  We introduce a spin-symmetry-broken extension of the connected determinant
algorithm [Phys. Rev. Lett. 119, 045701 (2017)]. The resulting systematic
perturbative expansions around an antiferromagnetic state allow for numerically
exact calculations directly inside a magnetically ordered phase. We show new
precise results for the magnetic phase diagram and thermodynamics of the
three-dimensional cubic Hubbard model at half-filling. With detailed
computations of the order parameter in the low to intermediate-coupling regime,
we establish the N{\'e}el phase boundary. The critical behavior in its vicinity
is shown to be compatible with the $O(3)$ Heisenberg universality class. By
determining the evolution of the entropy with decreasing temperature through
the phase transition we identify the different physical regimes at $U/t\!=\!4$.
We provide quantitative results for several thermodynamic quantities deep
inside the antiferromagnetic dome up to large interaction strengths and
investigate the crossover between the Slater and Heisenberg regimes.
