Multifractality of quantum wave functions in the presence of perturbations

We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a pseudointegrable system, the Anderson model, and a random matrix model. We apply several types of natural perturbations which can be relevant for experimental implementations. We construct an analytical theory for certain cases and perform extensive large-scale numerical simulations in other cases. The data are analyzed through refined methods including double scaling analysis. Our results confirm the recent conjecture that multifractality breaks down following two scenarios. In the first one, multifractality is preserved unchanged below a certain characteristic length which decreases with perturbation strength. In the second one, multifractality is affected at all scales and disappears uniformly for a strong-enough perturbation. Our refined analysis shows that subtle variants of these scenarios can be present in certain cases. This study could guide experimental implementations in order to observe quantum multifractality in real systems.

Figure 1

Multifractal dimensions $D_q$ (top left) and singularity spectrum $f\left(\alpha \right)$ (right) for the intermediate map for $\gamma =1/3$ and $N=2^{12}$; blue solid line: annealed exponent; red bold dashed line: typical. Bottom left: Correlation function $R_2\left(r\right)$; gray dashed line corresponds to the exponent $D_2-1$ obtained from top left.

Figure 2

Local multifractal dimension ${\tilde{D}}_2\left(\ell \right)$ as a function of the box size $\ell$ for the PRBM model (1) for two values of $b$. Top: Weak multifractality regime $b=2$. Bottom: Strong multifractality regime $b=0.01$. Black circles: $N=2^{10}$; red squares: $N=2^{11}$; green diamonds: $N=2^{12}$; blue triangles: $N=2^{13}$. Dotted dashed orange line: Analytical prediction $D_2=2b$ for $b\ll 1$ and $D_2=1-\frac{1}{\pi b}$ for $b\gg 1$ [6].

Figure 3

Local multifractal dimensions for the intermediate map for $\gamma =1/5$ and increasing system sizes $N=L$. The full lines correspond to $q=2$ while the dashed lines correspond to $q=1$. Black circles: $N=2^9$; red squares: $N=2^{10}$; green diamonds: $N=2^{11}$; blue up-triangles: $N=2^{12}$; orange left triangles: $N=2^{13}$; brown down-triangles: $N=2^{14}$. Left: Raw data. Right: All the box sizes are rescaled by $\mathrm{\Xi }=N/5$.

Figure 4

Intermediate map smoothed by a polynomial for $N=3^9$ and $\gamma =1/5$. Top: Smoothed potential. Dark brown full line: Exact potential. Dashed purple line: $\epsilon =0.05$. Dot-dashed magenta line: $\epsilon =0.1$. Bottom: Moment of the eigenvectors of the random intermediate map for $q=2$ as a function of box size. Dark brown full line: Exact potential; dashed green line: $\epsilon =0.005$; dashed-dotted purple line: $\epsilon =0.05$.

Figure 5

Scaling analysis of the local multifractal dimension ${\tilde{D}}_2$ for the perturbed random intermediate map for $N=3^9$ and $\gamma =1/5$. The perturbation is the smoothing of the singular potential over a length $\epsilon$. Left: Raw data. Right: Data after rescaling. Inset: Variation of the scaling length $\xi$ as a function of $\epsilon$; solid line is the fit corresponding to (13) with $\alpha =1.04$. Black circles: $\epsilon =0.002$; red squares: $\epsilon =0.003$; green diamonds: $\epsilon =0.005$; blue up-triangles: $\epsilon =0.008$; yellow left triangles: $\epsilon =0.01$; brown down-triangles: $\epsilon =0.02$; gray right triangles: $\epsilon =0.03$; purple plus: $\epsilon =0.05$; cyan crosses: $\epsilon =0.07$; magenta stars: $\epsilon =0.1$.

Figure 6

Two-point correlation function $R_2$ as defined in (8) for the random intermediate map smoothed by a polynomial for $N=3^9$ and $\gamma =1/5$. Left: Raw data. Right: Data after rescaling the variable $r$ by $\xi =1/\epsilon$ following (13). The same color code as in Fig. 5 is used.

Figure 7

Same as Fig. 5 for the nonrandom intermediate map for $N=3^9$ and $\gamma =1/5$. Left: Raw data of the local multifractal dimension ${\tilde{D}}_2$ for different perturbation strengths. Right: Data after finite-size scaling. Inset: Variation of the scaling length $\xi$ as a function of $\epsilon$; solid line is the fit corresponding to (13) with $\alpha =0.67$, differing from the result in Fig. 5. The same color code as in Fig. 5 is used.

