The high-temperature series expansion for quantum spin models is a well-established tool to compute thermodynamic quantities and equal-time spin correlations, in particular for frustrated interactions. We extend the scope of this expansion to the dynamic Matsubara spin-spin correlator and develop an algorithm that yields exact expansion coefficients in the form of rational numbers. We focus on Heisenberg models with a single coupling constant $J$ and spin lengths $S\in \{1/2,1\}$. The expansion coefficients up to 12th order in $J/T$ are precomputed on all possible $\sim {10}^6$ graphs embeddable in arbitrary lattices and are provided in a repository. This enables calculation of static momentum-resolved susceptibilities for arbitrary site pairs or wave vectors. We test our results for the antiferromagnetic $S=1/2$ chain and triangular lattice model. An important application that we discuss in a companion letter is the calculation of real-frequency dynamic structure factors. This is achieved by identifying the high-frequency expansion coefficients of the Matsubara correlator with frequency moments of the spectral function.

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• Frustrated magnetism
• Quantum field theory
• Graph theory
• Series expansions & exact enumeration