# Effective Potential for Scalar Field in Three Dimensions: Ising Model in the Ferromagnetic Phase

## TL;DR

This study computes the effective potential for a scalar field in 3D near the phase transition of the Ising model using Monte Carlo simulations, revealing a dominant $ ext{phi}^6$ term and unexpected negative $ ext{phi}^4$ coefficient.

## Contribution

It provides a detailed numerical analysis of the effective potential in the 3D Ising model, highlighting the significance of higher-order terms and their impact on the continuum limit.

## Key findings

- Effective potential well approximated by $ ext{phi}^2$, $ ext{phi}^4$, and $ ext{phi}^6$ terms.
- The $ ext{phi}^4$ term has a negative sign, an unexpected result.
- Universal coupling constants and ratios are computed, including critical amplitude combinations.

## Abstract

We compute the effective potential $V_{\rm eff}(\phi)$ for one-component real scalar field $\phi$ in three Euclidean dimensions (3D) in the case of spontaneously broken symmetry, from the Monte Carlo simulation of the 3D Ising model in external field at temperatures approaching the phase transition from below. We study probability distributions of the order parameter on the lattices from $30^3$ to $74^3$, at $L/\xi \approx 10$. We find that, in close analogy with the symmetric case, $\phi^6$ plays an important role: $V_{\rm eff}(\phi)$ is very well approximated by the sum of $\phi^2$, $\phi^4$ and $\phi^6$ terms. An unexpected feature is the negative sign of the $\phi^4$ term. As close to the continuum limit as we can get ($\xi \approx 7.2$), we obtain $$ {\cal L}_{\rm eff} \approx {1 \over 2} \partial_\mu \phi \partial_\mu \phi + 1.7 (\phi^2 - \eta^2)^2 (\phi^2 + \eta^2). $$ We also compute several universal coupling constants and ratios, including the combination of critical amplitudes $C^- (f_1^-)^{-3} B^{-2}$.