Figure 8

Same as Fig. 6 for the nonrandom intermediate map for $N=3^9$ and $\gamma =1/5$. Left: Two-point correlation function as defined in (8). Right: The variable $r$ is rescaled by $\xi \left(\epsilon \right)={\epsilon }^{-0.67}$, differing from the scaling used in Fig. 6. The same color code as in Fig. 5 is used.

Figure 9

Potential of the intermediate map obtained by truncation of Fourier series for $\gamma =1/5$ and $N=3^8=6561$ for different $N_f$. Black full line: $N_f=N=6561$; red squares: $N_f=6559$; green diamonds: $N_f=6557$; blue up-triangles: $N_f=6555$; orange left triangles: $N_f=6001$. Inset is a blow-up of the same data.

Figure 10

Two-point correlation function for the random intermediate map with potential approximated by a truncated Fourier series for $N=3^8$ and $\gamma =1/5$. Contrary to Fig. 6, we do not observe a systematic dependence on the truncation, indicating that the approximation is never perturbative. The same color code is used as in Fig. 9.

Figure 11

Potential of the intermediate map approximated by a trigonometric smoothing with $N_d$ derivatives fixed at $x=0.5$, $N=2^{12}$, and $\gamma =1/5$. From right to left on the left: $N_d=46$ (red), $N_d=80$ (green), $N_d=109$ (blue); black straight line is the exact potential.

Figure 12

Scaling analysis of the two-point correlation function for the random intermediate map with potential approximated by a trigonometric smoothing with $N_d$ derivatives fixed at $x=0.5$ for $N=2^{12}$ and $\gamma =1/5$. Black full line: Exact model. Red circles: $N_d=46$. Green squares: $N_d=80$. Blue diamonds: $N_d=109$. Left: Raw data; right: data rescaled with (16) with $\epsilon =1/N_d$.

Figure 13

Variation of the multifractal dimensions $D_{\pm 1}\left(\gamma \right)$ for the intermediate map (4) with random phases and $N=2^{12}$ in the vicinity of the rational values $\gamma =1/2$ (top) and $\gamma =1/5$ (bottom). The red solid line below 1 corresponds to $D_1\left(\gamma \right)$ and the blue one above 1 to $D_{-1}\left(\gamma \right)$. Black dash-dotted parabolas correspond to the theoretical expression Eq. (43) with $\kappa =1$ and 2 for $\gamma =1/2$ and $\kappa =0$ and 1 for $\gamma =1/5$. Inset: Zoom on the right part of the top plot, corresponding to $\kappa =2$.

Figure 14

Correlation functions for the random intermediate map of size $N=2^{12}$ in the vicinity of $\gamma =1/5$. Top: Two-point function (8); bottom: higher-order correlation function. The black dashed straight line corresponds to $1.68{(r/L)}^{3D_4-2D_2-1}$, as expected from Ref. [44]. The curves correspond to $\gamma =1/5+\epsilon /\left(5N\right)$ with (from top to bottom) $\epsilon =0$ (black solid line), $0.127$ (blue dashed line), $0.255$ (green dotted line), $0.382$ (orange dash-dotted line), $0.509$ (dark-red dash-dotted line), $0.637$ (light-blue double-dashed line), $0.764$ (purple dash-double dotted line), and $0.955$ (red dashed line).

Figure 15

Multifractal dimensions $D_1\left(\gamma \right)$ for the intermediate map with random phases (blue solid line) and deterministic phases (red dashed line) in the vicinity of $\gamma =1/2$ for $N=2^{12}$. Black dash-dotted parabolas correspond to the theoretical expression Eq. (43) for $\kappa =1,2$ while black dotted parabolas correspond to Eq. (43) with an additional prefactor 2 in front of $q$.

Figure 16

Left: Function $f_N\left(x\right)$ defined in (46) (see text) as a function of $x$ for $N=2^{11}$ (black), and function $x(N-x)$ (dashed/red). Right: Same plot with a different ordering of the abscissas. Points with abscissa $x=2^{n-q}(2y+1)$ with $q=1,2,...,n$ and $y=0,1,...,2^{q-2}$ are ordered by increasing $q$ and at fixed $q$ by increasing $y$. The last family (from 512 to 1024) corresponds to $q=n$, that is, odd $x$.

