Self-Similarity and Localization

TL;DR

This paper investigates the universal self-similar properties of localized eigenstates in the Harper equation and extends the analysis to generalized models with additional interactions, revealing fixed points and invariant sets.

Contribution

It introduces a renormalization framework to characterize the entire localized phase and generalizes the findings to models with next nearest neighbor interactions.

Findings

Localized eigenstates exhibit universal self-similar fluctuations.

A single strong coupling fixed point characterizes the localized phase.

Above a certain threshold, fluctuations are described by a strange invariant set.

Abstract

The localized eigenstates of the Harper equation exhibit universal self-similar fluctuations once the exponentially decaying part of a wave function is factorized out. For a fixed quantum state, we show that the whole localized phase is characterized by a single strong coupling fixed point of the renormalization equations. This fixed point also describes the generalized Harper model with next nearest neighbor interaction below a certain threshold. Above the threshold, the fluctuations in the generalized Harper model are described by a strange invariant set of the renormalization equations.