Critical quantum states and hierarchical spectral statistics in a Cantor potential

Abstract

We study the spectral statistics and wave-function properties of a one-dimensional quantum system subject to a Cantor-type fractal potential. By analyzing the nearest-neighbor level spacings, inverse participation ratio (IPR), and the scaling behavior of the integrated density of states (IDS), we demonstrate how the self-similar geometry of the potential is imprinted on the quantum spectrum. The energy-resolved level spacings form a hierarchical, filamentary structure, in sharp contrast to those of periodic and random systems. The normalized level-spacing distribution exhibits a bimodal structure, reflecting the deterministic recurrence of spectral gaps. A multifractal analysis of eigenstates reveals critical behavior: the generalized fractal dimensions $D_q$ lie strictly between the limits of extended and localized states, exhibiting a distinct $q$-dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.

Figures (6)

• Figure 1: Spatial profiles of the potentials investigated in this study. (a) Periodic potential $V_\mathrm{p}(x)$ composed of equally spaced barriers. The barrier count ($N_\mathrm{b}=128$) and width are matched to those of the seventh-generation Cantor potential to ensure a direct comparison. (b) Magnified view of a segment of the periodic potential. (c) Random potential $V_\mathrm{r}(x)$, where barrier positions are distributed uniformly at random. Although the limited plotting resolution may visually suggest overlapping barriers, the numerical model strictly imposes a non-overlapping constraint. (d) Magnified view of the random potential. (e) Iterative construction of Cantor potential $V_\mathrm{C}^n(x)$ for generations $n=1$--4. (f) The seventh-generation Cantor potential $V_\mathrm{C}^7(x)$ used in the main analysis. (g) Magnified view of $V_\mathrm{C}^7(x)$, highlighting the self-similar fractal structure.
• Figure 2: Density of states (DOS) for (a) the periodic, (b) the random, and (c) the Cantor potentials. The DOS is presented as a normalized histogram (with 60 bins) such that the total area integrates to unity. Solid curves represent smoothed distributions obtained via kernel density estimation (KDE). Note that the pronounced DOS peaks in the lowest-energy bins for the random and Cantor potentials exceed the vertical axis range and are truncated; the corresponding peak values are approximately $1.53\times 10^{-7}$ and $1.70\times 10^{-7}$, respectively.
• Figure 3: (a) Energy dependence of the nearest-neighbor level spacings $\Delta E_i$. Spacings smaller than $10^{-3}$ are regarded as numerically degenerate and excluded from the plot. (b)--(d) Histograms of the normalized level spacings $s_i$ for the periodic, random, and Cantor potentials, respectively. All distributions are normalized so that the total area equals unity.
• Figure 4: Dependence of the generalized fractal dimension $D_q$ on the moment order $q$ for the periodic, random, and Cantor potentials. The parameter $q$ ranges from $1.1$ to $5.0$. Solid lines are guides to the eye.
• Figure 5: (a)--(c) Histograms of the inverse participation ratio (IPR, $P_2$) calculated for the periodic, random, and Cantor potentials, respectively. The vertical axis represents the probability density $P(P_2)$ on a logarithmic scale, while the horizontal axis shows $P_2$ on a linear scale. The histograms are constructed using 80 bins over the range $0 \leq P_2 \leq 1200$. (d) Energy dependence of the IPR for the three potentials, plotted on a semi-logarithmic scale.