Observation of log-periodic oscillations in the quantum dynamics of electrons on the one-dimensional Fibonacci quasicrystal

Abstract

We revisit the question of quantum dynamics of electrons on the off-diagonal Fibonacci tight-binding model. We find that typical dynamical quantities, such as the probability of an electron to remain in its original position as a function of time, display log-periodic oscillations on top of the leading-order power-law decay. These periodic oscillations with the logarithm of time are similar to the oscillations that are known to exist with the logarithm of temperature in the specific heat of Fibonacci electrons, yet they offer new possibilities for the experimental observation of this unique phenomenon. Fibonacci ElectronsThe Fibonacci sequence of Long (L) and Short (S) intervals on the 1-dimensional line-generated by the simple substitution rules L → LS and S → L-is a favorite textbook model for demonstrating the peculiar nature of electrons in quasicrystals [1][2][3]. The wave functions of Fibonacci electrons are neither extended nor exponentially-localized, but rather decay algebraically; the spectrum of energies is neither absolutely-continuous nor discrete, but rather singular-continuous, like a Cantor set; and the quantum dynamics is anomalous. In recent years we have studied how these three electronic properties change as the dimension of the Fibonacci quasicrystal increases to two and three [4][5][6][7][8], by constructing square and cubic versions of the Fibonacci quasicrystal [9].The 1-dimensional off-diagonal Fibonacci tight-binding model is constructed by associating a unit hopping amplitude between sites connected by a Long interval, and a hopping amplitude T > 1 between sites connected by a Short interval, while assuming equal on-site energies that are taken to be zero. The resulting tight-binding Schrödinger equation, on an F N -site model, is 1

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