermediate map for $\gamma =1/11$. Parameter values are the same as in Fig. 3. The blue solid line shows a linear fit of the data $\left(y=1.14x-0.56\right)$. Dashed line is formula (6).

Figure 18

(Color online) Relation between ${\tau }_q$ and ${\tau }_q^{\text{typ}}$ for iterates of the Anderson map with $k=1.81$, $N=2^{11}$, $|\psi \left(0\right)⟩=|i⟩$, with $i=1024$, and $t={10}^8$. Parameter values are the same as in Fig. 4. The blue solid line corresponds to $y=x-0.29$. Dashed line is formula (6).

Figure 19

(Color online) Inverse of the tail exponents $x_q$ for eigenvectors of the random intermediate map. Top panel: $\gamma =1/3$; blue and black dashed lines are linear fits in two different $q$ ranges; dashed line (linear fit for large $q$) is $1/x_q=0.36q+0.23$; inset is a blowup of the small $q$ range. Bottom panel: $\gamma =1/11$; blue and black dashed lines are, respectively, quadratic and linear fits in two different $q$ ranges; dashed line (linear fit for large $q$) is $1/x_q=0.36q-0.55$; inset is a blowup of the small $q$ range, showing in red the best linear fit for the same data. In both cases, the two ends of the error bars correspond to two values of $x_q$ obtained from a fit of the moment probability distribution over two different intervals. Parameter values are the same as in Fig. 3.

Figure 20

(Color online) Anomalous exponents ${\Delta }_q$ (blue dashed curve) and ${\Delta }_{1-q}$ (red solid curve) for eigenvectors of the intermediate map with random phases, $\gamma =1/3$, and $N=2^{14}$. Parameter values are the same as in Fig. 3. The black thin line shows the parabola $q\left(1-q\right)/b$ corresponding to the linear regime.

Figure 21

(Color online) Anomalous exponents ${\Delta }_q$ (blue dashed curve) and ${\Delta }_{1-q}$ (red solid curve) for eigenvectors of the intermediate map (10) with $\gamma =1/3$ and $N=2^{14}$. The (light blue and light red) shaded region indicates standard error in the least-squares fitting. The multifractal analysis was done with box sizes ranging from 16 to 1024. The black thin line shows the parabola $q\left(1-q\right)/b$ corresponding to the linear regime.