Thermodynamic geometry of one-dimensional spin one lattice models

TL;DR

This paper explores the thermodynamic geometry of one-dimensional spin-1 lattice models, analyzing scalar curvatures related to order parameter fluctuations and their relation to correlation lengths near critical points.

Contribution

It introduces a geometric framework for understanding phase transitions in the Blume-Emery Griffiths model, linking scalar curvatures to physical fluctuations and correlation lengths.

Findings

Two complementary geometries with curvatures $R_m$ and $R_q$ relate to magnetic and quadrupole fluctuations.

Excellent agreement between curvatures and correlation lengths in significant parameter regions.

Scalar curvature $R_g$ encodes interactions and helps analyze critical behavior.

Abstract

State space geometry is obtained for the one dimensional Blume Emery Griffiths model and the associated scalar curvature(s) investigated for various parameter regimes, including the Blume-Capel limit and the Griffiths model limit. For the one-dimensional case two complementary geometries with their associated curvatures $R_m$ and $R_q$ are found which are related to the fluctuations in the two order parameters, namely the magnetic moment and the quadrupole moment. An excellent agreement is obtained in significant regions of the parameter space between the two curvatures and the two corresponding correlation lengths $\xi_1$ and $\xi_2$. The three dimensional scalar curvature $R_g$ is also found to efficiently encode interactions. The scaling function for the free energy near critical points and the tricritical point is obtained by making use of Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature.