Effective potential for a scalar field in three dimensions: Ising model in the ferromagnetic phase

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