Information-Geometric Signatures from Nonextensivity in the $1$-D Blume-Capel Model

TL;DR

This paper explores how nonextensive Tsallis statistics influence the thermodynamic geometry and correlation structures of the 1D Blume-Capel model, revealing that the Tsallis parameter $q$ alters pseudo-critical behavior and entropy surface geometry.

Contribution

It introduces an information-geometric analysis of the Blume-Capel model within the Tsallis framework, highlighting how nonextensivity modifies correlation measures and pseudo-critical features.

Findings

Finite peaks in scalar curvature indicate pseudo-critical crossovers.

Deviations from $q=1$ deform the curvature profile and correlation persistence.

Tsallis parameter $q$ geometrically reshapes the entropy surface.

Abstract

We study the thermodynamic geometry of the one-dimensional Blume--Capel model within the Tsallis nonextensive framework to understand how generalized statistics modify correlation structure and pseudo-critical behaviour. Using the transfer matrix method, we construct the Tsallis entropy based thermodynamic metric as its negative Hessian on the parameter space $(\beta, J)$, with the crystal-field anisotropy $D$ as a control parameter, and compute the associated scalar curvature $R(T)$ as a measure of correlations. Although no true phase transition occurs in one dimension, $R(T)$ exhibits finite peaks signaling pseudo-critical crossovers. We analyze both $D < J$ and $D > J$ regimes and show that deviations from the Boltzmann--Gibbs limit ($q=1$) systematically deform the curvature profile: for $q>1$ the peak shifts and correlations persist beyond the crossover, whereas for $q<1$ the peak is weakened or suppressed. Our results demonstrate that the Tsallis parameter $q$ geometrically reshapes the entropy surface, providing a clear information-geometric interpretation of nonextensive effects in spin-1 systems.