Local Density of States as a Probe of Multifractality in Quasiperiodic Moiré Materials

TL;DR

This work proposes a model-independent, experimentally feasible method to identify multifractality in quasiperiodic moiré materials by analyzing the local density of states (LDOS) through multifractal analysis. Using a 1D quasiperiodic model with mobility edges, the authors apply a canonical partition function approach to LDOS maps, extracting the multifractal spectrum via $\tau_q$, $\alpha_q$, and $f(\alpha_q)$ without requiring finite-size scaling. They demonstrate clear, robust signatures of multifractality in the critical bands (broad $f(\alpha)$ and $\alpha_0>1$) and distinguish them from Bloch-like extended states (point-like spectra) across a range of energy broadening $\eta$. The results imply that STM-based LDOS measurements can detect quasiperiodic multifractality in moiré materials, providing a practical diagnostic tool applicable to 2D quasiperiodic systems.

Abstract

Quasiperiodic moiré materials provide a new platform for realizing critical electronic states, yet a direct and experimentally practical method to characterize this criticality has been lacking. We show that a multifractal analysis of the local density of states (LDOS), accessible via scanning tunneling microscopy, offers an unambiguous signature of criticality from a single experimental sample. Applying this approach to a one-dimensional quasiperiodic model, a stringent test case due to its fractal energy spectrum, we find a clear distinction between the broad singularity spectra $f\left(α\right)$ of critical states and the point-like spectra of extended states. We further demonstrate that these multifractal signatures remain robust over a wide range of energy broadenings relevant to experiments. Our results establish a model-independent, experimentally feasible framework for identifying and probing multifractality in the growing family of quasiperiodic and moiré materials.

Figures (3)

• Figure 1: (a) Singularity spectrum $f(\alpha)$ of the LDOS at a fixed energy in the critical narrow band for different quasiperiodic approximants with total system sizes $L \simeq 1.2\times10^{5}$. Increasing $\varepsilon$ (larger unit cell size) enhances multifractality as we approach the quasiperiodic limit ($\varepsilon = 100\%$). Inset: corresponding nonlinear Legendre transform $\tau_{q}$. (b) Position of the spectral maxima $\alpha_{0}$ vs $\varepsilon$; $\alpha_{0}>1$ in the quasiperiodic limit and $\alpha_{0}\approx1$ for Bloch states. (c) Same as in (a) for extended states, yielding point-like singularity spectra and linear $\tau_{q}$. (d) Energy spectrum vs quasiperiodic modulation $V$; colors denote the inverse participation ratio, distinguishing the extended, localized, and critical narrow bands.
• Figure 2: Partition function $Z_{q = 2}$ as a function of the coarse-graining scale $\lambda$ in the critical narrow bands, for different quasiperiodic approximants $( L \simeq 1.2\times10^{5}$). The curves range from the most periodic (blue) to the most quasiperiodic (magenta) case. Vertical dashed lines mark the scale corresponding to each unit cell. The slopes of the linear regions yield the multifractal exponent $\tau_{q}$. Inset: same analysis for energies in the extended narrow bands, where all curves collapse with unit slope, consistent with Bloch-like behavior.
• Figure 3: (a) Fractal structure of the spectrum. The characteristic energy scale $\Delta$ corresponds to the mean level spacing of the smallest miniband for this system size and is used as our reference energy scale. The rightmost zoom shows the same energy window highlighted in Fig. 1(d). (b) Multifractal exponent $\tau_2$ as a function of the LDOS broadening $\eta$, normalized by $\Delta$, for a system of size $L = 10946$. Vertical dashed lines mark the broadening values analyzed in detail, corresponding to $\sim$1%, 5%, and 25% of states effectively contributing to the LDOS. (c) Multifractal exponents $\tau_q$ for representative broadening values $\eta/\Delta \sim 10^2$, $10^4$, and $10^6$. (c) Corresponding singularity spectra $f(\alpha)$, showing the crossover from narrow to broad multifractal behavior as the energy resolution increases (i.e., as $\eta$ decreases).