The Blume-Capel Model as a $\phi^6$ Field Theory: A Monte Carlo and Flow Model Study

The paper presents Monte Carlo simulations of a continuous-spin Blume‑Capel model and maps its phase diagram in the crystal‑field–temperature plane, varying the spin softness. It demonstrates that normalizing flows—generative models enabling direct, autocorrelation‑free sampling—serve as effective samplers throughout the diagram, including first‑order transitions and near the tricritical point. Finite‑size scaling of flow-generated samples accurately identifies first‑, second‑order transitions and the tricritical point, matching the precision of conventional Monte Carlo methods.

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