Multifractality without fine-tuning in a Floquet quasiperiodic chain

Abstract

Periodically driven, or Floquet, disordered quantum systems have generated many unexpected discoveries of late, such as the anomalous Floquet Anderson insulator and the discrete time crystal. Here, we report the emergence of an entire band of multifractal wavefunctions in a periodically driven chain of non-interacting particles subject to spatially quasiperiodic disorder. Remarkably, this multifractality is robust in that it does not require any fine-tuning of the model parameters, which sets it apart from the known multifractality of critical wavefunctions. The multifractality arises as the periodic drive hybridises the localised and delocalised sectors of the undriven spectrum. We account for this phenomenon in a simple random matrix based theory. Finally, we discuss dynamical signatures of the multifractal states, which should betray their presence in cold atom experiments. Such a simple yet robust realisation of multifractality could advance this so far elusive phenomenon towards applications, such as the proposed disorder-induced enhancement of a superfluid transition.

Drive-induced delocalization in the Aubry-André model

“…We emphasize that the dynamical instability and onset of chaos within the interval of T (as indicated by a straight line in Fig. 4(a) The existence of the staircase like structure in the spectrum together with the mobility edge observed in the delocalized band motivate us to explore the possible multifractal nature of the spectrum [22,31]. In general the local number of states ∆N within an interval ∆φ around the eigenphase φ follows the scaling law, ∆N ∼ (∆φ) α(φ) ; dependence of α on the eigenphase φ signifies the multifractality in the spectrum [22,32,33].…”

“…5(a). We also investigate the multifractality of the corresponding Floquet eigenstates from the scaling of their moments given by, I q = n |ψ(n)| 2q ∼ L −τq [31], where ψ(n) is the amplitude of the Floquet eigenstate |ψ at nth lattice site and L is the total number of lattice sites. Variation of τ q as a function of q is shown in Fig.…”

Renormalization to localization without a small parameter

“…10 Expressions Eqs. (31) and (32) are valid for quite large layers, Ri < r < R, with sufficiently small fluctuations of spatial density of sites. An actual probability to have no resonances, of course, is equal to k (1 − p i ε (r jk )) and depends on the particular realization of disorder as well as on the particular choice of r k .…”

“…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.…”

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

“…Multifractal statistics appears at the Anderson localization transition for single-particle lattice systems [17,[65][66][67][68][69][70][71]. In addition, recent examples have reported (multi)fractal phases extend-ing over a whole range of parameters [72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. Multifractal wavefunctions have been found for some quantum maps [68,70,87,88].…”