From the adiabatic theorem of quantum mechanics to topological states of matter. Issue 1 (8th January 2013)

• Record Type: Journal Article
• Authors: Budich, Jan Carl; Trauzettel, Björn
• Is Part Of: Physica status solidi. Volume 7:Issue 1/2(2013:Jan.)
• Page Start: 109
• Page End: 129
• Publication Date: 2013-01-08
• Subjects: Berry phase -- adiabatic theorem -- topological invariants -- Chern number -- topological insulators -- topological superconductors -- quantum Hall effect -- topological states of matter -- topological phases -- edge states; Solid state physics -- Periodicals; 530.4105
• DOI: 10.1002/pssr.201206416
• Languages: English
• ISSNs: 1862-6254

Abstract

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. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) Abstract : Topological states of matter that can be understood at the level of noninteracting insulators and mean field superconductors, respectively, have attracted enormous interest in recent years. The authors review the theoretical foundations of these topological band structures in all symmetry classes and discuss how interactions and disorder can be included into the classification scheme.