Simulating the interplay of dipolar and quadrupolar interactions in NMR by spin dynamic mean-field theory

TL;DR

This work introduces spinDMFT for simulating NMR spin dynamics in dense, high-temperature spin systems with both dipolar and local quadrupolar interactions. By replacing the environmental lattice with a Gaussian time-dependent mean-field and solving a time-evolving single-site problem, the method captures quantum local effects exactly while remaining computationally efficient. The authors demonstrate quantitative agreement with AlN experimental data across orientations, and show that classical dynamics miss essential features when quadrupolar couplings are significant. The framework offers a predictive tool for dipolar broadening in quadrupolar spectra and can be extended to more complex setups and time-dependent NMR experiments, providing practical impact for interpreting and predicting NMR signals in solid-state systems.

Abstract

The simulation of nuclear magnetic resonance (NMR) experiments is a notoriously difficult task, if many spins participate in the dynamics. The recently established dynamic mean-field theory for high-temperature spin systems (spinDMFT) represents an efficient yet accurate method to deal with this scenario. SpinDMFT reduces a complex lattice system to a time-dependent single-site problem, which can be solved numerically with small computational effort. Since the approach retains local quantum degrees of freedom, a quadrupolar term can be exactly incorporated. This allows us to study the interplay of dipolar and quadrupolar interactions for any parameter range, i.e., without the need for a perturbative treatment. We obtain a remarkable agreement with experimental data for an aluminium nitride monocrystal, which strongly suggests the use of spinDMFT as a prediction tool. Furthermore, we draw a comparison between a quantum-mechanical and a classical version of spinDMFT showing that local quantum effects are of great importance for the studied type of system.

Figures (11)

• Figure 1: Results for the normalized spin autocorrelation $G^{\alpha\alpha}$ from spinDMFT for different spin lengths and quadrupolar interaction strengths in the time domain. The top row shows the transverse and the bottom row the longitudinal results. The spin length is increased from left to right. Different quadrupolar strengths are indicated by different colors according to the provided legend. Numerical errors are of the order $1%$ or smaller of the signal amplitude at $t=0$.
• Figure 2: Results of the longitudinal spin autocorrelations in the time domain in logarithmic representation. The dashed lines display exponential fits, which work exceptionally well. Numerical errors are of the order $1%$ or smaller of the signal amplitude at $t=0$. The wiggles of the data, which become visible for small $G^{zz}$, result from the statistical error of the Monte-Carlo simulation.
• Figure 3: Plot of the extracted decay times of the longitudinal autocorrelations $G^{zz}$ in dependence of $\widetilde{\Omega}$ for different spin lengths. The corresponding exponential fits are shown in \ref{['fig:longexpfit']}. The error bars result from the finite time discretization, which is the dominant source of numerical error.
• Figure 4: Fourier transform $f^{xx}(\omega)\coloneqq \int_{-\infty}^{\infty} \mathrm{e}^{-\texttt{i} \omega t} g^{xx}(t) \mathrm{d}t$ of the transverse spin autocorrelation $g^{xx}(t)$ for different quadrupolar interaction strengths and spin lengths. The spin length is increased from left to right and the quadrupolar interaction from top to bottom. The orange dashed line corresponds to the Gaussian fit described in \ref{['eqn:fitfunction']}. Small deviations are seen at some of the peak maxima. These become smaller when allowing for an individual amplitude $A_i$ for each peak in the fit function. However, we prefer the shown fits because they require only a single parameter, namely, the standard deviation $\sigma$. The numerical errors of the simulation data are smaller than the width of the lines.
• Figure 5: Possible quadrupolar transitions with $\Delta m = \pm 1$ for different spin lengths. The quadrupolar energy is given by $E_{\text{Q}}=3\Omega m^2$.