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The recently introduced topological quantum chemistry (TQC) framework hasprovided a description of universal topological properties of all possible bandinsulators in all space groups based on crystalline unitary symmetries and timereversal. While this formalism filled the gap between the mathematicalclassification and the practical diagnosis of topological materials, an obviouslimitation is that it only applies to weakly interacting systems-which can bedescribed within band theory. It is an open question to which extent thisformalism can be generalized to correlated systems that can exhibit symmetryprotected topological phases which are not adiabatically connected to any bandinsulator. In this work we address the many facettes of this question byconsidering the specific example of a Hubbard diamond chain. This modelfeatures a Mott insulator, a trivial insulating phase and an obstructed atomiclimit phase. Here we discuss the nature of the Mott insulator and determine thephase diagram and topology of the interacting model with infinite densitymatrix renormalization group calculations, variational Monte Carlo simulationsand with many-body topological invariants. We then proceed by considering ageneralization of the TQC formalism to Green's functions combined with theconcept of topological Hamiltonian to identify the topological nature of thephases, using cluster perturbation theory to calculate the Green's functions.The results are benchmarked with the above determined phase diagram and wediscuss the applicability and limitations of the approach and its possibleextensions.

Towards a Topological Quantum Chemistry description of correlated systems: the case of the Hubbard diamond chain