Quantum Metric Localization and Quantum Metric Protection

TL;DR

This paper reveals a new regime called quantum metric localization where the localization length in disordered systems is protected by the quantum metric, deviating from traditional Anderson localization behavior.

Contribution

It introduces the concept of quantum metric localization and protection, supported by numerical, physical, and field theory analyses, applicable across various disordered systems.

Findings

Localization length plateaus at twice the quantum metric length with increasing disorder.

Quantum metric protection prevents localization length from decreasing until very high disorder.

Numerical and theoretical models explain the origin of quantum metric localization.

Abstract

The study of disorder effects in electronic systems is one of the central themes in physics. It is well established that in the Anderson localization regime, the localization length of electrons decreases monotonically as the disorder strength increases. Here, we demonstrate that the conventional Anderson localization paradigm fails completely in describing an isolated band with quantum metric, where the quantum metric of the band defines a length scale called the quantum metric length. For an isolated band with a finite bandwidth separated from other bands by a band gap $\Delta$, weak disorder results in conventional Anderson localization behavior. However, as the disorder increases, the localization length ceases to decrease and becomes pinned at a value approximately twice the quantum metric length, forming a localization length plateau. We term the regime within this localization length plateau as the quantum metric localization regime. Remarkably, the localization length does not deviate from the plateau until the disorder strength far exceeds $\Delta$. We refer to this strong protection against disorder, characterized by the quantum metric length, as quantum metric protection. In this work, we first numerically demonstrate quantum metric localization using a 1D Lieb lattice. We then provide a simple physical picture based on the properties of Wannier functions to explain the origin of the localization length plateau. A supersymmetric field theory approach explains why the localization length is approximately twice the quantum metric length and captures the crossover from Anderson localization to quantum metric localization. Our conclusions are broadly applicable to disordered electronic, photonic, and acoustic systems.