Dimensional Crossover in the Large N Limit

TL;DR

This paper investigates the universal behavior of an $O(N)$ model undergoing dimensional crossover in a film geometry at large $N$, deriving scaling forms and critical exponents with both renormalization group and direct methods.

Contribution

It provides explicit universal crossover scaling functions and critical exponents for the $O(N)$ model in a film geometry, comparing renormalization group and direct approaches.

Findings

Universal crossover scaling functions derived

Effective critical exponents calculated and related by scaling laws

Comparison between renormalization and direct methods

Abstract

We consider dimensional crossover for an $O(N)$ Landau-Ginzburg-Wilson model on a $d$-dimensional film geometry of thickness $L$ in the large $N$-limit. We calculate the full universal crossover scaling forms for the free energy and the equation of state. We compare the results obtained using ``environmentally friendly'' renormalization with those found using a direct, non-renormalization group approach. A set of effective critical exponents are calculated and scaling laws for these exponents are shown to hold exactly, thereby yielding non-trivial relations between the various thermodynamic scaling functions.