Quantized order parameters of approximate symmetry for metals and insulators
Yasuhiro Tada
TL;DR
The paper provides a rigorous foundation for distinguishing metals and insulators by introducing an approximate symmetry framework built around the polarization operator $U$ and, in higher dimensions, an accompanying tiny flux to render translation symmetry approximate. In one dimension, it proves that $|raket{Ψ_0|U^q|Ψ_0}|$ serves as a sharp indicator, vanishing for gapless metals and remaining finite for gapped insulators, thereby defining a many-body index ind$_0^q$. For higher dimensions, the authors construct approximate magnetic translation operators whose commutation with translations reproduces a quantized criterion in the thermodynamic limit via flux insertion, allowing $|raket{Ψ_0| ilde{U}^q| ilde{Ψ}_0}| o 1$ for insulators and $ o 0$ for metals. The framework unifies and generalizes polarization-based criteria, provides practical tools for finite-size numerics, and clarifies when and how the polarization operator can function as a genuine order parameter in gapless versus gapped phases.
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.