Classification of "Real" Bloch-bundles: Topological quantum systems of type AI

Abstract

We provide a classification of type AI topological quantum systems in dimension d=1,2,3,4 which is based on the equivariant homotopy properties of "Real" vector bundles. This allows us to produce a fine classification able to take care also of the non stable regime which is usually not accessible via K-theoretic techniques. We prove the absence of non-trivial phases for one-band AI free or periodic quantum particle systems in each spatial dimension by inspecting the second equivariant cohomology group which classifies "Real" line bundles. We also show that the classification of "Real" line bundles suffices for the complete classification of AI topological quantum systems in dimension d<4. In dimension d=4 the determination of different topological phases (for free or periodic systems) is fixed by the second "Real Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide explicit realizations of non trivial 4-dimensional free models for each given topological degree.

45 pages, 2 figures, 7 tables. key words: Topological insulators, Bloch-bundle, "Real" vector bundle, "Real" Chern class. In v.2 title, abstract, introduction and Appendix A have been changed; minor typos have been fixed. To appear in J. Geom. Phys

References in corpus (7)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Topology of crystalline insulators and superconductors
• Theory of the Three Dimensional Quantum Hall Effect in Graphite
• From the adiabatic theorem of quantum mechanics to topological states of matter
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII

Cited by in corpus (32)

• Classification of topological quantum matter with symmetries
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time reversal symmetry
• Controlled topological phases and bulk-edge correspondence
• Bott periodicity for $Z_2$ symmetric ground states of gapped free-fermion systems
• Topological Bloch oscillations
• Crystallographic splitting theorem for band representations and fragile topological photonic crystals
• Index pairings in presence of symmetries with applications to topological insulators
• Classification of "Real" Bloch-bundles: Topological quantum systems of type AI
• A noncommutative framework for topological insulators
• Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond
• Homotopy Theory of Strong and Weak Topological Insulators
• Spectral flows associated to flux tubes
• Coarse geometry and topological phases
• Topology and entanglement in quench dynamics
• No-go theorem for topological insulators and sure-fire recipe for Chern insulators
• K-Theory for Real C*-algebras via Unitary Elements with Symmetries
• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII
• Differential geometric invariants for time-reversal symmetric Bloch-bundles: the "Real" case
• On the Role of Symmetries in the Theory of Photonic Crystals
• On the C*-algebraic approach to topological phases for insulators
• The FKMM-invariant in low dimension
• Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology
• Bulk-edge correspondence and the cobordism invariance of the index
• On the K-theoretic classification of dynamically stable systems
• The Geometry of (non-Abelian) Landau levels
• Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions
• The cohomology invariant for class DIII topological insulators
• Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case
• Topology of Fermi Surfaces and anomaly inflows
• Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume
• A new light on the FKMM invariant and its consequences