From the adiabatic theorem of quantum mechanics to topological states of matter

TL;DR

This review explores the theoretical foundations of topological states of matter, emphasizing the geometrical and topological invariants derived from the adiabatic theorem, and discusses their classification, interactions, and distinctions from topological order.

Contribution

It provides a comprehensive theoretical framework for topological states of matter, including classification via K-Theory and extensions to interactions and disorder.

Findings

Topological invariants are defined through adiabatic curvature.

Classification of TSM in all symmetry classes using K-Theory.

Framework incorporates effects of interactions and disorder.

Abstract

Owing to the enormous interest the rapidly growing field of topological states of matter (TSM) has attracted in recent years, the main focus of this review is on the theoretical foundations of TSM. Starting from the adiabatic theorem of quantum mechanics which we present from a geometrical perspective, the concept of TSM is introduced to distinguish gapped many body ground states that have representatives within the class of non-interacting systems and mean field superconductors, respectively, regarding their global geometrical features. These classifying features are topological invariants defined in terms of the adiabatic curvature of these bulk insulating systems. We review the general classification of TSM in all symmetry classes in the framework of K-Theory. Furthermore, we outline how interactions and disorder can be included into the theoretical framework of TSM by reformulating the relevant topological invariants in terms of the single particle Green's function and by introducing twisted boundary conditions, respectively. We finally integrate the field of TSM into a broader context by distinguishing TSM from the concept of topological order which has been introduced to study fractional quantum Hall systems.