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

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.

v2: 61 pages, many revisions. To appear in Reviews in Mathematical Physics