Still life in a classic Blume-Capel model: pseudo-transitions in a spin-1 diamond chain

What the math gives to ML

The paper identifies a concrete pseudo-transition mechanism: a nearly degenerate low-energy state competes with a macroscopically degenerate excited manifold, so entropy can cause an abrupt but continuous crossover and exceptionally sharp yet finite response peaks. The characteristic scale is T* = ΔE / ln(g1/g0), where ΔE is the energy gap and g0,g1 are competing degeneracies. A transferable neural-network construction is to treat local parameter basins as competing states, inject calibrated weight noise, detect the predicted response peak, and use it to trigger optimizer or regularization changes.