Figure 17

Scaling analysis of the local multifractal dimension ${\tilde{D}}_1$ for the PRBM model (1) for $N=8192$ and $b=1$ for different changes of basis (55). Left: Raw data. Right: Data after vertical rescaling of ${\tilde{D}}_1$ by a factor $D_1\left(\epsilon \right)$. Each curve corresponds to a different value of the parameter $\epsilon$ in (54). It is chosen to be of the form $\epsilon ={10}^{0.1n-3}$; black circles: $n=1$; red squares: $n=3$; green diamonds: $n=5$; blue up-triangles: $n=7$; yellow left triangles: $n=9$; brown down-triangles: $n=11$; gray right triangles: $n=13$; purple pluses: $n=15$; cyan crosses: $n=17$; magenta stars: $n=19\mathrm{.}10$ realizations of GOE matrices in (54) are taken in order to average over $81920$ vectors.

Figure 18

Scaling parameter $D_1\left(\epsilon \right)$ extracted from the analysis of Fig. 17. It is normalized to correspond to the multifractal dimension in the PRBM model (1) with $N=2^{13}$, $b=1$ in the limit $\epsilon \to 0$. The perturbation parameter $\epsilon$ quantifies a generic change of basis (55).

Figure 19

Left: Moments $P_2$ of the Anderson model with basis change (57) for different values of $t/{\tau }_{\text{Th}}$, the Thouless time associated with the quasiperiodic kicked rotor. $W=16.53\approx W_c$, $L=120$, and $t/{\tau }_{\text{Th}}=0$ (black circles), $8.9{10}^{-11}$ (red squares), $1.1{10}^{-9}$ (green diamonds), $2.8{10}^{-7}$ (blue up-triangles), $8.7{10}^{-6}$ (purple tilted triangles), $2.3$ (brown down-triangles). Right: Local multifractal dimensions ${\tilde{D}}_2$ as a function of $\ell /L$ for different values of $t/{\tau }_{\text{Th}}$, same parameters and color code as in the left side. In both figures each color corresponds to a different couple of $t$ and $K$; unperturbed Anderson model (black circles); $t=10$, $K=21$ (red squares); $t=10$, $K=69.5$ (green diamonds); $t=21$, $K=760.9$ (blue up-triangles); $t=59$, $K=2517.7$ (purple tilted triangles); $t=100$, $K=998784$ (brown down-triangles). ${\tau }_{\text{Th}}\equiv N^2/D$ was determined by a numerical determination of the diffusion constant $D$ of the 1D quasiperiodic kicked rotor.

Figure 20

Multifractal dimension $D_2$ for the Anderson model with basis change (57) as a function of (a) $t/{\tau }_{\text{Th}}$ associated to the 1D quasiperiodic kicked rotor and (b) $t/T_{\text{Th}}$ associated with the 3D Anderson model for different sizes $L$ from $L=40$ to $L=120$. The scaling $t/T_{\text{Th}}$ puts the curves for different system sizes on top of each other. However, the ergodic limit $D_q=3$ is reached when $t/{\tau }_{\text{Th}}\approx 1$ associated with the 1D dynamics. This arises due to the 1D character of our rotation. Each symbol corresponds to a different size $L$; from right to left $L=40$ (circles), 60 (squares), 80 (diamonds), 100 (up-triangles), and 120 (left triangles). Each color (grayness) corresponds to a different value of $K\equiv \sqrt{2L^6{\hslash }^2/\tilde{\tau }}$ with $\tilde{\tau }\equiv {\tilde{\tau }}_{\text{max}}{{\tilde{\tau }}_{\text{min}}}^{n/9}/{{\tilde{\tau }}_{\text{max}}}^{n/9}$, $n$ an integer varying from $n=0$ to $n=9$ (bottom to top, i.e., black to magenta) and ${\tilde{\tau }}_{\text{max}}=L^6/({21}^2/2{\hslash }^2)$, ${\tilde{\tau }}_{\text{min}}=50$. For each couple of symbol and color (grayness), different points correspond to different times; $t=10$, 12, 16, 21, 27, 35, 46, 59, 77, and 100 from bottom to top. ${\tau }_{\text{Th}}\equiv N^2/D$ and $T_{\text{Th}}=L^2/D$ were determined by a numerical determination of the diffusion constant $D$ of the 1D quasiperiodic kicked rotor.

