Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals

Abstract

The emergence of a fractal energy spectrum is the quintessence of the interplay between two periodic parameters with incommensurate length scales. crystals can emulate such interplay and also exhibit a topological bulk-boundary correspondence, enabled by their nontrivial topology in virtual dimensions. Here we propose, fabricate and experimentally test a reconfigurable one-dimensional (1D) acoustic array, in which the resonant frequencies of each element can be independently fine-tuned by a piston. We map experimentally the full Hofstadter butterfly spectrum by measuring the acoustic density of states distributed over frequency while varying the long-range order of the array. Furthermore, by adiabatically changing the phason of the array, we map topologically protected fractal boundary states, which are shown to be pumped from one edge to the other. This reconfigurable crystal serves as a model for future extensions to electronics, photonics and mechanics, as well as to quasicrystalline systems in higher dimensions.

Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model

“…In order to accurately compare the stability of FCIs in higher Chern bands, we first need to systematically generate topological flat bands of arbitrary Chern number. To this end, we employ the Hofstadter model, as introduced in the previous section, due to its simplicity, versatility, and experimental relevance [35,[58][59][60][61][62][63][64]. Here, we build on work by Möller and Cooper [8,19], who showed evidence for novel fractional Chern insulators in higher Chern bands of the Hofstadter model [8] and later recast this result in the language of composite fermions [19], showing how the Jain series [65] generalizes to higher Chern bands.…”

Phase transitions in a non-Hermitian Aubry-André-Harper model

“…One-dimensional lattices with aperiodic order, i.e., displaying a long-range periodicity intermediate between ordinary periodic crystals and disordered systems, provide fascinating models to study unusual transport phenomena in a wide variety of classical and quantum systems, ranging from condensed matter systems to ultracold atoms, photonic, and acoustic systems [1][2][3][4][5][6][7]. Quasiperiodicity gives rise to a range of unusual behavior, including critical spectra, multifractal eigenstates, localization transitions at a finite modulation of the on-site potential, and mobility edges [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27].…”

Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

“…Topological states have been successfully observed in several platforms [13][14][15][16][17][18][19][20][21], and have been pursued to achieve robust, diffraction-free wave motion. Additional functionalities have been explored in the context of topological pumping [22][23][24][25][26], quasi-periodicity [27][28][29], and non-reciprocal wave propagation in active [30][31][32][33][34][35][36] or passive non-linear [37][38][39][40] systems. These works and the references therein illustrate a wealth of strategies for the manipulation of elastic and acoustic waves, and suggest intriguing possibilities for technological applications in acoustic devices, sensing, energy harvesting, among others.…”