Emergent criticality in the Aubry-André model with periodic modulation

Abstract

The Aubry-André model describes a system with quasiperiodic lattice modulation. In one dimension the AAH model is known to exhibit a sharp metal to insulator transition at a self-dual critical point at which all the states in the spectrum are critical or multifractal in nature. While such criticality is immediately destroyed by an additional onsite periodic modulation, we show an emergent criticality in the limit of strong periodic modulation strength under proper conditions. The resulting strong-modulation critical phase exhibits multifractal eigenstates and singular continuous spectra, belonging to the universality class of the critical Harper model. Moreover, we reveal that additional periodic potential of period N in the quasiperiodic chain folds the spectrum into N bands with quasiperiodicity being enhanced by a factor of N, producing N numbers of Hofstadter butterflies in each band. Our results reveal a general mechanism for engineering robust criticality and spectral replication in quasiperiodic systems.

Figures (6)

• Figure 1: (a) Emergent criticality is shown as a function of quasiperiodic potential strength $\lambda$ and 2-periodic potential strength $V$ in terms of fraction of critical states ($f$) (shown in the color bar). AAH type criticality emerges at $\lambda\approx 2/V$ for large values of $V$ (red diamonds). (b) $R[L,L']$ as a function of $\lambda$ for different system size-combinations $[L,L']$. The abscissa and ordinate of the crossing point gives the critical $\lambda_c\approx0.02$ and ratio $\gamma/ \nu\approx0.620 \pm 0.005$, respectively. The inset shows data collapse using finite-size scaling analysis for $\lambda_c$ and $\gamma/\nu$ obtained from figure (a). (c) $\log(\sigma_L)$ vs $\log(L)$ at $\lambda_c$ gives $\beta/\nu= 0.191 \pm 0.005$ from the slope of the fitted straight line. (d) Hausdorff dimension $D_H$ is calculated from the plot of $\log(N_l)$ vs $\log(1/l)$. The slope from the fitted line (blue dashed) gives $D_H = 0.478 \pm 0.004$.
• Figure 2: (a) The standard HB (energy spectrum as a function of $\beta$) at $V=0$ and $\lambda=2$, the self-dual AAH critical point.(b) HB doubling (shown for only the upper band) for 2-period superlattice modulation at large $V=100$ and $\lambda=0.02$ manifesting the emergent criticality at $\lambda=2/V$. For all plots, $L=2584$ and $\beta=\{1,2,...,L-1\}/L$.
• Figure 3: (a) The fractal dimension $D_2$ as a function of $\lambda$ and eigenstate index ($n/L$) at $V=100$ for 3-period superlattice modulation. $\lambda_{c1}=1/V^2$ (left vertical line) and $\lambda_{c2}=2/V^2$ (right vertical line) are the band-dependent critical points (b) Spectrum-averaged $\langle \rm{NPR}\rangle$ (circles) and $\langle \rm{IPR}\rangle$ (squares) are plotted against $\lambda$, corresponding to (a), to demonstrate the presence of an intermediate phase between $\lambda_{c1}$ and $\lambda_{c2}$ (the gray-shaded region). (c) Emergence of criticality is shown as a function of quasiperiodic potential strength $\lambda$ and 3-periodic potential strength $V$ in terms of fraction of critical states $f$ for the middle band with $L=6765$. AAH criticality emerges at $\lambda\approx 2/V^2$ for large values of $V$ (red diamonds). (d) Hausdorff dimension $D_H=0.468 \pm 0.004$ (fitted blue dashed line) for the emergent critical middle band at $V=100$ and $\lambda_{c_2}=0.0002$. For the localized upper band, $D_H=0.899 \pm 0.008$ (fitted red dashed line). Data points are shown as violet squares (blue circles) for the middle (upper) band.
• Figure 4: (a) $D_2$ after Hamiltonian-engineering for the $\lambda_{c1}$ where all eigenstates become critical. (c) Spectrum-averaged $\langle \rm{NPR}\rangle$ (circles) and $\langle \rm{IPR}\rangle$ (squares) are plotted against $\lambda$, corresponding to figure (a), which shows the shrinking of the intermediate phase to a single critical point at $\lambda_{c1}$.
• Figure S1: Evolution of Critical States and Hofstadter Butterfly with 2-period modulation: Energy-resolved fractal dimension $D_2$, shown for increasing values of $V$ from in figures (a-d). The vertical axis is the normalized eigenstate index $n/L\in[0,1]$; the horizontal axis is the AAH disorder strength $\lambda$. The system sizes used to estimate $D_2$ were $L=\{610,\,2584,\,10946\}$. Energy spectrum ($E$) as a function of $\beta$ is shown at various ($V,\lambda$) points in figures (e–h) mapping the transition from (e) close to standard AAH limit to (h) the same at the emergent AAH limit for large $V$ for system size $L=2584$.