An ab initio Calculation of the Universal Equation of State for the O(N) Model

TL;DR

This paper derives a universal equation of state for the O(N) model using an Environmentally Friendly Renormalization Group, providing a first-principles scaling function that captures critical behavior and crossover phenomena.

Contribution

It introduces a novel ab initio method to derive the universal scaling form of the equation of state for the O(N) model from the fundamental Hamiltonian, incorporating crossover effects.

Findings

Explicit one-loop results for N=2, 3, 4

Comparison with known results confirms validity

Provides a universal scaling function with correct analyticity properties

Abstract

Using an Environmentally Friendly Renormalization Group we derive an ab initio universal scaling form for the equation of state for the O(N) model, y=f(x), that exhibits all required analyticity properties in the limits $x\to 0$, $x\to\infty$ and $x\to -1$. Unlike current methodologies based on a phenomenological scaling ansatz the scaling function is derived solely from the underlying Landau-Ginzburg-Wilson Hamiltonian and depends only on the three Wilson functions $\gamma_\lambda$, $\gamma_\phi$ and $\gamma_{\phi^2}$ which exhibit a non-trivial crossover between the Wilson-Fisher fixed point and the strong coupling fixed point associated with the Goldstone modes on the coexistence curve. We give explicit results for N=2, 3 and 4 to one-loop order and compare with known results.