The Specific Heat of a Ferromagnetic Film.

TL;DR

This paper investigates the specific heat behavior of the $O(N)$ vector model in a film geometry, analyzing boundary effects and crossover phenomena between different scaling regimes using environmentally friendly renormalization.

Contribution

It introduces a novel renormalization approach to derive specific heat expressions for various boundary conditions in a film geometry, capturing crossover behaviors in the $d=3$, $N=1$ case.

Findings

Derived explicit formulas for specific heat under different boundary conditions.

Identified crossover from power-law to logarithmic behavior in the $N=1$, $d=3$ case.

Matched the effective exponent to the 2D Ising model results.

Abstract

We analyze the specific heat for the $O(N)$ vector model on a $d$-dimensional film geometry of thickness $L$ using ``environmentally friendly'' renormalization. We consider periodic, Dirichlet and antiperiodic boundary conditions, deriving expressions for the specific heat and an effective specific heat exponent, $\alpha\ef$. In the case of $d=3$, for $N=1$, by matching to the exact exponent of the two dimensional Ising model we capture the crossover for $\xi_L\ra\infty$ between power law behaviour in the limit ${L\over\xi_L}\ra\infty$ and logarithmic behaviour in the limit ${L\over\xi_L}\ra0$ for fixed $L$, where $\xi_L$ is the correlation length in the transverse dimensions.