  We systematically generate the perturbative expansion for the two-particle
spin susceptibility in the Feynman diagrammatic formalism and apply this
expansion to a model system - the single-band Hubbard model on a square
lattice. We make use of algorithmic Matsubara integration (AMI) [A.
Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 99, 035120
(2019)] to analytically evaluate Matsubara frequency summations, allowing us to
symbolically impose analytic continuation to the real frequency axis. We
minimize our computational expense by applying graph invariant transformations
[Amir Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 101,
125109 (2020)]. We highlight extensions of the random-phase approximation and
T-matrix methods that, due to AMI, become tractable. We present results for
weak interaction strength where the direct perturbative expansion is
convergent, and verify our results on the Matsubara axis by comparison to other
numerical methods. By examining the spin susceptibility as a function of
real-frequency via an order-by-order expansion we can identify precisely what
role higher order corrections play on spin susceptibility and demonstrate the
utility and limitations of our approach.
