[Paper Review] Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII

Giuseppe De Nittis, Kiyonori Gomi|arXiv (Cornell University)|Apr 23, 2014

Topological Materials and Phenomena21 references16 citations

TL;DR

This paper provides a complete classification of topological quantum systems of type AII in dimensions d=1,2,3,4 using a novel topological invariant called the FKMM-invariant, which fully classifies 'Quaternionic' (symplectic) vector bundles. The invariant, rooted in equivariant cohomology, reproduces the Fu-Kane-Mele index in physical cases and is proven to be a universal characteristic class, with the classification in d=4 requiring also the second Chern class.

ABSTRACT

We provide a classification of type AII topological quantum systems in dimension d=1,2,3,4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies "Quaternionic" vector bundles (a.k.a. "symplectic" vector bundles) in dimension d<4. This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d=4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we prove that the FKMM-invariant is a bona fide characteristic class for the category of "Quaternionic" vector bundles in the sense that it can be realized as the pullback of a universal topological invariant.

Motivation & Objective

• To provide a complete homotopy classification of 'Quaternionic' vector bundles underlying type AII topological quantum systems in dimensions d=1,2,3,4.
• To establish the FKMM-invariant as a complete topological invariant for classifying these bundles in low dimensions (d≤3), extending beyond K-theory to unstable cases.
• To prove that the FKMM-invariant is a bona fide characteristic class by constructing a universal version that pulls back to the invariant for any bundle.
• To resolve the gap in mathematical understanding of the Fu-Kane-Mele index by showing it arises naturally from the FKMM-invariant in physical systems.
• To generalize the classification to d=4 by combining the FKMM-invariant with the second Chern class, ensuring completeness.

Proposed method

• The FKMM-invariant is constructed via a determinant construction in equivariant Borel cohomology, leveraging the Z2-action induced by time-reversal symmetry.
• The classification in d≤3 relies on proving the injectivity of the FKMM-invariant, showing it distinguishes all non-isomorphic 'Quaternionic' bundles.
• For d=4, the classification combines the FKMM-invariant with the second Chern class, as the former alone is insufficient to classify all bundles.
• The paper proves universality of the FKMM-invariant by constructing a universal invariant in equivariant cohomology that realizes the FKMM-invariant via pullback.
• The framework uses KQ-theory and KR-theory, with 8-periodicity and suspension isomorphisms to compute KQ-groups for TR-spheres and TR-tori.
• The analysis is applied to standard physical models like TR-spheres and TR-tori, computing their KQ-groups and verifying consistency with known physical invariants.

Experimental results

Research questions

• RQ1 How can 'Quaternionic' vector bundles—central to class AII topological insulators—be completely classified in low dimensions using a single topological invariant?
• RQ2 Does the FKMM-invariant fully reproduce the Fu-Kane-Mele index in physical systems, and if so, under what conditions?
• RQ3 Is the FKMM-invariant a genuine characteristic class for the category of 'Quaternionic' bundles, and can it be universally realized via pullback?
• RQ4 Why is the second Chern class necessary for the classification in dimension d=4, and how does it complement the FKMM-invariant?
• RQ5 Can the classification be extended beyond spheres and tori to general involutive base spaces, and what are the topological obstructions?

Key findings

• The FKMM-invariant completely classifies 'Quaternionic' vector bundles in dimensions d≤3, with injectivity proven via homotopy classification.
• The FKMM-invariant reproduces the Fu-Kane-Mele index in all physically relevant cases, including TR-spheres and TR-tori.
• In dimension d=4, the classification requires both the FKMM-invariant and the second Chern class, as the former alone is insufficient.
• The FKMM-invariant is a universal characteristic class, meaning it arises as the pullback of a single universal invariant in equivariant cohomology.
• For TR-spheres, the reduced KQ-group is Z2 in d=2 and d=3, and Z in d=4, matching the expected topological invariants for quantum spin Hall systems.
• For TR-tori, the KQ-groups exhibit a complex structure with Z2-torsion and free abelian components, reflecting the interplay between dimension and time-reversal symmetry.