The one-dimensional Blume-Capel model has no phase transition. In one dimension, thermal fluctuations destroy long-range order at any nonzero temperature. There is no critical point, no symmetry breaking, no divergent correlation length.
But there are pseudo-critical crossovers — temperatures where the system changes character rapidly without truly transitioning. These crossovers leave signatures in the thermodynamic geometry.
Bhattacharjee, Bora, and Phukon (arXiv:2603.17483) apply Tsallis nonextensive statistics to the one-dimensional Blume-Capel model and compute the scalar curvature of the resulting thermodynamic metric. The curvature — which measures the information-geometric distance between neighboring thermodynamic states — develops peaks at pseudo-critical temperatures.
The nonextensivity parameter q controls what happens. At q = 1 (standard Boltzmann-Gibbs statistics), the peaks are modest. For q > 1, the peaks shift and the correlations extend beyond the crossover region — the nonextensivity stretches the pseudo-critical behavior. For q < 1, the peaks weaken or vanish entirely — the correlations are suppressed below what standard statistics would predict.
The geometry responds to a parameter that has no counterpart in the microscopic Hamiltonian. The spin interactions are the same; only the statistical framework changes. The peaks in curvature are not properties of the physical system alone but of the system viewed through a particular statistical lens. Change the lens, and the geometry reshapes.
This is information geometry doing what it does: treating the statistical manifold as a geometric object whose curvature encodes interaction structure. The curvature peaks mark where the system's states are maximally distinguishable — where small changes in temperature produce the largest changes in the probability distribution. The nonextensivity parameter tunes this distinguishability, amplifying or suppressing the geometric signal of incipient order.