Dynamic correlations of frustrated quantum spins from high-temperature expansion

For quantum spin systems in equilibrium, the dynamic structure factor (DSF) is among the most feature-packed experimental observables. However, from a theory perspective it is often hard to simulate in an unbiased and accurate way, especially for frustrated and high-dimensional models at intermediate temperature. To address this challenge, we compute the DSF from a dynamic extension of the high-temperature expansion to frequency moments. We focus on nearest-neighbor Heisenberg models with spin-lengths S=1/2 and 1. We provide comprehensive benchmarks and consider a variety of frustrated two- and three-dimensional antiferromagnets as applications. In particular we shed new light on the anomalous intermediate temperature regime of the S=1/2 triangular lattice model and reproduce the DSF measured recently for the S=1 pyrochlore material NaCaNi2F7. An open-source numerical implementation for arbitrary lattice geometries is also provided.

💡 Research Summary

The authors introduce a dynamic extension of the high‑temperature expansion (HTE), termed Dyn‑HTE, to compute the dynamical structure factor (DSF) of quantum spin systems at finite temperature. Traditional HTE is limited to equal‑time correlations and thermodynamic quantities, while the DSF—being the Fourier transform of the two‑point spin correlator in space and time—requires real‑frequency information that is notoriously difficult to obtain for frustrated or high‑dimensional models. The key conceptual step is to relate the frequency moments of the real‑frequency susceptibility to Matsubara‑time spin correlators. By expanding these Matsubara correlators in powers of x = J/T up to order n_max = 12, the authors obtain exact polynomial coefficients p^{(2r)}{ii’}(x) for each lattice bond. These coefficients encode the high‑temperature series for the even‑order moments m{k,2r}(x).

The moments are then used to construct a continued‑fraction representation of the relaxation function R_k(ω), whose real part yields the DSF via the fluctuation‑dissipation theorem. The continued‑fraction parameters δ_{k,r} are expressed as simple ratios of successive moments (e.g., δ_0 = m_0, δ_1 = m_2/m_0, etc.). Because only a finite number of moments (r_max ≈ 3–6 depending on dimension and spin) can be computed, the authors terminate the fraction with a linear extrapolation of δ_{k,r} for r > r_max and a Hermite‑polynomial based termination function Γ_{a,b}(s). This procedure yields a smooth spectral function without artificial delta‑function spikes.

Implementation relies on the graph‑based machinery of conventional HTE: the lattice is decomposed into a set of connected subgraphs (“graphs”), each evaluated up to n_max edges. For each graph the authors recursively generate the perturbative series and analytically perform all τ‑integrals using a kernel‑trick, producing the p‑polynomials for arbitrary geometries. The open‑source code (available on GitHub) automates this workflow for Heisenberg models with spin S = 1/2 or 1 and a single exchange constant J, but the framework can be extended to more complex interactions.

Benchmarking is performed on the well‑studied S = 1/2 antiferromagnetic Heisenberg chain. At infinite temperature (x = 0) the exact δ_{k,r} values are reproduced, and at finite temperatures x = 2 and 4 the Dyn‑HTE DSF agrees quantitatively with density‑matrix renormalization‑group (DMRG) data across the Brillouin zone, satisfying the sum rule Σ = ⟨S_i^z S_i^z⟩ to better than 1 %.

The method is then applied to two‑dimensional frustrated lattices. For the triangular lattice S = 1/2 antiferromagnet, the authors explore the enigmatic intermediate‑temperature regime (0.25 ≲ T/J ≲ 1) where static observables deviate from renormalized‑classical spin‑wave predictions. The DSF at the M‑point (zone edge) shows a persistent peak around ω ≈ J, indicating that the “roton‑like excitation” identified at zero temperature does not simply soften with temperature. Moreover, at the ordering wave vector K the DSF exhibits a scaling collapse S(K,ω) · (T/J)^α = Φ(ω/T) with α ≈ 1, consistent with a quantum critical fan scenario associated with a putative zero‑temperature quantum phase transition (e.g., from 120° order to a Dirac quantum spin liquid).

Finally, the authors address the S = 1 pyrochlore antiferromagnet NaCaNi₂F₇, a material that experimentally realizes a near‑ideal nearest‑neighbor Heisenberg model with J ≈ 2 meV. Using Dyn‑HTE at x ≈ 15 (the experimental temperature), they compute the DSF along the