Hybrid quantum-classical method for simulating high-temperature dynamics of nuclear spins in solids

First-principles calculations of high-temperature spin dynamics in solids in the context of nuclear magnetic resonance (NMR) are a long-standing problem, whose conclusive solution can significantly advance the applications of NMR as a diagnostic tool for material properties. In this work, we propose a hybrid quantum-classical method for computing NMR free induction decay(FID) for spin-$1/2$ lattices. The method is based on the simulations of a finite cluster of spins $1/2$ coupled to an environment of interacting classical spins via a correlation-preserving scheme. Such simulations are shown to lead to accurate FID predictions for one-, two-, and three-dimensional lattices with a broad variety of interactions. The accuracy of these predictions can be efficiently estimated by varying the size of quantum clusters used in the simulations.

Figure 1

Sketch of a hybrid lattice: a cluster of spins 1/2 surrounded by an environment of classical spins. The quantum cluster is described by a wave function $|\psi \rangle$. Classical spins are represented by three-dimensional vectors.

Figure 2

Correlation functions $C_{\alpha }\left(t\right)$ for one-dimensional periodic chains with nearest-neighbor interactions. The interaction constants are indicated above each plot. The left column compares the results of hybrid simulations with the reference plots obtained by direct quantum calculations. The right column does the same for purely classical simulations. For both hybrid and classical simulations, the full lattice size is 92. The sizes of quantum clusters in hybrid simulations and in reference quantum calculations are indicated in the legends.

Figure 3

Correlation functions $C_{\alpha }\left(t\right)$ for two-dimensional periodic lattices with nearest-neighbor interaction. The notations in (a)–(${\mathrm{b}}^{\prime }$) are the same as in Fig. 2. For both hybrid and classical simulations, the full lattice size is $9\times 9$. The shapes of quantum clusters for hybrid simulations are shown in (c).

Figure 4

FIDs in ${\mathrm{CaF}}_2$ for external magnetic field $B_0$ along the following crystal directions: (a) [001], (b) [011], and (c) [111]. Hybrid and classical simulations are compared with the experimental results of Ref. [20]. For both hybrid and classical simulations, the full lattice size is $9\times 9\times 9$. The quantum cluster in hybrid simulations was a chain extending along the [001] crystal direction in (a), a chain passing through the entire lattice and oriented along the [100] crystal direction in (b), and a chain along the [111] crystal direction in (c). The insets contain semilogarithmic plots of the respective FIDs.