  The recently introduced topological quantum chemistry (TQC) framework has
provided a description of universal topological properties of all possible band
insulators in all space groups based on crystalline unitary symmetries and time
reversal. While this formalism filled the gap between the mathematical
classification and the practical diagnosis of topological materials, an obvious
limitation is that it only applies to weakly interacting systems-which can be
described within band theory. It is an open question to which extent this
formalism can be generalized to correlated systems that can exhibit symmetry
protected topological phases which are not adiabatically connected to any band
insulator. In this work we address the many facettes of this question by
considering the specific example of a Hubbard diamond chain. This model
features a Mott insulator, a trivial insulating phase and an obstructed atomic
limit phase. Here we discuss the nature of the Mott insulator and determine the
phase diagram and topology of the interacting model with infinite density
matrix renormalization group calculations, variational Monte Carlo simulations
and with many-body topological invariants. We then proceed by considering a
generalization of the TQC formalism to Green's functions combined with the
concept of topological Hamiltonian to identify the topological nature of the
phases, using cluster perturbation theory to calculate the Green's functions.
The results are benchmarked with the above determined phase diagram and we
discuss the applicability and limitations of the approach and its possible
extensions.
