Symplectic Spin-Lattice Dynamics with Machine-Learning Potentials

TL;DR

TSPIN is proposed, a unified Nos\'e-Hoover chain (NHC) framework that augments the spin-lattice Lagrangian with spin kinetic terms and thermostat/barostat variables, yielding a symplectic Hamiltonian formulation for NVE, NVT, and NPT ensembles, ensuring energy conservation and reducing computational cost.

Abstract

Atomic-scale modeling of magnetic materials requires precise treatment of coupled spin-lattice degrees of freedom (DOFs). Traditional spin-lattice dynamics (SLD), employing Newtonian equation for lattice evolution and the Landau-Lifshitz-Gilbert (LLG) equation for spin, encounters severe limitations with machine-learning potentials (MLPs), including poor energy conservation and excessive computational costs due to non-symplectic integration. In this work, we propose TSPIN, a unified Nosé-Hoover chain (NHC) framework that augments the spin-lattice Lagrangian with spin kinetic terms and thermostat/barostat variables, yielding a symplectic Hamiltonian formulation for NVE, NVT, and NPT ensembles. The method integrates spin and lattice dynamics simultaneously, ensuring energy conservation and reducing computational cost. Benchmarks on harmonic models confirm its accuracy, while Fe simulations with the DeepSPIN potential demonstrate superior stability, efficiency, and the ability to capture magnetic phase transition. TSPIN thus provides a general and efficient framework for large-scale simulations of spin-lattice phenomena and multiple-DOF systems.

Figures (4)

• Figure 1: Benchmark of the TSPIN method for a harmonic lattice-spin system at $T = 100$, $200$, and $500\,\text{K}$. Panels (a) and (b) show the distributions of lattice coordinates ($R$) and momenta ($P_R$), while panels (c) and (d) display the corresponding spin distributions. Data points correspond to simulation results, and solid lines represent analytical solutions.
• Figure 2: Energy conservation comparison in FCC Fe simulations using TSPIN and LLG methods at different time steps: (a) $\Delta t = 0.1$ fs, (b) $\Delta t = 0.5$ fs, and (c) $\Delta t = 1.0$ fs. Red curve correspond to TSPIN simulations, while the blue curve corresponds to LLG result.
• Figure 3: Comparison of computational cost between classical MD and SLD simulations using TSPIN, MD+MCMC, and LLG methods as a function of system size. (a) Computational time (s) with linear and quadratic fittings indicated by dashed and dotted lines, respectively. (b) Zoom-in highlighting linear scaling behavior of TSPIN and MD.
• Figure 4: Spin ordering transition in FCC Fe across low and high temperature regimes. Panels (a) and (b) show representative spin configurations at 10K and 600K, with spheres and arrows representing atoms and spins, respectively. Panels (c) and (d) present the corresponding spin orientation distributions in $(\theta, \phi)$ space, where $\theta$ is the polar angle measured from the $z$ axis and $\phi$ is the azimuthal angle projected onto the $xy$ plane.