The Insulating State of Matter: A Geometrical Theory

TL;DR

This paper reviews the development of a geometrical framework for understanding the insulating state of matter, emphasizing the role of quantum metrics and topological concepts in classifying various insulators.

Contribution

It introduces a unified geometrical approach to characterize different types of insulators, highlighting the importance of quantum metrics and topological invariants.

Findings

Quantum metric sharply characterizes insulators.

Unified geometrical framework applies to various insulator types.

Topological invariants distinguish different insulating phases.

Abstract

In 1964 W. Kohn published the milestone paper "Theory of the insulating state'", according to which insulators and metals differ in their ground state. Even before the system is excited by any probe, a different organization of the electrons is present in the ground state and this is the key feature discriminating between insulators and metals. However, the theory of the insulating state remained somewhat incomplete until the late 1990s; this review addresses the recent developments. The many-body ground wavefunction of any insulator is characterized by means of geometrical concepts (Berry phase, connection, curvature, Chern number, quantum metric). Among them, it is the quantum metric which sharply characterizes the insulating state of matter. The theory deals on a common ground with several kinds of insulators: band insulators, Mott insulators, Anderson insulators, quantum Hall insulators, Chern and topological insulators