Landau level broadening, hyperuniformity, and discrete scale invariance

We study the energy spectrum of a two-dimensional electron in the presence of both a perpendicular magnetic field and a potential. In the limit where the potential is small compared to the Landau level spacing, we show that the broadening of Landau levels is simply expressed in terms of the structure factor of the potential. For potentials that are either periodic or random, we recover known results. Interestingly, for potentials with a dense Fourier spectrum made of Bragg peaks (as found, e.g., in quasicrystals), we find an algebraic broadening with the magnetic field characterized by the hyperuniformity exponent of the potential. Furthermore, if the potential is self-similar such that its structure factor has a discrete scale invariance, the broadening displays log-periodic oscillations together with an algebraic envelope.

Figure 1

Sketch of the LLL variance $w^2$ as a function of the magnetic field $B$ for different perturbing potentials. Red: Disordered $w^2\sim B$ (see Ref. [3]). Green: Hyperuniform with discrete scale invariance $w^2\sim B^{\frac{2+\alpha }{2}}+log$-periodic oscillations, where $\alpha$ is the hyperuniformity exponent (this work). Blue: Periodic $w^2\sim e^{-\#/B}$ (see Ref. [8]).

Figure 2

Integrated density function $Z$ of the Rauzy (left), Ammann-Beenker (center), and Penrose (right) tilings (see insets for illustrations). Top: Log-log plot (blue dots) together with the algebraic envelope $k^{2+\alpha }$ (green line). Bottom: $Z\left(k\right)/k^{2+\alpha }$ as a function of $lnk/ln\lambda$ ($lnk$) showing 1-periodic (nonperiodic) oscillations for tilings with (without) discrete scale invariance.

Figure 3

Integrated density function $Z$ of the Fibonacci chain (see the inset). Top: Log-log plot (blue dots) together with the power-law envelope $k^4$ (green line). Bottom: $Z\left(k\right)/k^4$ as a function of $lnk/ln\tau$ showing periodic oscillations (with period 1).