Scaling of the bulk polarization in extended and localized phases of a quasiperiodic model

TL;DR

This paper investigates the finite size scaling of bulk polarization in a quasiperiodic Aubry-André model, demonstrating the effectiveness of a geometric Binder cumulant method in identifying phase transitions and localization properties.

Contribution

It introduces a geometric Binder cumulant approach to accurately determine phase transition points and scaling exponents in the Aubry-André model.

Findings

Binder cumulant reproduces known distribution values in delocalized phase

Phase transition point is accurately identified at half-filling

Sign change in Binder cumulant indicates the localization transition

Abstract

We study the finite size scaling of the bulk polarization in a quasiperiodic (Aubry-Andr\'{e}) model using the geometric analog of the Binder cumulant. As a proof of concept we show that the geometric Binder cumulant method described here can reproduce the known literature values for the flat and raised cosine distributions, which are the two distributions that occur in the delocalized phase. For the Aubry-Andr\'{e} model at half-filling the phase transition point is accurately reproduced. Not only is the correct size scaling exponent of the variance obtained in the extended and the localized phases, but the geometric Binder cumulant undergoes a sign change at the phase transition. We also calculate the state resolved Binder cumulant as a function of disorder strength to gain insight into the mechanism of the localization transition.