Quantized order parameters of approximate symmetry for metals and insulators

TL;DR

This paper introduces a new method using approximate symmetry order parameters to distinguish metals from insulators, providing rigorous proofs in 1D and extending the concept to higher dimensions with flux control.

Contribution

It develops a simple scheme based on approximate symmetry operators to classify gapless and gapped phases, with rigorous proofs for 1D and a novel approach for higher dimensions.

Findings

Exact criteria for 1D metals and insulators confirmed

Approximate symmetry operators serve as quantized order parameters

Method applicable to higher-dimensional systems with flux control

Abstract

We develop a simple scheme to distinguish between metals and insulators, or more generally gapless and gapped phases, by introducing the notion of an approximate symmetry order parameter. For one dimensional systems, we provide an explicit proof for the known criteria of metals and insulators based on the polarization operator which have been widely accepted for decades without a rigorous proof. For higher dimensions, we introduce a tiny magnetic flux to control the system, where the translation symmetry becomes approximate. We show that insulators and metals can be well distinguished with use of the approximate symmetry operators and they work as quantized order parameters in the thermodynamic limit characterizing gapless and gapped nature of the system.