Abstract
A dynamic mean-field theory for spin ensembles (spinDMFT) at infinite temperatures on arbitrary lattices is established. The approach is introduced for an isotropic Heisenberg model with $S=\frac{1}{2}$ and external field. For large coordination numbers, it is shown that the effect of the environment of each spin is captured by a classical time-dependent random mean field which is normally distributed. Expectation values are calculated by averaging over these mean fields, i.e., by a path integral over the normal distributions. A self-consistency condition is derived by linking the moments defining the normal distributions to spin autocorrelations. In this framework, we explicitly show how the rotating-wave approximation becomes a valid description for increasing magnetic field. We also demonstrate that the approach can easily be extended. Exemplarily, we employ it to reach a quantitative understanding of a dense ensemble of spins with dipolar interaction which are distributed randomly on a plane including static Gaussian noise as well.
Highlights
Nuclear magnetic resonance (NMR) has been an extremely important field for a long time
While we cannot provide a comprehensive review over all techniques applicable to spin systems, we present a brief overview of the most commonly used techniques
Figures (C = 10.0, B = 10.0) and (C = 10.0, B = 50.0) confirm this conclusion displaying better and better agreement between the results of both approaches. We argue that this behavior is physically highly plausible because the rotating-wave approximation (RWA) is justified if the energy scale of the magnetic field is larger than the energy scales of any other interaction in the system, including the static noise
Summary
Nuclear magnetic resonance (NMR) has been an extremely important field for a long time. One useful observation is that the dynamics of spins and in particular the effect of a larger number of spins can be captured fairly well by their classical equations of motion [36,37,38,39,40] This can be understood as an application of the ideas of the truncated Wigner approximation [41,42] whose foundations date back to the idea of Wigner that a part of the quantumness is captured by averaging over distributions of initial conditions [43]. A dynamic mean-field approach has been used for ordered magnetic phases [49] for which, the couplings between the spins have to be scaled differently The Appendixes provide technical details of the derivation and the numerical implementation of spinDMFT including an analysis of the achievable accuracy in the numerical simulations
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