Dynamic correlations of frustrated quantum spins from high-temperature expansion

Abstract

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.

Figures (7)

• Figure 1: Heisenberg $S=1/2$ AFM chain at $T=\infty$. (a) Continued fraction parameters $\delta_r$ for two momenta $\mathbf{k}$. The large dots denote the exact results for $r=0,1,...,6$ from Dyn-HTE, the small markers depict various linear extrapolation schemes $\delta_{r>6} = (r-6)a + b$ to which the DSF in (b), obtained from the infinite continued fraction Eq. \ref{['eq:linear_Approx']}, is largely insensitive as seen from the overlap of various linestyles.
• Figure 2: DSF for the Heisenberg $S=1/2$ AFM chain at $x=J/T \in \{0,2,4\}$, left to right column. Top row: Dyn-HTE results based on extrapolation of moments with $r\leq6$ ($r\leq3$ for $x>0$). Middle row: DMRG data reproduced from Ref. kish_high-temperature_2024. Bottom row: Lineshape for $S(\mathbf{k},\omega)$ at fixed $\mathbf{k}\in \{0.2 \pi, \pi\}$. The lattice spacing is set to unity. The values for $\Sigma \equiv \frac{J}{V_{BZ}} \! \int_{-\infty}^\infty \! \mathrm{d} \omega \! \int_{BZ} \! \mathrm{d}\mathbf{k} \:S(\mathbf{k},\omega)$ with $V_{BZ}$ the BZ-volume are also indicated and Dyn-HTE fulfills the sum-rule $\Sigma=\langle S_i^z S_i^z \rangle = S(S+1)/3=1/4$ within $<1\%$.
• Figure 3: DSF of triangular lattice $S=1/2$ Heisenberg AFM at momenta $M$ (top) and $K$ (bottom) from Dyn-HTE with $r_{max}=3$ and $f=0.55$. Inset: $\omega/T$-scaling collapse for $\omega \leq J$ and $0.43 \leq T/J \leq 0.95$ with scaling exponent $\alpha=1.10(2)$.
• Figure 4: (a) DSF for the pyrochlore lattice $S=1$ Heisenberg AFM for $\mathbf{k} = (2,2,l)$, $J=2.4\meV$ at $x=J/T=4$ from Dyn-HTE (via u-Padé with $f=0.6$). (b) Experimental INS data for $\mathrm{NaCaNi}_2\mathrm{F}_7$ at $x\simeq 15$, from Ref. plumb_continuum_2019.
• Figure 5: Heisenberg $S=1/2$ AFM chain: First four normalized moments $x \, m_{\mathbf{k},2r}(x)/m_{\mathbf{k},2r}(0)$, $r=0,1,2,3$ from Dyn-HTE (top) and corresponding continued fraction parameters $\delta_{\mathbf{k},r}$ (bottom) at momenta $\mathbf{k}=0.2 \pi$ (left) and $\mathbf{k}=\pi$ (right). The solid lines represent the bare series, while the dotted and dashed lines are different Padé approximants in the variable $u= \tanh(fx)$. We chose $f=0.48$ such that the different Padé approximants agree with each other. The bottom row shows the corresponding $\delta_r$ of the continued fraction expansion at various temperatures (dots) and their linear extrapolation for $r\geq 4$ (crosses).