A theorem regarding families of topologically non-trivial fermionic systems

TL;DR

This paper introduces a Hamiltonian model for fermions on a lattice, establishing a topological criterion based on a $ ext{Z}_2$ invariant, which unifies various topological phases like Chern insulators and superconductors.

Contribution

It proves a theorem linking the topological properties of fermionic systems to a $ ext{Z}_2$ invariant derived from the Pfaffian polynomial, providing a unified framework for different topological phases.

Findings

The topological invariant is given by the sign of the Pfaffian polynomial.

The theorem applies to models including Chern insulators and superconductors.

Explicit examples demonstrate the computation of topological invariants.

Abstract

We introduce a Hamiltonian for fermions on a lattice and prove a theorem regarding its topological properties. We identify the topological criterion as a $\mathbb{Z}_2-$ topological invariant $p(\textbf{k})$ (the Pfaffian polynomial). The topological invariant is not only the first Chern number, but also the sign of the Pfaffian polynomial coming from a notion of duality. Such Hamiltonian can describe non-trivial Chern insulators, single band superconductors or multiorbital superconductors. The topological features of these families are completely determined as a consequence of our theorem. Some specific model examples are explicitly worked out, with the computation of different possible topological invariants.