One-Dimensional Quasicrystals with Power-Law Hopping

Abstract

One-dimensional quasi-periodic systems with power-law hopping, 1/r a , differ from both the standard Aubry-Azbel-Harper (AAH) model and from power-law systems with uncorrelated disorder. Whereas in the AAH model all single-particle states undergo a transition from ergodic to localized at a critical quasi-disorder strength, short-range power-law hops with a > 1 can result in mobility edges. We find that there is no localization for long-range hops with a ≤ 1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but non-ergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

Renormalization to localization without a small parameter

“…Probabilities (30) help us to define the typical probability 10 to have no resonances in the layer R i < r < R,…”

“…There are many other surprises such as emergence of multifractality in long-range static[28][29][30] or shortrange driven[31] models with quasiperiodic potentials but we focus on the one relevant for our consideration 2. In this case a top energy level keeps delocalized even at strong disorder due to its energy diverging with the system size and shields the rest levels from the hopping terms.…”

Dynamical observation of mobility edges in one-dimensional incommensurate optical lattices

“…We average 2000 quasi-disorder realizations by choosing different phases δ for all the data. In the extended phase, the initial state at the center of the lattice expands rapidly and after some long-time intervals, the wave function presents a ergodic character To further distinct the dynamics of the system in different phases, we observe the long-time survival probability P(r) [77]. The probability of detecting the wave packet in sites within the region (−r/2, r/2) after a given time,…”

“…When the parameters are in the extended regime (b=0.2, λ<1.48), since the probability of finding the wave packet at each site is the same, it linearly increases with r, P(r)∝r/L. For the localized phase (b=0.2, λ>2.52), P(r) presents exponential rise and rapidly reaches to (r/L) 0 =1 [77]. In the intermediate regime, P(r) exponentially increases for r/L=1 and for finite r, the increase of P(r) is proportional to r/L again.…”

Anomalous transport through algebraically localized states in one dimension

“…The most common characterization of a single-particle eigenstate as localized or delocalized is done in terms of the scaling of the Inverse Participation Ratio (IP R) with system-size [6,34,35]. For a localized state, the IP R does not scale with system size.…”