# On $\mathbb{Z}_2$-indices for ground states of fermionic chains

## TL;DR

This paper reviews how to assign and interpret $bZ_2$-indices to ground states of fermionic chains, revealing topological obstructions and phase distinctions related to spectral flow and gap closing.

## Contribution

It introduces a $bZ_2$-index framework for fermionic ground states, linking spectral flow to topological phases and providing a new phase label concept.

## Key findings

- $bZ_2$-indices classify ground states and detect topological differences.
- Spectral flow acts as an obstruction to connecting states with different indices.
- Phase labels indicate distinct phases separated by spectral gap closures.

## Abstract

For parity-conserving fermionic chains, we review how to associate $\mathbb{Z}_2$-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the $\mathbb{Z}_2$-valued spectral flow provides a topological obstruction for two systems to have the same $\mathbb{Z}_2$-index. A rudimentary definition of a $\mathbb{Z}_2$-phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.