Figure 21

Anderson transition for the Anderson model with change of basis. Finite-size scaling analysis of ${\tau }_2=ln{P}_2^{\text{typ}}/ln\lambda$ for fixed $\lambda =\ell /L=0.1$ as a function of disorder at various system sizes $L\in [20,120]$. The results for the standard Anderson model are represented in (a) while (b) corresponds to the rotated eigenstates with $K=115$ and fixed $t/T_{\text{Th}}\approx 40$. The error bars are standard deviations. The lines are the fit with a Taylor expansion of the scaling function ${\tau }_2=F(L/\xi )+L^{-y}{F}_{\text{irr}}(L/\xi )$, and $\xi \sim |W-W_c{|}^{-\nu }$ the scaling length. The polynomial fit has 10 adjustable parameters, $\nu$ and $W_c$ being fixed to their known values $\nu =1.6$ and $W_c=16.53$ (see Ref. [12]). In both cases the goodness of fit is quite acceptable (larger than 0.1). We find $y\approx 1.8$ for the unperturbed Anderson model and $y\approx 1.5$ for the rotated one. This compares well with the simulations of Ref. [12].

Figure 22

Top: Local multifractal exponent for the random intermediate map, $\gamma =1/5$, $N=2^{13}$, $q=2$ under a generic change of basis. Left: Raw data. Right: Rescaled data. The points corresponding to $\ell =N/2$, $N/4$, and $N/8$ have been dropped. Each curve corresponds to a different value of the parameter $\epsilon$ in (54). It is chosen to be of the form $\epsilon ={10}^{0.1n-3}$; red squares: $n=4$; blue up-triangles: $n=8$; yellow left triangles: $n=9$; brown down-triangles: $n=10$; gray right triangles: $n=11$; purple pluses: $n=12$; cyan crosses: $n=13$. Bottom left: Variation of the scaling length $\xi$ as a function of $\epsilon$. Bottom right: variation of the second scaling parameter $D_2\left(\epsilon \right)$ as a function of $\epsilon$. Black full symbols are exact data, and dashed lines are fitting functions (respectively, exponential and second-order polynomial).

Figure 23

Discrete Wigner function of an eigenstate of the random intermediate map for $\gamma =1/3$, color (grayness) varies from blue (dark gray) (minimal value) to red (light gray) (maximal value). The symbols represent lines $a_{1}P-a_{2}X=a_3$ (mod $2N$) ($x=X/2N$ and $p=P/2N$). Here $N=128$, $a_2=1$ and $a_1=64$ (circles) and 65 (squares).

Figure 24

Left: ${\tilde{D}}_2$ as a function of the box size $\ell$ for the random intermediate map with momentum to position basis change for $\gamma =1/3$ and different values of $N/a_1=2^5$ (pentagons), $N/a_1=2^4$ (squares), $N/a_1=2^3$ (circles), $N/a_1=2^2$ (triangles), $N/a_1=2^1$ (down-triangles), $N/a_1=1$ (diamonds), with $N=2^{13}$ (size of the Wigner function is $2N$). Right: Rescaled ${\tilde{D}}_2$ as a function of the rescaled box size for the same data.

Figure 25

Left: Scaling length $\xi$ as a function of $\epsilon$ for the random intermediate map with momentum to position basis change for $\gamma =1/3$ and $N=2^{13}$. Right: Second scaling parameter $D_2\left(\epsilon \right)$ as a function of $\epsilon ={log}_2(N/a_1)$. The dashed lines are fitting functions (respectively, exponential and second-order polynomial). Same data as in Fig. 24.

Figure 26

Double scaling analysis of the local multifractal dimensions for the Anderson model with isotropic basis change. Left: ${\tilde{D}}_2$ as a function of the box size $\ell$ for different values of $\epsilon \equiv t/T_{\text{Th}}$: from bottom to top, $\epsilon =0$, $0.000125$, $0.00025$, $0.00062$, $0.0013$, $0.0023$, $0.0035$, $0.005$, $0.0068$, $0.0089$, and $0.011$. Here $T_{\text{Th}}=L^2/D$, $L=80$, and $D\approx 0.403$ the diffusion constant determined numerically for the 3D isotropic periodic Kicked Rotor (61) with $K=10$ and $\hslash =2.89$. Right: Rescaled ${\tilde{D}}_2$ as a function of the rescaled box size for the same values of $\epsilon$.

Figure 27

Left: Box size scaling $\xi \left(\epsilon \right)$ as a function of $\epsilon$ for the Anderson model with isotropic basis change. Right: Scaling parameter $D_2\left(\epsilon \right)$ as a function of $\epsilon$. Same data as in Fig. 26.