Phase transitions in the two-dimensional soft-spin Blume–Capel model
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How does the phase diagram of the two-dimensional soft-spin φ⁶ Blume–Capel model change as spin softness varies, including continuous, first-order and tricritical transitions?
Why it matters and what counts as progress
Why it matters: One tunable model lets sampling methods be compared across different phase-transition bottlenecks and toward the discrete spin-one limit.
What counts as progress: Phase boundaries and finite-size scaling with controlled errors and comparison to Monte Carlo on the same model. Flow-sampling evidence is numerical rather than an exact infinite-volume phase theorem.
How the problem developed
- 1966 · Model [1]
Martin Blume and Hans Willem Capel Introduced the spin-one Ising model family that later became known as the Blume–Capel model; this work concerns discrete spin variables, not the later soft-spin variant.
- 14 Oct 2026 · Computational study [2]
Ejaaz Merali, Ryan Larson, Ehsan Khatami and Richard T. Scalettar Released Monte Carlo and normalizing-flow phase diagrams for a two-dimensional continuous-spin φ⁶ variant, using finite-size scaling to locate continuous, first-order and tricritical transitions.
AI contribution
The Blume-Capel Model as a φ⁶ Field Theory: A Monte Carlo and Flow Model Study
Merali, Larson, Khatami and Scalettar report phase diagrams for a two-dimensional soft-spin φ⁶ Blume–Capel model as spin softness varies. Normalizing-flow samples with finite-size scaling locate continuous, first-order and tricritical transitions with accuracy comparable to canonical and extended-ensemble Monte Carlo on the same model.
Sources
Website timeline
Publication dates and editorial changes.
Normalizing-flow study tracks tricriticality in a soft-spin Blume–Capel model