Self-similarity among energy eigenstates

In a quantum system, different energy eigenstates have different properties or features, allowing us to define a classifier to divide them into different groups. We find that the ratio of each type of energy eigenstate in an energy shell $[E_c-\mathrm{\Delta }E/2,E_c+\mathrm{\Delta }E/2]$ is invariant with changing width $\mathrm{\Delta }E$ or Planck constant $\hslash$ as long as the number of eigenstates in the shell is statistically large enough. We give an argument that such self-similarity in energy eigenstates is a general feature for all quantum systems, which is further illustrated numerically with various quantum systems, including circular billiard, double top model, kicked rotor, and Heisenberg XXZ model.

Figure 1

Circular billiard. (a) A typical quantum energy eigenfunction $|{\psi }_{nm}+{\psi }_{n,-m}{|}^2/2$, (b) a typical classical trajectory, and (c) the square of a Bessel function $J_m\left(k_{nm}r\right)$ with $R_b$ indicated. In this figure, $n=7$ and $m=12$.

Figure 2

(a) Ratio $f(R_0/2;E_c,\mathrm{\Delta }E)$ as a function of $\mathrm{\Delta }E$. (b) Ratio $f(R_0/2;E_c,\mathrm{\Delta }E)$ as a function of $R_b/R$. In this numerical results, we take $\hslash =1$ and fix $k_c=515$ (corresponding $E_c=k_c^2/2$), while changing $\mathrm{\Delta }k$ from 0.1 to 5 (corresponding $\mathrm{\Delta }E=2k_c\mathrm{\Delta }k$). The narrowest energy window (i.e., $\mathrm{\Delta }k=0.2$) has merely 28 energy eigenstates, and thus $\mathrm{\Delta }k$ cannot be further narrowed. Here we plot the ratio of energy eigenstates with “blank region radius” less than $R_b$ vs. $R_b$. Except for $\mathrm{\Delta }k=0.1$ and 0.25 (i.e., energy window is too narrow), other curves (with moderate or large energy window) embody a perfect self-similarity. The black line represents the theoretical result Eq. (8).

Figure 3

As the Planck constant $\hslash$ becomes smaller, more energy eigenstates enter a given energy shell. In the semiclassical limit $\hslash \to 0$, the energy spectrum becomes continuous.

Figure 4

In a classical circular billiard, for a particle starting at $\mathbf{r}=(x,y)$, only when its direction of motion is limited inside the angle of $\alpha$ can it enter a circle of radius of $R_b$. Otherwise, it always stays outside of the circle.

Figure 5

Mushroom billiard which is made of a quarter of circle and a rectangle. For the motion with the blank region radius $R_b$ smaller than $r$, the horizontal length of the rectangle, it is chaotic; otherwise, it is integrable.

Figure 6

The schematic of a quantum system approaching its semiclassical limit $\hslash \to 0$ in phase space. The two black curves represent two constant energy surfaces with difference $\mathrm{\Delta }E$; the area enclosed by them are the volume of a given energy shell $\mathcal{S}(E_c,\mathrm{\Delta }E)$. The dark shaded area is a region where the system has a certain physical property $A$. As $\hslash$ decreases toward zero, the Planck cells (squares in the figure) become smaller. As a result, there are more energy eigenstates in the energy shell $\mathcal{S}(E_c,\mathrm{\Delta }E)$ and more eigenstates have property $A$. However, the proportional of energy eigenstates having property $A$ in the energy shell $\mathcal{S}(E_c,\mathrm{\Delta }E)$ quickly approach a limit, which is the overlap of the energy shell and the dark area.

Figure 7

(a) Poincaré sections for the classical coupled top model with energy $E_c=-0.9$. [(b)–(f)] are the 3194, 3196, 3198, 3202, and 3207th energy eigenstates (energy eigenvalues are $-0.9018,-0.9007,-0.8998,-0.8970,-0.8939$), respectively, in the quantum phase space. The parameters are $\mu =0.5$ and $Q_1=\pi$.

Figure 8

Histograms the variance $\mathrm{Var}\left({\mathbf{P}}^2\right)$ at three different widths (a) $\mathrm{\Delta }E=0.2$, ($\mathrm{b})\mathrm{\Delta }E=0.1$, and ($\mathrm{c})\mathrm{\Delta }E=0.04$. (d) The ratio function $f(E_c,\mathrm{\Delta }E)$ with the threshold $\delta =0.16$. The energy shells are all centered at $E_c=-0.9$.

Figure 9

The Poincaré section of the classical kicked rotor. The initial points are uniformly sampled along $p$ axis with fixed $q$ near $\pi$.

Figure 10

The typical chaotic (a) and integrable (c) trajectories of classical kicked rotor and their corresponding quantum Floquet states [(b) and (d)]. The parameter $K=1.1$ and the effective Planck constant is ${\hslash }_{\mathrm{eff}}\approx 0.0039$, i.e., there are $40\times 40$ Planck cells in total.

Figure 11

The self-similarity with quantum kicked rotor. (a) The ratio $f(E_c,\mathrm{\Delta }E)$ as a function of the inverse of the effective Planck constant with $\mathrm{\Delta }E=2$ and $E_c=0$. (b) The ratio as the function of the width of energy shell $\mathrm{\Delta }E$ with ${\hslash }_{\text{eff}}\approx 0.0039$ and $E_c=0$. (c) The number of Floquet states as a function of the width of energy shell $\mathrm{\Delta }E$. (d) The dependence of the generalized Wigner–von Neumann entropy $\mathrm{\Gamma }(E_c,\mathrm{\Delta }E)$ on the center pseudoenergy $E_c$. The pseudoenergy shell width $\mathrm{\Delta }E=0.4$ here.

Figure 12

The ratio $f$ for various classifiers: (a) magnetization for a single spin (sixth spin), (b) total magnetization, (c) two-spin-correlation for third spin and fourth spin, (d) global-X-correlation, and (e) second Rényi entropy of first spin. (f) The total number of energy eigenstates in energy shell $[E_c-\mathrm{\Delta }E/2,E_c+\mathrm{\Delta }E/2]$. We take the length of spin chain to be $N=12$, parameter $\rho =0.4$ and central energy $E_c=4$.

Figure 13

The coupled-top's classical Poincaré sections of various $Q_1$ with $E=-0.9$ fixed. The partition we take is $\{0.1,0.7,1.0,1.3,1.6,1.9,2.2,2.5,2.8,3.1,3.4,3.7,4.0,4.3,4.6,4.9,5.2,5.8\}$ (from left to right, up to down).

Figure 14

Plot of integral $A(\mathrm{\Omega };Q_1)$. As for black line, $\mathrm{\Omega }$ represents “total isoenergetic surface”; while for dashed line $\mathrm{\Omega }$ denotes integrable islands.

Figure 15

The mobility edge and $E_c$ dependence of ratio in AAH model. (a) The mobility edge indicated by inverse participation ratio of eigenstates at different energies. The parameters are $1/b=(\sqrt{5}-1)/2$, $\alpha =0.2$, $L=3000$, and $\lambda =-1.1$. Theoretically the mobility edge is at $\mathcal{E}=1$ as in Ref. [60]. (b) Different ratio $f(E_c,\mathrm{\Delta }E)$ of $E_c$ being above and below the mobility edge. Some points are missing for specific $E_c$ and $\mathrm{\Delta }E$ since there are gaps in this model.