# Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory

## TL;DR

This paper explores two-dimensional topological insulators with time-reversal symmetry, introducing a new edge invariant and proving its equivalence to the bulk invariant using Fredholm theory, thereby advancing understanding of topological phases.

## Contribution

It introduces a novel edge topological invariant for 2D time-reversal-invariant insulators and establishes its equivalence to the bulk invariant through homotopy, using Fredholm theory.

## Key findings

- Established the bulk-edge correspondence for 2D topological insulators.
- Defined a new topological invariant for the edge states.
- Revisited and confirmed known bulk topological invariants.

## Abstract

We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy.