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

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 $(β, J)$, 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.

Figures (5)

• Figure 1: (left) Scaling of the finite-size correction term $\Omega_N$ as a function of temperature for various system sizes $N$ and decay of $S_{corr}$ in $q<1$ (centre) and $q>1$ (right) regimes as a function of temperature for various system sizes $N$. As $N$ increases, the finite-size correction term $\Omega_N$ vanish rapidly and $S_{corr}$ approaches 0, isolating the bulk geometry.
• Figure 2: Temperature dependence of the thermodynamic scalar curvature $R(T)$ for the one-dimensional Blume Capel model using the Tsallis entropy for (D $<$ J)$D=0.935, J=1$ and $N=60$. (a)For the $q>1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) ($q=1$) lies around $T=0.24$ and is negative, but for $q>1$ the curvature peak though still negative shifts towards lower temperatures and the scalar curvature is non zero after the transition takes place for $q>1$ case whereas it was zero in the previous BG limit. (b)For the $q<1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) also lies around $T=0.24$ and is negative but for $q>1$ the curvature peak becomes positive and is seen to be suppressed for larger q values but they shift towards higher temperatures and here the scalar curvature is zero after the transition takes place in both the the BG limit and $q<1$ regime.
• Figure 3: Temperature dependence of the thermodynamic scalar curvature $R(T)$ for the one-dimensional Blume Capel model using the Tsallis entropy for (D $<$ J)$D=0.935, J=1$ and $N=600$. (a)For the $q>1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) ($q=1$) lies around $T=0.24$ and is negative, but for $q>1$ the curvature peak though still negative shifts towards lower temperatures and the scalar curvature is non zero after the transition takes place for $q>1$ case whereas it was zero in the previous BG limit. (b)For the $q<1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) also lies around $T=0.24$ and is negative but for $q>1$ the curvature peak is completely suppressed for the entire $q<1$ regime.
• Figure 4: Temperature dependence of the thermodynamic scalar curvature $R(T)$ for the one-dimensional Blume Capel model using the Tsallis entropy for (D $>$ J)$D=1.065, J=1$ and $N=60$. (a)For the $q>1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) ($q=1$) lies around $T=0.24$ and is negative, but for $q>1$ the curvature peak becomes positive and shifts towards lower temperatures and the scalar curvature is non zero after the transition takes place for $q>1$ case whereas it was zero in the previous BG limit. (b)For the $q<1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) also lies around $T=0.24$ and is negative but for $q>1$ the curvature peak becomes positive and is seen to be suppressed for larger q values but they shift towards higher temperatures and here the scalar curvature is zero after the transition takes place in both the the BG limit and $q<1$ regime.
• Figure 5: Temperature dependence of the thermodynamic scalar curvature $R(T)$ for the one-dimensional Blume Capel model using the Tsallis entropy for (D $>$ J)$D=1.065, J=1$ and $N=600$. (a)For the $q>1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) ($q=1$) lies around $T=0.24$ and is negative, but for $q>1$ the curvature peak becomes positive and shifts towards lower temperatures and the scalar curvature is non zero after the transition takes place for $q>1$ case whereas it was zero in the previous BG limit. (b)For the $q<1$ regime, the scalar curvature for the Boltzmann -Gibbs entropy (BG limit) also lies around $T=0.24$ and is negative but for $q>1$ the curvature peak is completely suppressed for the entire $q<1$ regime.