Dynamic correlations of frustrated quantum spins from high-temperature expansion

Technical Summary: Dynamic Correlations of Frustrated Quantum Spins from High-Temperature Expansion

Problem Statement
The dynamic structure factor (DSF), $S(k, \omega)$, is a primary experimental observable for probing collective spin dynamics, quasiparticles, and entanglement in quantum spin systems via inelastic neutron scattering (INS) and Raman spectroscopy. However, theoretical calculation of the DSF for generic frustrated quantum spin models at intermediate temperatures is notoriously difficult. Existing methods face significant limitations: exact diagonalization and finite-temperature Lanczos are restricted to small system sizes; quantum Monte Carlo is hindered by the sign problem in frustrated systems; and tensor-network methods (e.g., DMRG) are currently limited to $T=0$, small spin lengths, and suffer from finite-size and entanglement effects in higher dimensions. Furthermore, diagrammatic approaches providing imaginary-frequency correlations often fail upon analytic continuation to real frequencies due to instability.

Methodology: Dynamic High-Temperature Expansion (Dyn-HTE)
To address these challenges, the authors introduce a dynamic extension of the high-temperature expansion (HTE). The core of the method involves computing the frequency moments of the real-frequency dynamic susceptibility and reconstructing the DSF via a continued fraction expansion.

1. Moment Expansion: The DSF is reconstructed from the frequency moments $m_{k,2r} = \int_{-\infty}^{\infty} dw \, w^{2r} R_k(w)$, where $R_k(w)$ is the relaxation function. These moments are related to the coefficients $\delta_{k,r}$ of a continued fraction representation of the susceptibility.
2. Mapping to Matsubara Correlators: A key conceptual result is that the HTE of the real-frequency moments $m_{k,2r}$ is encoded in the HTE of the imaginary-time (Matsubara) spin correlator $G_{ii'}(i\nu_m)$. This allows the authors to leverage the well-established, efficient graph-based expansion techniques of conventional HTE (which are oblivious to frustration, dimensionality, and spin length) to calculate dynamic properties.
3. Computational Implementation: The authors utilize an open-source numerical implementation that pre-evaluates HTE coefficients on generic lattice "graphs" (snippets) and embeds them into arbitrary lattices. The method computes exact HTE coefficients up to order $n_{max}=12$ for spin lengths $S \in \{1/2, 1\}$.
4. Reconstruction and Resummation:
   • Continued Fraction: The first $r_{max}$ moments determine the first $r_{max}$ continued fraction parameters $\delta_{k,r}$.
   • Extrapolation: Since the expansion order is finite, the continued fraction is terminated using a linear extrapolation scheme for higher-order parameters ($\delta_{r>r_{max}}$), which is shown to be largely insensitive to the specific linear scheme chosen.
   • Resummation: For finite temperatures ($T < \infty$), the raw HTE series for moments diverge at intermediate $J/T$. The authors employ Padé approximants on a transformed variable $u = \tanh(fx)$ (where $x=J/T$) to extend the validity of the expansion to lower temperatures.

Key Results and Benchmarks
The paper validates the Dyn-HTE method through several benchmarks and applications:

• 1D Heisenberg $S=1/2$ Chain: At infinite temperature ($T=\infty$), the method reproduces exact results for moments up to $r=6$. The reconstructed DSF shows excellent agreement with DMRG data across the entire Brillouin zone. At finite temperatures ($x=2, 4$), the Dyn-HTE results match DMRG data with only minor deviations in line shapes, demonstrating the method's accuracy beyond the convergence range of bare HTE.
• Triangular Lattice ($S=1/2$): The method sheds new light on the anomalous intermediate-temperature regime ($0.25 \lesssim T/J \lesssim 1$).
  • The DSF does not show significant softening at the roton-like excitation minimum ($M$-point) across this regime, suggesting that the anomalous static properties are not driven by simple thermal excitations of these modes.
  • At the ordering wavevector ($K$-point), the DSF exhibits a temperature-frequency scaling relation $JS(k, \omega)(T/J)^\alpha = \Phi(\omega/T)$ with an exponent $\alpha \approx 1.10(2)$. This supports a scenario involving a critical fan of a quantum phase transition (potentially to a Dirac quantum spin liquid), though the scaling range is narrow.
• Pyrochlore Lattice ($S=1$): The authors apply Dyn-HTE to the material NaCaNi$_2$F$_7$. The calculated DSF along the $[22l]$ momentum direction shows fair agreement with recent experimental INS data. Notably, the method captures the "V" shape and the height of the dome-shape features, which were missed by previous semiclassical approximations. A slight discrepancy in peak position is attributed to potential beyond-Heisenberg perturbations in the material and a temperature mismatch between the simulation ($x=4$) and experiment ($x \approx 15$).
• Kagome and Square Lattices: The paper provides preliminary DSF results for the $S=1/2$ Kagome and square lattice models, showing the emergence of paramagnons and broad features consistent with the lack of long-range order in the Kagome case.

Significance and Claims
The paper claims that Dyn-HTE provides a versatile, unbiased, and accurate numerical tool for calculating dynamic correlations in frustrated quantum magnets at intermediate temperatures ($T \gtrsim J/4$).

• Methodological Gap: It bridges the gap between static HTE (which is robust but limited to equal-time correlations) and dynamic observables, avoiding the instability of analytic continuation from imaginary frequencies.
• Experimental Relevance: The method directly targets experimental observables (DSF) and is applicable to arbitrary lattice geometries, making it a direct tool for interpreting INS and other spectroscopic data.
• Scalability: Unlike tensor networks, the method is not limited by entanglement entropy or system size (working in the thermodynamic limit) and is not hindered by the sign problem.
• Future Utility: The authors state that the method can inform experiments using local probes (muon spin relaxation, tunneling spectroscopy) and can be extended to compute finite-$T$ spin diffusion, quantum Fisher information, and higher-order or chiral correlators.

The work emphasizes that while the method is currently limited to $S=1/2$ and $S=1$ with single couplings, its open-source implementation and algorithmic framework offer a robust pathway for studying dynamic properties in complex, frustrated quantum systems where other methods fail.