Breakdown of Avila's theory in the diamond chain with quasiperiodic disorder

Abstract

The mobility edges (MEs) that separate localized, multifractal and ergodic states in energy are a central concept in understanding Anderson localization. In this work we study the effect of several mutually commensurate quasiperiodic frequencies on the mobility-edge formation. We focus on the example of the addition of a constant offset to the quasiperiodic potential of the one-dimensional all-bands-flat diamond chain. We show that this additional offset can transform the anomalous mobility edges (AMEs), i.e. the energies, separating localized and multifractal states, into conventional mobility edges, separating localized from delocalized states. Also this appears to be the first example which shows the failure of Avila's global theory to analytically predict the ME location. We observe this violation both quantitatively, through the ME location mismatch, and qualitatively, via the formation of multiple MEs, not predicted by the theory.

Figures (14)

• Figure 1: Schematic of diamond lattice with $u$ (top), $c$ (center), and $d$ (bottom) sites. Solid grey (orange) lines between lattice sites have the same hopping amplitude and negative (positive) sign. Dashed rectangle shows the unit cell containing three sites.
• Figure 2: A schematic showing the presence of AMEs in $\epsilon_1<\lambda$ regime and MEs in $\epsilon_1>\lambda$ regime.
• Figure 3: (a)-(c) The fractal dimension $D_2$ (whose magnitude is captured as shown in the colorbar) as a function of increasing $\epsilon_1$ and energy $E$ for (a) $\lambda=10$, (b) $\lambda=5$, and (c) $\lambda=2$. (d)-(f) The Lyapunov exponent $\gamma$ (whose magnitude in the logscale is captured as shown in the colorbar) as a function of increasing $\epsilon_1$ and energy $E$ for (d) $\lambda=10$, (e) $\lambda=5$, and (f) $\lambda=2$. Here averaging is done over $50$ values of $\theta$. The system size is $N=12543$ and $J=1$. Green dashed lines are Avila's theory predictions given by Eq. \ref{['Final_ME']} and red dashed lines correspond to the modified analytical expression using fitting, given by Eq. \ref{['Final_ME_fitting']}.
• Figure 4: Spatial decay profiles of two consecutive eigenstates. (a) A localized state with $E=-0.26$ exhibiting clear exponential decay, with $\gamma(E)=0.006$ and a good linear fit ($R^2=0.856$). (b) The neighboring delocalized state with $E=-0.22$, characterized by $\gamma(E)\approx 0$ and poor linear fitting quality ($R^2=0.326$), indicating the absence of exponential localization. The dashed horizontal line marks the reference level $-\ln N$ of the ergodic wave function. Here averaging is done over 50 values of $\theta$. Number of unit cells $L=4181$, system size $N=3L=12543$, $\lambda=10$, and $\epsilon_1=16.8$.
• Figure 5: (a), (b) Lyapunov exponent $\gamma(E)$ for all eigenstates with (a) $\epsilon_1 = 1.5$, and (b) $\epsilon_1 = 16.8$. The green dashed lines indicate analytically derived mobility edges using Avila's theory [Eq. \ref{['Final_ME']}]. The red dashed lines are plotted using Eq. \ref{['Final_ME_fitting']} obtained after the fitting procedure. (c)–(f) Averaged $\ln(|\psi_n(E)|^2)$ over eigenstates with energies within the specified intervals, plotted as a function of the distance $n$ from the site of maximum amplitude $\psi_{\text{max}}$. (c) For $\epsilon_1 = 1.5$, data averaged over $E \in [-0.3, -0.25]$; (d) for $\epsilon_1 = 16.8$, data averaged over $E \in [-0.28, -0.25]$, where the black dashed line denotes an exponential fit with $\langle \gamma\rangle_E = 0.104$ and $\langle \gamma\rangle_E = 0.014$, respectively showing exponential decay. (e) For $\epsilon_1 = 1.5$, data averaged over $E \in [-0.004, 0.004]$ where the black dashed line shows a power-law decay with exponents $\mu = 0.789$ indicating multifractal nature. (f) For $\epsilon_1 = 16.8$, data averaged over $E \in [-0.06, 0.06]$ with $\langle \gamma\rangle_E=0$ showing delocalized behavior of states. The corresponding energy windows are highlighted by blue rectangles in panels (a) and (b). In all panels, averages are taken over 50 realizations of the phase parameter $\theta$. The system parameters are fixed at $N = 12543$, $\lambda=10$ and $J=1$.