Quantum phase transitions from analysis of the polarization amplitude

TL;DR

This paper explores the polarization amplitude as a geometric phase in quantum systems, deriving formulas for its cumulants, and uses these to identify quantum phase transitions in interacting fermionic models.

Contribution

It introduces a new interpretation of the polarization amplitude as a discretized characteristic function and applies it to detect phase transitions in interacting systems.

Findings

Finite size scaling of variance identifies gap closure transitions.

Cumulants reveal phase transition points in interacting models.

Rich phase diagram emerges with next nearest neighbor interactions.

Abstract

In the modern theory of polarization, polarization itself is given by a geometric phase. In calculations for interacting systems the polarization and its variance are obtained from the polarization amplitude. We interpret this quantity as a discretized characteristic function and derive formulas for its cumulants and moments. In the case of a non-interacting system, our scheme leads to the gauge-invariant cumulants known from polarization theory. We study the behavior of such cumulants for several interacting models. In a one-dimensional system of spinless fermions with nearest neighbor interaction the transition at which gap closure occurs can be clearly identified from the finite size scaling exponent of the variance. When next nearest neighbor interactions are turned on a model with a richer phase diagram emerges, but the finite size scaling exponent is still an effective way to identify the localization transition.