Localisation in quasiperiodic chains: a theory based on convergence of local propagators

Abstract

Quasiperiodic systems serve as fertile ground for studying localisation, due to their propensity already in one dimension to exhibit rich phase diagrams with mobility edges. The deterministic and strongly-correlated nature of the quasiperiodic potential nevertheless offers challenges distinct from disordered systems. Motivated by this, we present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators; exploiting the fact that the imaginary part of the associated self-energy acts as a probabilistic order parameter for localisation transitions and, importantly, admits a continued-fraction representation. Analysing the convergence of these continued fractions, localisation or its absence can be determined, yielding in turn the critical points and mobility edges. Interestingly, we find anomalous scalings of the order parameter with system size at the critical points, consistent with the fractal character of critical eigenstates. Self-consistent theories at high orders are also considered, shown to be conceptually connected to the theory based on continued fractions, and found in practice to converge to the same result. Results are exemplified by analysing the theory for three quasiperiodic models covering a range of behaviour.

11 pages, 9 figures, version published in Phys. Rev. B, see also related article Phys. Rev. B 103 L060201 (2020)

References in corpus (13)

• Many-Body Localization in a Quasiperiodic System
• Nearest neighbor tight binding models with an exact mobility edge in one dimension
• One dimensional quasiperiodic mosaic lattice with exact mobility edges
• Localization in one dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-André models
• Localization in one-dimensional incommensurate lattices beyond the Aubry-André model
• Flat Bands Under Correlated Perturbations
• Self-dual quasiperiodic systems with power-law hopping
• Non-equilibrium phase diagram of a 1D quasiperiodic system with a single-particle mobility edge
• Duality between two generalized Aubry-Andre models with exact mobility edges
• Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
• Localisation on certain graphs with strongly correlated disorder
• Self-consistent theory of mobility edges in quasiperiodic chains
• Quantum walks in commensurate off-diagonal Aubry-André-Harper model

Cited by in corpus (6)

• Hidden dualities in 1D quasiperiodic lattice models
• Single-Particle Mobility Edge without Disorder
• Local Integrals of Motion in Quasiperiodic Many-Body Localized Systems
• Numerical Investigation of Localization in Two-Dimensional Quasiperiodic Mosaic Lattice
• Many-body localization in the infinite-interaction limit and the discontinuous eigenstate phase transition
• Quantum Fidelity of the Aubry-André Model and the Exponential Orthogonality Catastrophe