We generalize the formalism of the dynamical vertex approximation ($\mathrm{D}\mathrm{\Gamma }\mathrm{A}$)—a diagrammatic extension of the dynamical mean-field theory (DMFT)—to treat magnetically ordered phases. To this aim, we start by concisely illustrating the many-electron formalism for performing ladder resummations of Feynman diagrams in systems with broken SU(2) symmetry associated to ferromagnetic (FM) or antiferromagnetic (AF) order. We then analyze the algorithmic simplifications introduced by taking the local approximation of the two-particle irreducible vertex functions in the Bethe-Salpeter equations, which defines the ladder implementation of $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ for magnetic systems. The relation of this assumption with the DMFT limit of large coordination-number/high dimensions is explicitly discussed. As a last step, we derive the expression for the ladder $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ self-energy in the FM- and AF-ordered phases of the Hubbard model. The physics emerging in the AF-ordered case is explicitly illustrated by means of approximated calculations based on a static mean-field input for $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ equations. The results obtained capture fundamental aspects of both metallic and insulating ground states of two-dimensional antiferromagnets, providing a reliable compass for future, more extensive applications of our approach. Possible routes to further develop diagrammatic-based treatments of magnetic phases in correlated electron systems are briefly outlined in the Conclusions.

• Received 21 December 2020
• Revised 13 July 2021
• Accepted 15 July 2021

DOI:https://doi.org/10.1103/PhysRevB.104.085120