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

TL;DR

This paper introduces a novel theory for localisation in quasiperiodic chains based on the convergence of local propagators and continued fractions, providing insights into critical points and mobility edges.

Contribution

The paper develops a new theoretical framework using local propagators and continued fractions to analyze localisation and mobility edges in quasiperiodic systems.

Findings

Identifies critical points and mobility edges through continued-fraction convergence.

Finds anomalous scaling of the order parameter at critical points.

Shows self-consistent theories converge to the same results as the continued-fraction approach.

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.