Dynamic Simulations of Strongly Coupled Spin Ensembles for Inferring Nature of Electronic Correlations from Nuclear Magnetic Resonance

TL;DR

The paper develops Spin Echo Sim, a GPU-accelerated mean-field NMR simulator to study how strongly correlated electronic phases imprint long-range hyperfine interactions on a nuclear spin ensemble. By deriving a rotating-frame mean-field Hamiltonian and Liouville-space dynamics, it reveals phase-locking and pulse-dependent Knight shifts that produce distinctive spin-echo and spectral features, whose character depends on interaction strength $\omega$, range $\xi$, and anisotropy $\eta$. The authors demonstrate robust numerical performance improvements over CPU approaches, validate accuracy against high-precision baselines, and provide convergence guidelines for $n$ and $dt$. These results offer a practical tool to infer electronic correlation length and anisotropy from NMR measurements in unconventional superconductors and related strongly correlated materials, with direct relevance to FFLO-like physics in systems such as CeCoIn$_5$.

Abstract

We develop an efficient package for the simulation of nuclear magnetic resonance spin echo experiments to study the effects of strong electronic spin correlations on the dynamics of the nuclear spin ensemble. A mean-field model is used to study correlated electronic phases through their hyperfine interaction with nuclear spins. We explore the dynamics of the interacting nuclear ensemble and discuss the key behaviors of the system. In particular, we classify the types of temporal asymmetry that the interaction induces in the system as well as a pulse-dependent shift in the spectral domain. Us- ing these results, we discuss how careful measurement of the pulse-dependent shiftcanbeusedtoextractinformationabouttheanisotropyoftheelectronic interaction and how these results represent a novel tool for the examination of exotic NMR signatures in strongly correlated materials. Finally, we re- view specific aspects of the simulation package developed for our exploration and give explicit examples where package can be used to infer range and anisotropy of electronic correlations. In particular, we discuss its structure, accuracy, and the technical merits of the various approximations used to model the nuclear spin ensemble.

Figures (15)

• Figure 1: $\langle I_z \rangle$ vs $\langle I_y \rangle$ for each nuclear spin at $t = 2/\Gamma$ for an echo with $\tau = 2.5/\Gamma$ and $T_2^* = 1/\Gamma$. Each marker indicates the $(y,z)$ components of a single spin. To generate this figure we sample the spins' frequency distribution uniformly, but each spin's effective contribution to the mean field is proportional to the distribution's probability at that frequency. Marker transparency and size follows the distribution's probability at each spin's $\Delta \nu$, with larger and more opaque markers indicating higher effective contribution.
• Figure 2: Echo phases labeled by color for a simulation with total planar interaction weight $\omega = 3.05\Gamma$. The $\pi$ pulse occurs at $\tau = 2.5\Gamma^{-1}$, causing an echo at $2\tau = 5\Gamma^{-1}$.
• Figure 3: (a) Spin echoes for a global interaction with varying planar weight $\omega$. The $\omega$ value for each simulation is given on the right of each curve. (b) Fourier transform (spectrum) of the FID region ($t < \tau$). (c) Fourier transform (spectrum) of a region centered on the echoes.
• Figure 4: (a) Spin echoes for a global interaction with $\omega = 3.15\Gamma$ but varying flip angles $\theta$. (b-d) Spectra dependence on flip angle $\theta$ for interaction $(\omega, \omega_z)$ equal to (b) $(0, 3.15\Gamma)$, (c) $(3.15\Gamma, 3.15\Gamma)$, and (d) $(3.15\Gamma, 0)$.
• Figure 5: (a) Spin echoes for a local interaction with $\xi \in [2,6,20]$. The planar interaction weight $\omega$ is given on the right of each set of curves, and $\omega_z = 0$. (b) Spectra for the same echoes.