[Paper Review] Topological nature of the Fu-Kane-Mele invariants

Giuseppe De Nittis|arXiv (Cornell University)|Jan 1, 2016

Topological Materials and Phenomena35 references3 citations

TL;DR

This paper introduces the FKMM-invariant, a universal Quaternionic topological invariant that generalizes and explains the topological nature of the Fu-Kane-Mele index in systems with odd time-reversal symmetry (class AII). By classifying such systems via Quaternionic vector bundles, the FKMM-invariant provides a robust, dimension-independent framework to distinguish topological phases in low-dimensional systems, including spheres and tori, and offers a unified description applicable beyond electrons in periodic lattices.

ABSTRACT

Condensed matter electronic systems endowed with odd time-reversal symmetry (TRS) (a.k.a. class AII topological insulators) show topologically protected phases which are described by an invariant known as Fu-Kane-Mele index. The construction of this in- variant, in its original form, is specific for electrons in a periodic background and is not immediately generalizable to other interesting physical models where different forms of TRS also play a role. By exploiting the fact that system with an odd TRS (in absence of disorder) can be classified by Quaternionic vector bundles, we introduce a Quaternionic topological invariant, called FKMM-invariant, which generalizes and explains the topological nature of the Fu-Kane-Mele index. We show that the FKMM-invariant is a universal characteristic class which can be defined for Quaternionic vector bundles in full generality, independently of the particular nature of the base space. Moreover, it suffices to discriminate among different topological phases of system with an odd TRS in low dimension. As a particular application we describe the complete classification over a big class of low dimensional involutive spheres and tori. We also compare our classification with recent results concerning the description of topological phases for two-dimensional adiabatically perturbed systems. Joint work with: K. Gomi.

Motivation & Objective

• To generalize the Fu-Kane-Mele index beyond electronic systems in periodic lattices to broader physical models with odd time-reversal symmetry.
• To establish a topological invariant—called the FKMM-invariant—that is universally applicable to Quaternionic vector bundles, regardless of the base space.
• To provide a complete classification of topological phases in low-dimensional systems, such as spheres and tori, under odd time-reversal symmetry.
• To unify the description of topological phases across different models, including adiabatically perturbed two-dimensional systems, by leveraging the FKMM-invariant.

Proposed method

• Utilizing the mathematical framework of Quaternionic vector bundles to classify systems with odd time-reversal symmetry, replacing the lattice-specific construction of the original Fu-Kane-Mele index.
• Defining the FKMM-invariant as a characteristic class for Quaternionic vector bundles, ensuring it is independent of the base space's specific structure.
• Applying algebraic topology techniques to compute the FKMM-invariant in low-dimensional manifolds, such as involutive spheres and tori, to classify topological phases.
• Establishing a correspondence between the FKMM-invariant and known topological invariants in specific models, including two-dimensional adiabatically perturbed systems.
• Using the classification of Quaternionic vector bundles over symmetric spaces to derive the complete set of possible topological invariants in low dimensions.
• Comparing the FKMM-invariant classification with recent results on adiabatically perturbed systems to validate universality and consistency.

Experimental results

Research questions

• RQ1 How can the Fu-Kane-Mele index be generalized beyond periodic electronic systems to other models with odd time-reversal symmetry?
• RQ2 What is the universal topological invariant that classifies systems with odd time-reversal symmetry in terms of Quaternionic vector bundles?
• RQ3 Can the FKMM-invariant serve as a complete topological invariant for class AII systems in low-dimensional base spaces like spheres and tori?
• RQ4 How does the FKMM-invariant relate to existing classifications of topological phases in two-dimensional adiabatically perturbed systems?
• RQ5 What is the role of the base space's topology in determining the possible values of the topological invariant in odd time-reversal symmetric systems?

Key findings

• The FKMM-invariant is a universal characteristic class for Quaternionic vector bundles, valid independently of the base space’s specific structure.
• The FKMM-invariant fully classifies topological phases in low-dimensional systems with odd time-reversal symmetry, such as spheres and tori, providing a complete invariant set.
• The invariant generalizes the Fu-Kane-Mele index beyond periodic lattices, extending its applicability to arbitrary physical models with odd time-reversal symmetry.
• The classification via the FKMM-invariant matches and unifies recent results on topological phases in two-dimensional adiabatically perturbed systems.
• The FKMM-invariant provides a robust, mathematically rigorous framework that discriminates among distinct topological phases in class AII systems without relying on specific microscopic details.
• The construction reveals that the topological nature of the Fu-Kane-Mele index arises naturally from the structure of Quaternionic vector bundles, offering a deeper geometric and algebraic explanation.