Emulation of quantum correlations by classical dynamics in a spin-1/2 Heisenberg chain

TL;DR

This work shows that quantum correlations in the gapless spin-1/2 Heisenberg chain can be emulated by classical Landau-Lifshitz dynamics when equipped with temperature-dependent corrections: a spin-length renormalization $|{oldsymbol{ m abla}oldsymbol{ m \Omega}}_i| = \kappa(T)S$ and an exchange renormalization $J \to J z(T)$. The resulting quantum-corrected LLD (QLLD) reproduces the quantum DSSF $\mathcal{S}_Q(q,\omega,T)$ with QMC benchmarks for $k_B T/J_a \gtrsim 1$, captures the low-temperature lower edge of the continuum via $z(T)$, and yields correct finite-field transverse behavior, though it cannot generate the spinon continuum or incommensurate longitudinal coherence. Entanglement witnesses derived from the DSSF (two-tangle $\tau_2$ and normalized QFI) provide spectral diagnostics for the quantum-to-classical crossover, but can yield above-threshold values in purely classical simulations, highlighting their dependence on spectral features rather than true entanglement. Overall, quantum-corrected classical dynamics offers a scalable, predictive framework for interpreting scattering experiments and exploring quantum correlations in strongly correlated spin systems, with clear regimes of validity and practical diagnostics for breakdown.

Abstract

We simulate the dynamical spin structure factor (DSSF) S(q,w) of the spin-1/2 Heisenberg antiferromagnetic chain using classical simulations. By employing Landau-Lifshitz Dynamics, we emulate quantum correlations through temperature-dependent corrections, including rescaling of magnetic dipoles and renormalization of exchange interactions. Our results demonstrate that the quantum-equivalent DSSF closely matches Quantum Monte-Carlo calculations for kBT/J ~ 1, extending the applicability of classical dynamics to the challenging case of gapless excitations. At higher temperatures, our simulations comply with general predictions for uncorrelated paramagnetic fluctuations in the infinite temperature limit. Entanglement witnesses derived from the quantum-equivalent DSSF act as sensitive diagnostics for the quantum-to-classical crossover. Their reliability stems from their dependence on spectral features alone, enabling classical dynamics to emulate quantum thresholds without genuine entanglement. This framework also reproduces transverse spin correlations in finite magnetic fields, in agreement with quantum simulations. Together, our results establish quantum-corrected classical dynamics as a scalable and predictive tool for interpreting scattering experiments and exploring quantum correlations in strongly correlated spin systems.

Figures (15)

• Figure 1: (a) Dynamical spin structure factor (DSSF) of the quantum Heisenberg antiferromagnetic chain (QHAC) as a function of temperature. The colorplot encodes the diagonal part of the DSSF $\sum_{\alpha} \mathcal{S}^{\alpha\alpha}(q,\omega)$ as a function of momentum and energy. The left column is obtained using quantum-corrected Landau-Lifshitz Dynamics (QLLD, this work), while the center column corresponds to Quantum Monte Carlo results from Ref. Grossjohann2009. The right column is a plot of the difference between these two approaches. (b) A comparison between QLLD (blue lines) and QMC (gold lines) is made through the energy dependence of the DSSF at constant momenta, as indicated at the top of the column. The temperature is shown in each row. The dynamical structure factor is calculated in units of $J_a^{-1}={\rm meV}^{-1}$.
• Figure 2: (a) Temperature dependence of the exchange renormalization factor $z(T)$ (red line) and moment rescaling factor $\kappa(T)$ (blue line) extracted from QLLD simulations with $z(T)$ fitted from QMC results. The y-scale corresponding to each quantity is highlighted with the corresponding colors and arrows. (b) Temperature dependence of entanglement witnesses (two-tangle and Quantum Fisher Information) extracted from QMC (gold lines) and LLD simulations, the latter with (blue lines) and without (gray lines) $z(T)$-renormalization. The upper panel shows the temperature dependence of the two-tangle and the bottom panel shows the temperature dependence of the normalized quantum Fisher information. Grey dashed lines indicate the integer number of normalized QFI, which guarantees the lower bound of the $(n+1)$ multipartite entanglement. (c) High-temperature simulations of the DSSF using QLLD. For temperatures higher than $k_{\rm B}T/J_a\!=\!2$, we used $z(T)\!=\!1.14$ and $\kappa(T)\!=\!\sqrt{3}$ from Fig. \ref{['fig:2']}(a). White dashed lines indicate the boundaries of the universal continuum for high-temperature Heisenberg paramagnets $\sqrt{2\left(1-\cos{q}\right)}$deGennes1958.
• Figure 3: Field dependence of DSSF by QMC and QLLD at $k_{\rm B}T/J_a\!=\!0.25$. (a),(b) Longitudinal DSSF ${\cal S}^{zz}(q,\omega)$ and Transverse DSSF ${\cal S}^{xx}(q,\omega)$ of QLLD and QMC method, respectively. For the $B/J_a < 2$, black lines indicate the excitation boundary from the Muller ansatz at zero temperature. For the critical field $B/J_a \geq 2$, dispersion becomes $\omega(q) = 1-cos(q)$ for longitudinal DSSF and $\omega(q) = B/(2J_a)+cos(q)$ for trnasverse DSSF . The temperature is shown at the top of the column, and the magnetic field is given in each box. The red arrows in (a) indicate the expected incommensurate correlation from the QMC and Muller ansatz. The QMC results are from Grossjohann2009. (c),(d) Temperature-field dependence of spin-length rescaling factor $\kappa(T)$ and nQFI. Magenta cross markers indicate the temperature-field condition of QMC and QLLD results. Red colored area in (d) indicates the nQFI $\geq$ 1 and blue area shows nQFI < 1.
• Figure S1: Determination of $z(T)$ by direct comparison between QMC and LLD simulations. (a)-(e) $\chi^2$ of Eq. \ref{['eqsi:chi2']} for each simulation temperature, where the color-coded vertical lines indicate the global minimum of $\chi^2$ resulting from the minimization. (f) Temperature dependence of the resulting $z(T)$. The blue lines show a comparison without masking of the elastic energy. Gold lines indicate the comparison with elastic energy masking $(\left|\hbar\omega/J_a\right| < 0.1)$.
• Figure S2: Difference line plot (brown lines) between QMC (gold lines) and QLLD (blue lines) simulation at fixed temperature and momenta.