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We introduce a spin-symmetry-broken extension of the connected determinantalgorithm [Phys. Rev. Lett. 119, 045701 (2017)]. The resulting systematicperturbative expansions around an antiferromagnetic state allow for numericallyexact calculations directly inside a magnetically ordered phase. We show newprecise results for the magnetic phase diagram and thermodynamics of thethree-dimensional cubic Hubbard model at half-filling. With detailedcomputations of the order parameter in the low to intermediate-coupling regime,we establish the N{\'e}el phase boundary. The critical behavior in its vicinityis shown to be compatible with the $O(3)$ Heisenberg universality class. Bydetermining the evolution of the entropy with decreasing temperature throughthe phase transition we identify the different physical regimes at $U/t\!=\!4$.We provide quantitative results for several thermodynamic quantities deepinside the antiferromagnetic dome up to large interaction strengths andinvestigate the crossover between the Slater and Heisenberg regimes.

Symmetry-broken perturbation theory to large orders in antiferromagnetic phases