Scalable Canonical and Isothermal-Isobaric Sampling of Coupled Spin-Lattice Systems with Machine-Learning Potentials

Magnetic machine-learning potentials (MLPs) now reach near-first-principles accuracy on the spin-lattice potential energy surface, but the dynamics and sampling frameworks that convert this accuracy into quantitative finite-temperature thermodynamics have lagged behind. Landau-Lifshitz-Gilbert spin-lattice dynamics fixes the local moment magnitude and incurs $O(N)$ MLP evaluations per integration step, while hybrid molecular-dynamics/Monte-Carlo lacks rigorous isothermal-isobaric sampling and remains expensive. We introduce TSPIN, which promotes the spin to a canonical pair $(\mathbf{S}_i,\boldsymbol{\pi}_i)$ alongside the lattice $(\mathbf{R}_i,\mathbf{p}_i)$ within a Nos\'e-Hoover-chain / Martyna-Tobias-Klein construction, delivering rigorous canonical and isothermal-isobaric sampling, native access to longitudinal spin fluctuations through an unconstrained spin amplitude, and one MLP evaluation per integration step. Applied to itinerant Co and localized multiferroic BiFeO$_3$, TSPIN matches the MD/MC reference thermodynamics of Co at substantially lower cost and reproduces the Curie and N\'eel temperatures within $\sim 7\%$ and $\sim 2\%$ of experiment, respectively. The same unconstrained-amplitude dynamics resolves contrasting spin-amplitude behavior: pronounced spin-modulus softening in Co, but a nearly temperature-independent high-spin Fe$^{3+}$ moment in BiFeO$_3$. TSPIN thereby promotes magnetic MLPs from accurate energy models to predictive finite-temperature simulation engines.