Irrational Mode Locking in Quasiperiodic Systems

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous-time models with irrational mode locking and possible experimental realizations are discussed.

Figure 1

The activity staircase, velocity $v$ vs drive $F$, for the quasiperiodic automaton of size $L_x\times L_y=169\times 153$ with pinning given by the rational approximants ($70/169$, $112/153$) to ($\sqrt{2}-1$, $\sqrt{3}-1$). Fractions with small denominators generally have the widest mode-locked steps, but the step at $86/1611$ is an anomalous step which does not fit this pattern. Inset: results for the FK model, Eq. (2), with $h=10$ and the same size and quasiperiodic pinning; an anomalous step with similar $v\approx 0.053$ is seen.

Figure 2

The size dependence of (a) activity $v(L_x,L_y)$, and (b) width, $w(L_x,L_y)$, of the anomalous step. Each curve is for fixed $L_y$. The convergence of $v$ and $w$ as a function of system size strongly suggests that there is a well-defined activity $\widehat{v}$ and step width $\widehat{w}$ in the limit $(L_x,L_y)\to (\infty ,\infty )$; we conjecture that $\widehat{v}\mathrm{\in ̸}\mathbb{Q}$, Eq. (9), and $\widehat{w}\ne 0$.

Figure 3

The effect of thermal noise on the anomalous step for dimensionless noise values $\eta ={10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$ for system sizes defined in the key. The horizontal axis is the scaled force $f$ which is linear in $F$ with each zero-temperature step starting at $f=0$ and ending at $f=1$. For easier comparison, the zero-temperature activity $v(L_x,L_y)$ has been subtracted. For each value of noise, there is similar rounding of the plateau for all system sizes.