Thermodynamic geometry of the spin-1 model. II. Criticality and coexistence in the mean field approximation

TL;DR

This paper explores the thermodynamic geometry of the spin-1 BEG model and its limiting cases, revealing how curvature divergences and crossing diagrams encode criticality and phase coexistence in mean field approximation.

Contribution

It introduces a geometric approach with dual curvatures for two order parameters, providing new insights into critical phenomena and phase coexistence in spin models.

Findings

Curvatures diverge at critical lines with correct scaling.

Geometric crossing diagrams predict phase coexistence accurately.

Comparison of geometric and Ornstein-Zernicke correlation lengths.

Abstract

We continue our study of the thermodynamic geometry of the spin one model from paper I by probing the state space geometry of the Blume Emery Griffiths (BEG) model, and its limiting case of the Blume Capel model, in their mean field approximation. By accounting for the stochastic variables involved we construct from the thermodynamic state space two complimentary two-dimensional geometries with curvatures $R_m$ and $R_q$ which are shown to encode correlations in the model's two order parameters, namely, the magnetization $m$ and the quadrupole moment $q$. The geometry is investigated in the zero as well as the non zero magnetic field region. We find that the relevant scalar curvatures diverge to negative infinity along the critical lines with the correct scaling and amplitude. We then probe the geometry of phase coexistence and find that the relevant curvatures predict the coexistence curve remarkably well via their respective $R$-crossing diagrams. We also briefly comment on the effectiveness of the geometric correlation length compared to the commonly used Ornstein-Zernicke type correlation length vis-a-vis their scaling properties.