High-temperature series expansion of the dynamic Matsubara spin correlator

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=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 wavevectors. 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.

💡 Research Summary

The manuscript introduces a novel extension of the high‑temperature series expansion (HTE) to the dynamic Matsubara (imaginary‑frequency) spin‑spin correlator, a method the authors term “Dyn‑HTE”. Traditional HTE has been highly successful for thermodynamic quantities and equal‑time spin correlations, even in frustrated quantum magnets, but it has never been applied to dynamical observables. By focusing on Heisenberg models with a single exchange constant J and spin length S = ½ or 1, the authors develop an exact algorithm that produces the expansion coefficients of the Matsubara correlator as rational numbers up to 12th order in the dimensionless coupling x = J/T.

The theoretical framework starts by splitting the Hamiltonian into a non‑interacting part H₀ = 0 (no magnetic field) and an interacting part V = J∑V_{ij}. Using standard perturbation theory for the Matsubara correlator, the authors write the n‑th order coefficient c^{(n)}(iνₘ) as a (n + 2)-fold imaginary‑time integral over a product of n bond operators V_{b} and two spin operators Sᶻ_i(τ), Sᶻ_{i’}(τ′). The crucial technical advance is the use of the “Kernel trick”, a recently developed analytic method that evaluates these multi‑dimensional integrals in closed form, yielding rational numbers for every term.

To handle the combinatorial explosion of bond configurations, the authors adopt a graph‑based linked‑cluster approach. Each set of n bonds defines a graph g^{(n)}; the contribution of a given graph, c_g^{(n)}(iνₘ), is computed once and then embedded into any lattice L with a weight e(L,i,i′,g) that counts the number of embeddings of the graph consistent with the chosen site pair (i,i′). This re‑organization mirrors conventional HTE for static correlators but now includes the additional time‑ordering structure. The authors pre‑computed the contributions for all ≈10⁶ distinct graphs up to order 12 and stored them as exact rational numbers in a public repository, together with open‑source tools for lattice generation and graph embedding.

The final series takes the double‑expansion form

 T G_{ii′}(iνₘ) = ∑{r=0}^{⌊n_max/2⌋} p^{(2r)}{ii′}(x) ·