On Z₂-indices for ground states of fermionic chains

For parity-conserving fermionic chains, we review how to associate Z2-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 Z2-valued spectral flow provides a topological obstruction for two systems to have the same Z2-index. A rudimentary definition of a Z2-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.