Critical Behavior and Fractality in Shallow One-Dimensional Quasiperiodic Potentials

Abstract

Quasiperiodic systems offer an appealing intermediate between long-range ordered and genuine disordered systems, with unusual critical properties. One-dimensional models that break the socalled self-dual symmetry usually display a mobility edge, similarly as truly disordered systems in dimension strictly higher than two. Here, we determine the critical localization properties of single particles in shallow, one-dimensional, quasiperiodic models and relate them to the fractal character of the energy spectrum. On the one hand, we determine the mobility edge and show that it separates the localized and extended phases, with no intermediate phase. On the other hand, we determine the critical potential amplitude and find the universal critical exponent ν 1/3. We also study the spectral Hausdorff dimension and show that it is nonuniversal but always smaller than unity, hence showing that the spectrum is nowhere dense. Finally, applications to ongoing studies of Anderson localization, Bose-glass physics, and many-body localization in ultracold atoms are discussed. arXiv:1904.01463v3 [cond-mat.quant-gas]

Universality and quantum criticality in quasiperiodic spin chains

“…We remark that although this flow to nearly discrete sequences is a general property of the RG rules, and occurs for any irrational number, the self-similarity of these discrete sequences is a special property of the Golden Ratio and other irrational numbers with recurring continued fraction expansions, such as the metallic means 1/(n + 1/(n + …)) (the Golden Ratio is the case n = 1). For more general irrational numbers, the sequences do not repeat under the RG: instead, each level of the RG is governed by the coefficient of the continued fraction expansion at the corresponding level 21,28,42,43 .…”

Metal-insulator phase transition in a non-Hermitian Aubry-André-Harper model

“…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]. Typical dynamical variables that characterize single-particle transport are the largest propagation speed v of excitation in the lattice, which is bounded (to form a light cone) for short-range hopping according to the Lieb-Robinson bound [28], and the quantum diffusion exponent δ, which measures the asymptotic spreading of wave packet variance σ 2 (t) in the lattice according to the power law σ 2 (t) ∼ t 2δ .…”

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

“…To measure the localization of the eigenstates of the system, we study the inverse participation ratio (IPR) of the eigenstate y ñ n | corresponding to the eigenenergy E n , = å C IPR n j j n 4 | | ( ) ( ) [31,48,52], containing information of the eigenstate y ñ = å ñ C j n j j n | | ( ) with the Wannier basis ñ j | being chosen at each lattice site j. The IPR shows the scaling behavior with respect to the system size L,…”

“…One can obtain a one-dimensional model displaying the mobility edge when the so-called self-dual symmetry is broken, such as a system with a shallow one-dimensional quasi-periodic potential [47][48][49][50][51]. Another class of systems with the mobility edge by introducing a long-range hopping term [31] or a special form of the on-site incommensurate potential [52] present the energy-dependent self-duality in the compactly analytic form.…”