Thermodynamic calculation of spin scaling functions

TL;DR

This paper explores calculating spin scaling functions in critical phenomena using information geometry, providing explicit solutions for different thermodynamic scaling variables and confirming results with the 1D Ising model.

Contribution

It introduces a method to compute the scaling function $Y(z)$ via information geometry for various thermodynamic scaling variables, connecting geometric insights with critical phenomena.

Findings

Derived explicit forms of $Y(z)$ for different $t(eta)$ functions

Confirmed the geometric approach with the 1D Ising model

Identified conditions where $Y(z)= extstylerac{1}{2}ig( ext{e}^{z}+ ext{e}^{-z}ig)$

Abstract

Critical phenomena theory centers on the scaled thermodynamic potential per spin $\phi(\beta, h)=|t|^{p}Y(h|t|^{-q})$, with inverse temperature $\beta=1/T$, $h=-\beta H$, ordering field $H$, reduced temperature $t=t(\beta)$, critical exponents $p$ and $q$, and function $Y(z)$ of $z=h|t|^{-q}$. I discuss calculating $Y(z)$ with the information geometry of thermodynamics. Scaled solutions obtain with three admissible functions $t(\beta)$: 1) $t=e^{-J\beta}$, 2) $t=\beta^{-1}$, and 3) $t=\beta_C-\beta$, where $J$ and $\beta_C$ are constants. For $p=q$, information geometry yields $Y(z)=\sqrt{1+z^2}$, consistent with the one-dimensional (1D) ferromagnetic Ising model.